Sheaves and symplectic geometry of cotangent bundles 9782856299722

Table of contents :
Introduction
Acknowledgments
Part I. Microlocal theory of sheaves
Chapter I.1. Notations
Chapter I.2. Microsupport
I.2.1. Definition and first properties
I.2.2. Functorial operations
I.2.3. Constructibility
Chapter I.3. Sato's microlocalization
Chapter I.4. Simple sheaves
Chapter I.5. Composition of sheaves
Part II. Sheaves associated with Hamiltonian isotopies
Chapter II.1. Homogeneous case
Chapter II.2. Local behavior
Chapter II.3. Non homogeneous case
Part III. Cut-off lemmas
Chapter III.1. Global cut-off
Chapter III.2. Local cut-off—special case
Chapter III.3. Local cut-off—general case
Chapter III.4. Cut-off and -topology
Chapter III.5. Remarks on projectors—Tamarkin projector
Part IV. Constructible sheaves in dimension 1
Chapter IV.1. Gabriel's theorem
Chapter IV.2. Constructible sheaves on the real line
Chapter IV.3. Constructible sheaves on the circle
Chapter IV.4. Cohomological dimension 1
Part V. Graph selectors
Part VI. The Gromov nonsqueezing theorem
Chapter VI.1. Cut-off in fiber and space directions
Chapter VI.2. Nonsqueezing results
VI.2.1. Invariance of the displacement energy
VI.2.2. Nonsqueezing for a flying saucer
VI.2.3. Nonsqueezing for L0
VI.2.4. Nonsqueezing for the ball
Part VII. The Gromov-Eliashberg theorem
Chapter VII.1. The involutivity theorem
Chapter VII.2. Approximation of symplectic maps
Chapter VII.3. Degree of a continuous map
Chapter VII.4. The Gromov-Eliashberg theorem
Part VIII. The three cusps conjecture
Chapter VIII.1. Examples
Chapter VIII.2. Simple sheaf at a generic tangent point
VIII.2.1. Local cohomology
VIII.2.2. Generic tangent point—notations and hypotheses
VIII.2.3. Local study around C0
Chapter VIII.3. Microlocal linked points
Chapter VIII.4. Examples of microlocal linked points
Chapter VIII.5. Generic tangent point—global study
Chapter VIII.6. Front with one cusp
Chapter VIII.7. Proof of the three cusps conjecture
Chapter VIII.8. The four cusps conjecture
Part IX. Triangulated orbit categories for sheaves
Chapter IX.1. Definition of triangulated orbit categories
Quick reminder on localization
Definition of the orbit category
Internal tensor product and homomorphism
Morphisms in the triangulated orbit category
Direct sums
Direct and inverse images
Chapter IX.2. Microsupport in the triangulated orbit categories
IX.2.1. Definition and first properties
IX.2.2. Functorial behavior
IX.2.3. Microsupport in the zero section
Part X. The Kashiwara-Schapira stack
Chapter X.1. Definition of the Kashiwara-Schapira stack
Link with microlocalization
Chapter X.2. Simple sheaves
Chapter X.3. Obstruction classes
Chapter X.4. The Kashiwara-Schapira stack for orbit categories
Chapter X.5. Microlocal germs
Chapter X.6. Monodromy morphism
Part XI. Convolution and microlocalization
Chapter XI.1. The functor
Chapter XI.2. Adjunction properties
Chapter XI.3. Link with microlocalization
Chapter XI.4. Doubled sheaves
Part XII. Quantization
Chapter XII.1. Quantization for the doubled Legendrian
Chapter XII.2. The triangulated orbit category case
Chapter XII.3. Translation of the microsupport
Chapter XII.4. Restriction at infinity
Part XIII. Exact Lagrangian submanifolds in cotangent bundles
Chapter XIII.1. Fundamental groups
Chapter XIII.2. Vanishing of the Maslov class
Chapter XIII.3. Restriction at infinity
Chapter XIII.4. Vanishing of the second obstruction class
Chapter XIII.5. Homotopy equivalence
Bibliography

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440

ASTÉRISQUE 2023

SHEAVES AND SYMPLECTIC GEOMETRY OF COTANGENT BUNDLES Stéphane GUILLERMOU

SOCIÉTÉ MATHÉMATIQUE DE FRANCE

Astérisque est un périodique de la Société Mathématique de France. Numéro 440, 2023

Comité de rédaction Marie-Claude Arnaud Alexandru Oancea Christophe Breuil Nicolas Ressayre Philippe Eyssidieux Rémi Rhodes Colin Guillarmou Sylvia Serfaty Fanny Kassel Sug Woo Shin Eric Moulines Nicolas Burq (dir.) Diffusion Maison de la SMF AMS Case 916 - Luminy P.O. Box 6248 13288 Marseille Cedex 9 Providence RI 02940 France USA [email protected] http://www.ams.org Tarifs Vente au numéro : 54 e ($ 81) Abonnement Europe : 761 e, hors Europe : 830 e ($ 1 245) Des conditions spéciales sont accordées aux membres de la SMF. Secrétariat Astérisque Société Mathématique de France Institut Henri Poincaré, 11, rue Pierre et Marie Curie 75231 Paris Cedex 05, France Fax: (33) 01 40 46 90 96 [email protected] • http://smf.emath.fr/ © Société Mathématique de France 2023 Tous droits réservés (article L 122–4 du Code de la propriété intellectuelle). Toute représentation ou reproduction intégrale ou partielle faite sans le consentement de l’éditeur est illicite. Cette représentation ou reproduction par quelque procédé que ce soit constituerait une contrefaçon sanctionnée par les articles L 335–2 et suivants du CPI.

ISSN: 0303-1179 (print) 2492-5926 (electronic) ISBN 978-2-85629-972-2 doi:10.24033/ast.1199 Directeur de la publication : Fabien Durand

440

ASTÉRISQUE 2023

SHEAVES AND SYMPLECTIC GEOMETRY OF COTANGENT BUNDLES Stéphane GUILLERMOU

SOCIÉTÉ MATHÉMATIQUE DE FRANCE

Stéphane Guillermou Laboratoire de Mathématiques Jean Leray 2 Chemin de la Houssinière BP 92208 F-44322 Nanes Cedex 3, France

The author is also partially supported by the ANR project MICROLOCAL (ANR-15CE400007-01). Part of this paper was written during a stay at UMI 3457 in Montréal (CNRS – CRM – Université de Montréal). Texte soumis le 19 août 2019, révisé le 3 septembre 2020, accepté le 28 avril 2022. Mathematical Subject Classification (2010). — 18F20, 35A27, 53D12. Keywords. — Symplectic geometry, exact Lagrangian, microlocal sheaves, microsupport. Mots-clefs. — Géométrie symplectique, lagrangienne exacte, faisceaux microlocaux, microsupport.

SHEAVES AND SYMPLECTIC GEOMETRY OF COTANGENT BUNDLES by Stéphane GUILLERMOU

Abstract. — The aim of this paper is to apply the microlocal theory of sheaves of Kashiwara-Schapira to the symplectic geometry of cotangent bundles, following ideas of Nadler-Zaslow and Tamarkin. We recall the main notions and results of the microlocal theory of sheaves, in particular the microsupport of sheaves. The microsupport of a sheaf F on a manifold M is a closed conic subset of the cotangent bundle T ∗ M which indicates in which directions we can modify a given open subset of M without modifying the cohomology of F on this subset. An important theorem of KashiwaraSchapira says that the microsupport is coisotropic and recent works of Nadler-Zaslow and Tamarkin study in the other direction the sheaves which have for microsupport a given Lagrangian submanifold Λ, obtaining information on Λ in this way. Nadler and Zaslow made the link with the Fukaya category but Tamarkin only made use of the microlocal sheaf theory. We go on in this direction and recover several results of symplectic geometry with the help of sheaves. In particular we explain how we can recover the Gromov nonsqueezing theorem, the Gromov-Eliashberg rigidity theorem, the existence of graph selectors. We also prove a three cusps conjecture of Arnol’d about curves on the sphere. In the last sections we recover more recent results on the topology of exact Lagrangian submanifolds of cotangent bundles. Résumé. (Faisceaux et géométrie symplectique des fibrés cotangents) — Le but de cet article est d’appliquer la théorie microlocale des faisceaux de Kashiwara-Schapira à la géométrie symplectique des fibrés cotangents, suivant des idées de Nadler-Zaslow et Tamarkin. Nous rappelons les notions principales de la théorie microlocale des faisceaux, en particulier le microsupport des faisceaux. Le microsupport d’un faisceau F sur une variété M est un sous-ensemble conique fermé du fibré cotangent T ∗ M qui indique dans quelles directions on peut modifier un ouvert donné de M sans modifier la cohomologie de F sur cet ouvert. Un théorème important de Kashiwara-Schapira dit que le microsupport est coisotrope et des travaux récents de Nadler-Zaslow et Tamarkin étudient dans l’autre sens les faisceaux qui ont pour microsupport une sous-variété lagrangienne donnée Λ, obtenant de cette façon des informations sur Λ. Nadler et Zaslow ont fait le lien avec la catégorie de Fukaya mais Tamarkin a utilisé seulement la théorie microlocale des faisceaux. Nous poursuivons dans cette direction

© Astérisque 440, SMF 2023

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et retrouvons plusieurs résultats de géométrie symplectique à l’aide des faisceaux. En particulier nous expliquons comment retrouver le théorème de non plongement de Gromov, le théorème de rigidité de Gromov-Eliashberg, l’existence de sélecteurs de graphes. Nous démontrons aussi une conjecture des trois cusps d’Arnol’d au sujet de courbes sur la sphère. Dans les dernières sections nous retrouvons des résultats plus récents sur la topologie des sous-variétés lagrangiennes compactes exactes des fibrés cotangents.

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CONTENTS

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 4

Part I. Microlocal theory of sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5

I.1. Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7

I.2. Microsupport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I.2.1. Definition and first properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I.2.2. Functorial operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I.2.3. Constructibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11 11 13 17

I.3. Sato’s microlocalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

21

I.4. Simple sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

25

I.5. Composition of sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

29

Part II. Sheaves associated with Hamiltonian isotopies . . . . . . . . . . . . . . . . . . . . . .

31

II.1. Homogeneous case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

33

II.2. Local behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

37

II.3. Non homogeneous case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

41

Part III. Cut-off lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

43

III.1. Global cut-off . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

45

III.2. Local cut-off—special case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

53

III.3. Local cut-off—general case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

59

III.4. Cut-off and γ-topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

61

III.5. Remarks on projectors—Tamarkin projector . . . . . . . . . . . . . . . . . . . . . . . . . .

63

Part IV. Constructible sheaves in dimension 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

67

IV.1. Gabriel’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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IV.2. Constructible sheaves on the real line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

71

IV.3. Constructible sheaves on the circle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

75

IV.4. Cohomological dimension 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

79

Part V. Graph selectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

81

Part VI. The Gromov nonsqueezing theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

85

VI.1. Cut-off in fiber and space directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

87

VI.2. Nonsqueezing results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VI.2.1. Invariance of the displacement energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . VI.2.2. Nonsqueezing for a flying saucer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VI.2.3. Nonsqueezing for L0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VI.2.4. Nonsqueezing for the ball . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

95 95 96 97 98

Part VII. The Gromov-Eliashberg theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

101

VII.1. The involutivity theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

103

VII.2. Approximation of symplectic maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

105

VII.3. Degree of a continuous map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

107

VII.4. The Gromov-Eliashberg theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

109

Part VIII. The three cusps conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

113

VIII.1. Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

117

VIII.2. Simple sheaf at a generic tangent point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VIII.2.1. Local cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VIII.2.2. Generic tangent point—notations and hypotheses . . . . . . . . . . . . . . VIII.2.3. Local study around C0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

121 121 122 124

VIII.3. Microlocal linked points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

133

VIII.4. Examples of microlocal linked points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

139

VIII.5. Generic tangent point—global study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

143

VIII.6. Front with one cusp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

147

VIII.7. Proof of the three cusps conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

151

VIII.8. The four cusps conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

155

Part IX. Triangulated orbit categories for sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . .

161

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CONTENTS

IX.1. Definition of triangulated orbit categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Quick reminder on localization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Definition of the orbit category . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Internal tensor product and homomorphism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Morphisms in the triangulated orbit category . . . . . . . . . . . . . . . . . . . . . . . . . . . . Direct sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Direct and inverse images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

163 163 165 167 168 171 171

IX.2. Microsupport in the triangulated orbit categories . . . . . . . . . . . . . . . . . . . . . . . IX.2.1. Definition and first properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IX.2.2. Functorial behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IX.2.3. Microsupport in the zero section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

175 175 177 178

Part X. The Kashiwara-Schapira stack . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

181

X.1. Definition of the Kashiwara-Schapira stack . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Link with microlocalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

183 184

X.2. Simple sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

187

X.3. Obstruction classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

189

X.4. The Kashiwara-Schapira stack for orbit categories . . . . . . . . . . . . . . . . . . . . . .

191

X.5. Microlocal germs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

193

X.6. Monodromy morphism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

199

Part XI. Convolution and microlocalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

207

XI.1. The functor Ψ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

209

XI.2. Adjunction properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

217

XI.3. Link with microlocalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

219

XI.4. Doubled sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

225

Part XII. Quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

239

XII.1. Quantization for the doubled Legendrian . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

241

XII.2. The triangulated orbit category case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

245

XII.3. Translation of the microsupport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

247

XII.4. Restriction at infinity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

251

Part XIII. Exact Lagrangian submanifolds in cotangent bundles . . . . . . . . . . . . .

255

XIII.1. Fundamental groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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XIII.2. Vanishing of the Maslov class . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

259

XIII.3. Restriction at infinity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

263

XIII.4. Vanishing of the second obstruction class . . . . . . . . . . . . . . . . . . . . . . . . . . . .

265

XIII.5. Homotopy equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

267

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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INTRODUCTION

As the title suggests this paper explains some applications of the microlocal theory of sheaves of Kashiwara and Schapira to the symplectic geometry of cotangent bundles. The main notions of the microlocal theory of sheaves are Sato’s microlocalization, introduced in the 70’s, and the notion of microsupport of a sheaf, introduced by Kashiwara and Schapira in the 80’s. These notions were motivated by the study of modules over the ring of (micro-)differential operators. The link with the symplectic geometry was noticed (a deep result of [27] says that the microsupport of any sheaf is coisotropic) but not used to study global aspects of symplectic geometry until the papers [39] of Nadler-Zaslow and [45] of Tamarkin. The paper [39], together with [37], show that the dg-category of constructible sheaves on a real analytic manifold M is equivalent to the triangulated envelope of a version of the Fukaya category of T ∗ M . The paper [45] proves non-displaceability results in symplectic geometry using the properties of the microsupports of sheaves. Building on the ideas of this paper it is explained in [21] how to associate a sheaf with a Hamiltonian isotopy of a cotangent bundle. In this paper we go on in this direction and use sheaves to recover some classical results of symplectic geometry (the Gromov non-squeezing theorem and the Gromov-Eliashberg rigidity theorem) and a more recent result, which says that a compact exact Lagrangian submanifold of a cotangent bundle is homotopically equivalent to the base. We also prove a result about cusps of curves on the sphere (Arnol’d three cusps conjecture). Before we give more details we recall some facts about the microsupport. Let M be a manifold of class C ∞ and let k be a ring. We denote by D(kM ) the derived category of sheaves of k-modules over M . The microsupport SS(F ) of an object F of D(kM ) is introduced in [26]. It is a closed subset of the cotangent bundle T ∗ M , conic for the action of R+ on T ∗ M . It is defined as the closure of the set of singular directions with respect to F , where (x; ξ) ∈ T ∗ M is said non singular if the restriction maps from a neighborhood B of x to B ∩ {f < f (x)} induce isomorphisms between H i Fx and limB∋x H i (B ∩ {f < f (x)}; F ), for all functions f with df (x) = ξ and all i ∈ Z. The −→ easiest example is SS(kN ) = TN∗ M , where kN is the constant sheaf on a submanifold N of M . In general the microsupport can be a very singular set but it is coisotropic in some sense (see Theorem I.3.6). If M is real analytic and F is constructible, then SS(F ) is Lagrangian. Any smooth conic Lagrangian submanifold of T ∗ M is locally the microsupport of some sheaf on M .

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The microsupport is well-behaved with respect to the standard sheaf operations. An important example is the composition. Let Mi , i = 1, 2, 3, be three manifolds and let qij be the projection from M1 × M2 × M3 to Mi × Mj . For K1 ∈ D(kM1 ×M2 ) L

−1 −1 and K2 ∈ D(kM2 ×M3 ) we set K1 ◦ K2 = Rq13 ! (q12 K1 ⊗ q23 K2 ). We can define a set theoretic analog of the composition where direct and inverse images are replaced by the same set operations and the tensor product by the intersection. Then, under some geometric hypotheses, we have SS(K1 ◦ K2 ) ⊂ SS(K1 ) ◦ SS(K2 ). In [45] a sheaf version of the Chekanov-Sikorav theorem (see [10] and [44]) is given. A more functorial version is given in [21] as follows. Let Φ be an R>0 -homogeneous ∗ Hamiltonian isotopy of T˙ ∗ M = T ∗ M \ TM M . Then there exists a sheaf KΦ on M 2 which is invertible for the composition (there exists K ′ such that KΦ ◦ K ′ = k∆M ) ˙ ˙ and such that SS(K Φ ) is the graph of Φ. Here we let SS(F ) be SS(F ) with the zero ˙ ˙ section removed. We then have SS(KΦ ◦ F ) = Φ(SS(F )) for any F ∈ D(kM ) and F 7→ KΦ ◦ F is an auto-equivalence of D(kM ). We recall this in Part II. Since the microsupport is conic it is rather related with the contact geometry of the sphere cotangent bundle than the symplectic geometry of the cotangent bundle. We can also consider a Legendrian submanifold of the 1-jet space J 1 (M ) as a conic Lagrangian submanifold in T ∗ (M × R) contained in {τ > 0}, where we use the coordinates (t; τ ) on T ∗ R. In [45] Tamarkin also remarks that a sheaf F on M × R with a microsupport in {τ ≥ 0} comes with natural morphisms τc : F → − Tc∗ (F ), where c ≥ 0 and Tc is the vertical translation in M × R, Tc (x, t) = (x, t + c). A useful invariant of F is then e(F ) = sup{c ≥ 0; τc (F ) ̸= 0}. It is introduced in [45] and used in [7] to obtain displacement energy bounds. We use it in Part VI to prove classical nonsqueezing results. Our proof is a baby case of the proof of Chiu of a contact nonsqueezing theorem in [11]. In Part VI we use some operations on sheaves introduced in [28] (cut-off lemmas) to reduce the size of a microsupport. These operations are compositions with the constant sheaf on a cone. We recall them in Part III, where we also prove that a sheaf F whose microsupport can be decomposed into two disjoint (and unknotted) ˙ subsets, say SS(F ) = S1 ⊔S2 , can itself be locally decomposed, up to constant sheaves, ˙ i ) = Si . as F1 ⊕ F2 with SS(F We also use the cut-off results in Part V to prove that a Legendrian submanifold of J 1 (M ) has a graph selector as soon as it is the microsupport of a sheaf F satisfying some conditions at infinity. The graph selector is given by the boundary of the support of a section of F . In Part VII we prove the Gromov-Eliashberg rigidity theorem as a consequence of the involutivity theorem of Kashiwara-Schapira. The starting point is very simple. Let M be a manifold and let ϕn be a sequence of homogeneous Hamiltonian isotopies of T˙ ∗ M which converges in C 0 norm to a diffeomorphism ϕ∞ of T˙ ∗ M . Let Kn ∈ D(kM 2 ) be the sheaf associated with ϕn as recalled above. Then we can consider a kind of limit K∞ of Kn and the microsupport of K∞ is contained in the graph of ϕ∞ . We deduce from the involutivity theorem that this graph is Lagrangian,

ASTÉRISQUE 440

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3

hence that ϕ∞ is a symplectic map. This idea does not work directly to prove a local statement but we can cut-off the microsupport (using the cut-off results recalled in Part III) and make it work. The main result of this paper is a sheaf theoretic proof that a compact exact Lagrangian submanifold L of a cotangent bundle T ∗ M is homotopically equivalent to M . This is done in Parts IX-XIII. This result was previously obtained with Floer homology methods (see the beginning of Part XIII for references). However we do not recover the more precise results of Abouzaid and Kragh, who proved in [4] that the map L → − M is a simple homotopy equivalence and gave some conditions on the higher Maslov classes in [3] (we only prove the vanishing of the first two classes; for the other classes we should use sheaves of spectra – see [24, 23]). An important tool for our proof is the Kashiwara-Schapira stack µSh(kΛ ) of a Lagrangian submanifold Λ of a cotangent bundle T ∗ M . In [28] Kashiwara and Schapira consider the “microlocal” category D(kM ; Ω) where Ω is a subset of T ∗ M . It is defined as the quotient of D(kM ) by the subcategory formed by the F such that SS(F )∩Ω = ∅. When Ω runs over the open subsets of T ∗ M this gives a prestack on T ∗ M and we consider its associated stack, say µSh(kT ∗ M ). In [28] it is proved that the Hom sheaf in µSh(kT ∗ M ) is H 0 µhom where µhom is a variant of Sato’s microlocalization (our stack has a very poor structure because the triangulated structure does not survive in the stackification and we only obtain H 0 µhom, not µhom). The stack µSh(kΛ ) is the substack of µSh(kT ∗ M ) formed by the objects with microsupport contained in Λ. One step in the proof of the homotopy equivalence L ≃ M is the construction of a sheaf representing a given global object of µSh(kΛ ), where Λ is a conic Lagrangian submanifold of T ∗ (M × R) deduced from L by adding one variable. This is done in Part XII. A similar result is obtained by Viterbo in [52] using Floer homology methods. More precisely, any given F ∈ µSh(kΛ )(Λ) is represented by F ∈ D(kM ×R ) such that F− := F | , t ≪ 0, vanishes and F+ := F | , t ≫ 0, is locally M ×{t} M ×{t} constant. Then we prove that F 7→ F+ gives an equivalence between the category ˙ D[Λ],+ formed by the F such that SS(F ) ⊂ Λ and F− ≃ 0 and the subcategory Dlc (kM ) of D(kM ) of locally constant sheaves. We prove another equivalence between D[Λ],+ and Dlc (kΛ ). Hence Dlc (kM ) is equivalent to Dlc (kΛ ) and it follows that L → − M is a homotopy equivalence. Actually the proof is not so straightforward. We first prove the fully faithfulness of F 7→ F+ in Part XII, then use this fully faithfulness in the beginning of Part XIII to prove some result on the fundamental groups and the vanishing of the first Maslov class. For this we need to know that µSh(kΛ )(Λ) has many objects. But the first Maslov class is an obstruction to the existence of a global object in µSh(kΛ ). To bypass this problem we first work in the orbit category of sheaves (see Part IX) where there is no such obstruction. The orbit category contains less information than D(kM ) but the above argument works well enough in this framework to obtain the vanishing of the Maslov class. Then we can go back to the usual category of sheaves and we can prove Dlc (kM ) ≃ Dlc (kΛ ), as claimed.

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INTRODUCTION

Acknowledgments The starting point of this paper is a discussion with Claude Viterbo. He explained me his construction of a quantization in the sense of this paper using Floer homology and asked whether it was possible to obtain it with the methods of algebraic analysis. Masaki Kashiwara gave me the idea to glue locally defined simple sheaves under the assumption that the Maslov class vanishes. Claire Amiot explained me that we can use the orbit category to work without this vanishing assumption. The idea of applying the involutivity theorem to the C 0 -rigidity emerged after several discussions with Claude Viterbo, Pierre Schapira and Vincent Humilière. The work on the three cusps conjecture was motivated by discussions with Emmanuel Giroux and Emmanuel Ferrand. I also thank Sylvain Courte, Pierre Schapira and Nicolas Vichery for several remarks and many stimulating discussions. I thank the anonymous referees for their careful reading of the paper, for pointing out several mistakes and for their constructive suggestions, making the exposition more clear.

ASTÉRISQUE 440

PART I MICROLOCAL THEORY OF SHEAVES

We recall here some results of [28] that we will use very often. The notion of microsupport of a sheaf is in particular very important. We recall its definition and its behavior under sheaf operations. We also recall quickly the definition of Sato’s microlocalization and the µhom functor, later introduced by Kashiwara and Schapira; it will be used in particular in §X to define a category of sheaves associated with a Lagrangian submanifold of a cotangent bundle. When [28] was written, it was not well understood how to deal with unbounded derived categories. In particular the theory of microsupport is written for bounded derived categories of sheaves. Moreover one of the fundamental lemmas was proved using an induction on the cohomological degree and its extension to the unbounded case could be unclear. However this problem has been solved in [42] and we will state the results on the microsupport for unbounded categories (although the sheaves we consider later are always locally bounded).

CHAPTER I.1 NOTATIONS

We mainly follow the notations of [28].

Geometry. — Unless otherwise specified the manifolds we consider here are not “manifolds with boundary”. (The reason is that the definition of the microsupport is not so meaningful at the boundary—if we want to deal with a sheaf F on a manifold with boundary M , we consider an embedding i : M → − M + , with M + a usual manifold, and look at the microsupport of the direct image Ri∗ (F ).) When we say that a submanifold N of M is locally closed, closed or compact, we mean that it is locally closed (intersection of a closed subset and an open subset), closed or compact as a subset of M . We denote by πM : T ∗ M → − M the cotangent bundle of M . If N ⊂ M is a submanifold, we denote by TN∗ M its conormal bundle; it is the subbundle of the restriction of T ∗ M over N whose fiber over a point x ∈ N is the orthogonal space of Tx N , that is, (TN∗ M )x = {θ ∈ Tx∗ M ; ⟨Tx N, θ⟩ = 0}. The zero-section of T ∗ M will usually be ∗ ∗ M and M , or M , if there is no ambiguity. We set T˙ ∗ M = T ∗ M \ TM denoted by TM ∗ ∗ ˙ we denote by π˙ M : T M → − M the projection. For any subset A of T M we define its antipodal Aa = {(x; ξ) ∈ T ∗ M ; (x; −ξ) ∈ A}. We usually denote by ∆M the diagonal of M 2 . We denote the normal bundle of N by TN M . It is defined as the quotient bundle (N ×M T M )/T N , where N ×M T M is the restriction of the bundle T M to N . The cotangent bundle T ∗ M carries an exact symplectic structure. We denote the P Liouville 1-form by αM . It is given in local coordinates (x; ξ) by αM = ξ dx i i. i The symplectic structure is then given by the non-degenerate 2-form dαM . Since it is ∼ non-degenerate, it induces an isomorphism Hp : Tp∗ T ∗ M −− → Tp T ∗ M , p ∈ T ∗ M , such ∗ ∗ ∗ that ⟨v, θ⟩ = (dαM )p (v, Hp (θ)), v ∈ Tp T M , θ ∈ Tp T M . We obtain in this way an ∼ isomorphism H : T ∗ T ∗ M −− → T T ∗ M , the Hamiltonian isomorphism. For the sections ∗ ∗ of T T M it gives H(dxi ) = −∂/∂ξi and H(dξi ) = ∂/∂xi . The Hamiltonian vector field of a function h : T ∗ M → − R is Xh = H(dh); in local coordinates we have X ∂h ∂ ∂h ∂ Xh (x; ξ) = − . ∂ξi ∂xi ∂xi ∂ξi i

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A submanifold Λ of T ∗ M is Lagrangian if, for each p ∈ Λ, the tangent space Tp Λ is its own orthogonal with respect to the symplectic structure of Tp T ∗ M . If N ⊂ M is a submanifold, then TN∗ M is a Lagrangian submanifold of T ∗ M . Let f : M → − N be a morphism of manifolds. It induces morphisms on the cotangent bundles: (I.1.1)

fd

fπ

T ∗ M ←− M ×N T ∗ N −→ T ∗ N.

Let N ⊂ M be a submanifold and A ⊂ M any subset. We denote by CN (A) ⊂ TN M the cone of A along N . We recall its definition in §I.3 using the normal deformation of N in M ; in the case where M is a vector space, x0 ∈ N and q : M → − TN,x0 M denotes the natural quotient map, then \ [ (I.1.2) CN (A) ∩ TN,x0 M = q([x0 , x)), U x∈A∩(U \{x0 })

where U runs over the neighborhoods of x0 and [x0 , x) denotes the half line starting at x0 and containing x. If A, B are two subsets of M , we set C(A, B) = C∆M (A × B). Identifying T∆M (M × M ) with T M through the first projection, we consider C(A, B) as a subset of T M . If M is a vector space and x0 ∈ M , we have [ \ q([y, x)), (I.1.3) C(A, B) ∩ Tx0 M = U x∈A∩U, y∈B∩U, x̸=y

where U runs over the neighborhoods of x0 .

Sheaves. — We consider a commutative unital ring k of finite global dimension (we will use k = Z or k = Z/2Z). We denote by Mod(k) the category of k-modules and by Mod(kM ) the category of sheaves of k-modules on M . We denote by D(kM ) (resp. Db (kM ), Dlb (kM )) the derived category (resp. bounded derived category, locally bounded derived category) of Mod(kM ). (Hence Dlb (kM ) is the subcategory of D(kM ) formed by the F such that F | ∈ Db (kC ), for any compact subset C ⊂ M .) We recall C that Mod(kM ) has a fully faithful embedding (that is, which preserves the Hom sets) into D(kM ) by sending a sheaf to a complex of sheaves concentrated in degree 0. We refer to [28] for results about derived categories and sheaves and also [29] for the unbounded case. We recall some standard notations (following mainly [28]). For a morphism of manifolds f : M → − N , we denote by Rf∗ , Rf! : D(kM ) → − D(kN ) the direct image and proper direct image functors. We denote by f −1 , f ! : D(kN ) → − D(kM ) their adjoint functors. We thus have adjunctions (f −1 , Rf∗ ) and (Rf! , f ! ). For L ∈ D(k) we denote by LM = a−1 M (L) the constant sheaf with stalk L on M , where aM : M → − {pt} is the map to a point. For the inclusion j : Z → − M of a subset of M and F ∈ D(kM ) we often write F | = j −1 F. Z

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CHAPTER I.1. NOTATIONS

We say that F ∈ D(kM ) is locally constant if any point of M has a neighborhood U such that F | is constant. If F is concentrated in degree 0, we often say “F is a local U system” instead of “F is a locally constant sheaf”. When Z is locally closed we define FZ = j! j −1 F,

RΓZ (F ) = Rj∗ j ! F.

When F = LM is the constant sheaf with stalk L ∈ D(k), we set for short LZ = (LM )Z = j! (LZ ). (In case of ambiguity we write LM,Z = (LM )Z , but in general the ambient manifold M is understood.) We recall that, if Z is closed and U is open, then Γ(U ; kZ ) = {f : Z ∩ U → − k; f is locally constant}. If Z ′ is locally closed and Z is a closed subset of Z ′ , we have an exact sequence in Mod(kM ) 0→ − kZ ′ \Z → − kZ ′ → − kZ → − 0.

(I.1.4)

We let ⊗ and Hom denote the tensor product and internal Hom in Mod(kM ). We recall that, for F, G ∈ Mod(kM ), their internal Hom is the sheaf Hom (F, G) whose sections over an open subset U is given by Γ(U ; Hom (F, G)) = HomMod(kU ) (F | , G| ). U

U

L

The derived functors of ⊗ and Hom are denoted ⊗ and RHom (we sometimes write L

L

F ⊗ G for F ⊗ G when F or G has free stalks over k, typically F ⊗ kZ for F ⊗ kZ ). L

We have an adjunction (⊗, RHom ). We have natural isomorphisms (I.1.5)

FZ ≃ F ⊗ k Z ,

RΓZ (F ) ≃ RHom (kZ , F )

and (I.1.4) gives the excision distinguished triangles, for F ∈ D(kM ), +1

FZ ′ \Z → − FZ ′ → − FZ −−→, +1

RΓZ (F ) → − RΓZ ′ (F ) → − RΓZ ′ \Z (F ) −−→ . If U is open, we let RΓ(U ; −) be the derived section functor. We set H i (U ; F ) = H i RΓ(U ; F ). We let Γc (−) be the functor of sections with compact support and RΓc (−) its derived functor. If aM : M → − {pt} is the map to a point, we thus have RΓ(M ; F ) ≃ RaM ∗ (F ) and RΓc (M ; F ) ≃ RaM ! (F ). We also set (I.1.6)

RΓZ (U ; F ) = RΓ(U ; RΓZ (F )),

HZi (U ; F ) = H i RΓZ (U ; F ).

We denote by ωM the dualizing complex on M . Since M is a manifold, ωM is actually the orientation sheaf shifted by the dimension, that is, ωM ≃ orM [dM ]. (In general it is defined by ωM = a!M (k{pt} ).) We recall that the sheaf orM is locally constant with stalks k and, for a connected orientable open subset U of M , orM (U ) is the rank 1 free module ⟨o, o′ ; o = −o′ ⟩ where o, o′ are the two possible orientations of U . The duality functors are defined by (I.1.7)

DM ( • ) = RHom ( • , ωM ),

D′M ( • ) = RHom ( • , kM ).

For f : M → − N we also use the notation ωM |N = f ! (kN ). We have ωM |N ≃ ωM ⊗ f −1 (D′N (ωN )) ≃ orM ⊗ f −1 (orN )[dM − dN ] (we remark ∼→ kN that D′N (orN ) ≃ orN ; indeed there exists a canonical isomorphism u : orN ⊗ orN −−

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such that, for any x ∈ N and any choice of orientation o around x, we have ux (o ⊗ o) = 1). L

For two manifolds M, N and F ∈ D(kM ), G ∈ D(kN ) we define F ⊠ G ∈ D(kM ×N ) by L

L

F ⊠ G = q1−1 F ⊗ q2−1 G, where qi (i = 1, 2) is the i-th projection defined on M × N . We recall some useful facts (see [28, §2, §3]). Proposition I.1.1. — Let f : M → − N be a morphism of manifolds, F, G, H ∈ D(kM ), F ′ , G′ ∈ D(kN ). Then we have for any i ∈ Z (a) (b) (c) (d) (e)

H i RHom(F, G) ≃ Hom(F, G[i]), RHom(kU , F ) ≃ RΓ(U ; F ), for U ⊂ M open, RΓ(U ; RHom (F, G)) ≃ RHom(F | , G| ), for U ⊂ M open, U U H i F is the sheaf associated with V 7→ H i (V ; F ), H i RHom (F, G) is the sheaf associated with V 7→ Hom(F | , G| [i]), V

V

L

(f) RHom (F ⊗ G, H) ≃ RHom (F, RHom (G, H)), L

L

(g) Rf! (F ⊗ f −1 F ′ ) ≃ (Rf! F ) ⊗ F ′ , (projection formula), (h) f ! RHom (F ′ , G′ ) ≃ RHom (f −1 F ′ , f ! G′ ), (i) if f is an embedding, Rf∗ RHom (F, G) ≃ RHom (Rf! F, Rf∗ G), M (j) for a Cartesian diagram

f

N g′

g

we have the base change formulas

f′

f

′−1

′

′

Rg ! (F ) ≃ Rg! f

−1

M′ N′ ′! ′ (F ) and f Rg ∗ (F ′ ) ≃ Rg∗ f ! (F ′ ). ′

L

The adjunction between ⊗ and RHom together with kU ⊗ kU ≃ kU give Hom(kU , D′ (kU )) ≃ Hom(kU , kM ) ≃ H 0 (U ; kM ). The canonical section of this last group gives a morphism kU → − D′ (kU ). Similarly we have a natural mor′ phism kU → − D (kU ). In the following case they are isomorphisms. If the inclusion U ⊂ M is locally homeomorphic to the inclusion ]−∞, 0[ × Rn−1 ⊂ Rn (for example, if ∂U is smooth), then we have the first isomorphism in (I.1.8) below. Indeed, to prove that some morphism is an isomorphism we can work locally and thus assume that U = ]−∞, 0[ × Rn−1 ⊂ Rn . For a point x ∈ ∂U and an open ball B around x, we have RΓ(B; D′ (kU )) ≃ RHom(kU | , kM | ) B

B

≃ RHom(kU ∩B , kB ) ≃ RΓ(U ∩ B; kB ) ≃ k using (b) and (c). It follows that (D′ (kU ))x ≃ (kU )x . This also holds for x ∈ U or x ∈ M \ U and we obtain the claim. The second isomorphism in (I.1.8) follows from the first one applied to M \ U and the exact sequence 0 → − kM \U → − kM → − kU → − 0. (I.1.8)

ASTÉRISQUE 440

∼ → D′ (kU ), kU −−

∼ kU −− → D′ (kU ).

CHAPTER I.2 MICROSUPPORT

I.2.1. Definition and first properties We recall the definition of the microsupport (or singular support) SS(F ) of a sheaf F , introduced by M. Kashiwara and P. Schapira in [26] and [27]. Let F ∈ D(kM ) and p = (x0 ; ξ0 ) ∈ T ∗ M be given. We choose a real C 1 -function ϕ on M satisfying dϕ(x0 ) = ξ0 and we consider the restriction morphism “in the direction p” for a given degree i ∈ Z: (I.2.1)

H i Fx0 ≃ lim H i (U ; F ) → − lim H i (U ∩ {x; ϕ(x) < ϕ(x0 )}; F ), −→ −→ U

U

where U runs over the open neighborhoods of x0 . We are interested in the points p where this morphism is not an isomorphism (for some ϕ and i). Taking the cone of the restriction morphism we obtain the following definition. Definition I.2.1 (See [28, Def. 5.1.2]). — Let F ∈ D(kM ). We define SS(F ) ⊂ T ∗ M as the closure of the set of points (x0 ; ξ0 ) ∈ T ∗ M such that there exists a real C 1 -function ϕ on M satisfying dϕ(x0 ) = ξ0 and (RΓ{x; ϕ(x)≥ϕ(x0 )} (F ))x0 ̸≃ 0. ˙ We set SS(F ) = SS(F ) ∩ T˙ ∗ M . The following properties are easy consequences of the definition: (a) the microsupport is closed and R>0 -conic, that is, invariant by the action of (R>0 , ×) on T ∗ M , ∗ (b) SS(F ) ∩ TM M = πM (SS(F )) = supp(F ), +1 (c) the microsupport satisfies the triangular inequality: if F1 → − F2 → − F3 −−→ is a distinguished triangle in D(kM ), then SS(Fi ) ⊂ SS(Fj ) ∪ SS(Fk ) for all i, j, k ∈ {1, 2, 3} with j ̸= k. Notation I.2.2. — Following [28, §6.1], for an R>0 -conic subset S of T ∗ M , we denote by DS (kM ) the full subcategory of D(kM ) formed by the F with SS(F ) ⊂ S. We ∗ M (kM ). We denote by D also set D[S] (kM ) = DS∪TM (S) (kM ) the full subcategory of D(kM ) formed by the F for which there exists a neighborhood Ω of S in T ∗ M such that SS(F ) ∩ Ω ⊂ S. By the property (c) above, all these subcategories are triangulated.

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Example I.2.3. — (i) If F is a non-zero local system on a connected manifold M , ∗ ∗ then SS(F ) = TM M , the zero-section. Conversely, if SS(F ) ⊂ TM M , then the i cohomology sheaves H (F ) are local systems, for all i ∈ Z. (We say that F is locally constant for short.) This follows for example from Proposition I.2.9 below. (ii) If N is a closed submanifold (we only mean N is a closed subset of M —see §I.1) of M and F = kN , then SS(F ) = TN∗ M . (iii) Let U ⊂ M be an open subset with smooth boundary. Then ∗,out ∗ SS(kU ) = (U ×M TM M ) ∪ T∂U M, ∗,out ∗ SS(kU ) = (U ×M TM M ) ∪ (T∂U M )a , ∗,out where T∂U M = {(x; λdf (x)); f (x) = 0, λ ≤ 0}, if U = {f > 0} and df ̸= 0 on ∂U (see Fig. I.2.1). To remember the directions of the microsupports we can take ϕ = f in Definition I.2.1 and use RΓ{x; ϕ(x)≥0} (F ) ≃ F for both F = kU and F = kU (since supp(F ) ⊂ {x; ϕ(x) ≥ 0}). This implies

(RΓ{x; ϕ(x)≥0} (kU ))x0 ≃ (kU )x0 ≃ 0 and (RΓ{x; ϕ(x)≥0} (kU ))x0 ≃ (kU )x0 ≃ k, for x0 ∈ ∂U , and both formulas are consistent with the above description of the microsupport.

SS(kU )

SS(kU )

Figure I.2.1. We identify covectors with vectors to draw the microsupport. The microsupport of the constant sheaf on an open (resp. closed) subset with smooth boundary points outward (resp. inward).

(iv) Let λ be a closed convex cone with vertex at 0 in E = Rn . By [28, Prop. 5.3.1] we have SS(kλ ) ∩ T0∗ E = λ◦ , where λ◦ is the polar cone of λ: (I.2.2)

λ◦ = {ξ ∈ E ∗ ; ⟨v, ξ⟩ ≥ 0 for all v ∈ λ}.

For a given a > 0 we define ha : E → − E, x 7→ ax. We clearly have h−1 a (kλ ) ≃ kλ . Hence SS(kλ ) is invariant by the map induced by ha on T ∗ E. Since the micro− 0 support is closed, we deduce the rough bound SS(kλ ) ⊂ E × λ◦ by letting a → (see Fig. I.2.2). ˙ (v) Let F ∈ D(kR ) be such that SS(F ) = {(0; ξ); ξ > 0}. On one hand, it follows from (i) that F | is locally constant, hence F | is constant, where R\{0}

U±

U± = {±x > 0}. In particular (RΓ{x; ϕ(x)≥ϕ(x0 )} (F ))x0 ≃ 0 for any x0 ̸= 0 and any function ϕ with dϕ(x0 ) ̸= 0. On the other hand, by the definition of the microsupport there exist points x1 arbitrarily close to 0 (maybe equal to 0) together with functions ϕ such that dϕ(x1 ) > 0 and (RΓ{x; ϕ(x)≥ϕ(x1 )} (F ))x1 ̸≃ 0.

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13

Figure I.2.2. At the vertex SS(kλ ) is the polar cone of λ.

We have seen that the case x1 ̸= 0 is excluded. Hence (RΓ{x; ϕ(x)≥ϕ(0)} (F ))0 ̸≃ 0 for some function ϕ with dϕ(0) > 0. This means (RΓZ F )0 ̸= 0, where Z = U+ . Let us set Br = ]−r, r[. Since F | is constant, RΓ(Br ; RΓZ F ) is indepenU± dent of r > 0. Hence we have in fact (RΓZ F )0 ≃ RΓ(R; RΓZ F ). Let us set E = (RΓZ F )0 . By the adjunction (a−1 , Ra∗ ) where a : R → − {pt} is the pro∼ jection, the isomorphism E −− → RΓ(R; RΓZ F ) = Ra∗ RΓZ F gives a morphism L

u : ER → − RΓZ F . By the adjunction (⊗, RHom ) we obtain from u another morphism v : EZ → − F . We have (RΓZ (EZ ))0 ≃ E and the morphism v induces an ∼ isomorphism (RΓZ (EZ ))0 −− → (RΓZ F )0 . In other words, defining G by a disv +1 tinguished triangle G → − EZ − → F −−→, we have (RΓZ G)0 ≃ 0. By the triangle ˙ inequality we deduce that SS(G) = ∅, hence G is a constant sheaf on R and we ′ can write G = ER for some E ′ ∈ D(k). In conclusion there exist E, E ′ ∈ D(k) and a distinguished triangle +1

ER′ → − EZ → − F −−→ . Conversely a sheaf F defined by such a distinguished triangle satisfies ˙ SS(F ) = {(0; ξ); ξ > 0} and (RΓZ F )0 ≃ E.

I.2.2. Functorial operations Proposition I.2.4 (See [28, Prop. 5.4.4]). — Let f : M → − N be a morphism of manifolds and let F ∈ D(kM ). We assume that f is proper on supp(F ). Then SS(Rf! F ) ⊂ fπ fd−1 SS(F ), with equality when f is a closed embedding (see the beginning of §I.1 for “closed”). Example I.2.5. — With the notations of Proposition I.2.4 we assume that f is an ∗ embedding. Let F ∈ D(kN ) be such that SS(F ) ⊂ TM N . Then there exists a locally constant G ∈ D(kM ) such that F ≃ Rf∗ G. Indeed, since SS(F ) ∩ T ∗ (N \ M ) = ∅, we have F | ≃ 0, by the property (b) after Definition I.2.1. Hence F ≃ Rf∗ G N \M −1 where G = f F . Then SS(F ) = fπ fd−1 SS(G) by Proposition I.2.4 and we deduce SS(G) ⊂ TN∗ N . Now the result follows from Example I.2.3-(i). We recall some notations of [28, Def. 6.2.3].

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Let M and N be two manifolds and f : M → − N a morphism. Let Γf ⊂ M × N be the graph of f . We have TΓ∗f (M × N ) ≃ M ×N T ∗ N . If X is a manifold and Λ a Lagrangian submanifold of T ∗ X, the Hamiltonian isomorphism identifies TΛ T ∗ X with T ∗ Λ. Applying this to X = M × N and Λ = TΓ∗f (M × N ), we obtain a natural identification TTΓ∗ (M ×N ) T ∗ (M × N ) ≃ T ∗ (M ×N T ∗ N ). f

We also remark that T ∗ M has a natural embedding in T ∗ (M ×N T ∗ N ) as T ∗ M = T ∗ (M ×N TN∗ N ). For a conic subset A ⊂ T ∗ N we set f ♯ (A) = T ∗ M ∩ CTΓ∗

(I.2.3)

f

(M ×N ) (T

∗

M × A).

Example I.2.6. — Let f be the embedding of Rm in Rn+m . We take coordi′′ nates (x′ , x′′ ; ξ ′ , ξ ′′ ) on T ∗ Rn+m such that Rm = {x′ = 0}. Then (x′′∞ , ξ∞ ) ∈ f ♯ (A) if ′ ′′ ′ ′′ ′ and only if there exists a sequence {(xn , xn ; ξn , ξn )} in A such that xn → − 0, x′′n → − x′′∞ , ′′ ′′ ′ ′ ξn → − ξ∞ and |xn | |ξn | → − 0. Example I.2.7. — With the notations (I.1.1) we say that f and A are noncharacteristic if ∗ fπ−1 (A) ∩ fd−1 (TM M ) ⊂ M ×N TN∗ N. If A ⊂ T ∗ N is closed conic and f and A are non-characteristic, then f ♯ (A) = fd fπ−1 (A). Theorem I.2.8 (See [28, Cor. 6.4.4])). — Let f : M → − N be a morphism of manifolds and let F ∈ D(kN ). Then, using the notation f ♯ of (I.2.3) (see also Example I.2.7), SS(f −1 F ) ⊂ f ♯ (SS(F ))

and

SS(f ! F ) ⊂ f ♯ (SS(F )).

If f is smooth, these inclusions are equalities. If f is non-characteristic for SS(F ), then the natural morphism f −1 F ⊗ ωM |N → − f ! (F ) is an isomorphism, where ωM |N ≃ orM ⊗ f −1 (orN )[dM − dN ] is the relative dualizing complex. Proposition I.2.9 (See [28, Prop. 5.4.5])). — Let N, I be manifolds. We assume that I is contractible. Let f : N × I → − N be the projection. Let F ∈ D(kN ×I ). Then ∼ SS(F ) ⊂ T ∗ N × TI∗ I if and only if f −1 Rf∗ (F ) −− → F. Example I.2.10. — We set M = Rn , S = {xn = 0}, Z = {xn ≥ 0} and ˙ Λ = {(x1 , . . . , xn−1 , 0; 0, ξn ); ξn > 0}. Let F ∈ D(kM ) be such that SS(F ) = Λ. ′ Then there exist E, E ∈ D(k) and a distinguished triangle +1

′ EM → − EZ → − F −−→ .

Indeed, we can identify M with a product I × N , where I = Rn−1 and N = R. Then Λ = TI∗ I × {(0; ξn ); ξn > 0}. By Proposition I.2.9 we can write F ≃ f −1 G for

ASTÉRISQUE 440

I.2.2. FUNCTORIAL OPERATIONS

15

some G ∈ D(kN ), where f : N × I → − N is the projection. By Theorem I.2.8 we must ˙ have SS(G) = {(0; ξn ); ξn > 0} and we conclude with Example I.2.3-(v). Example I.2.11 (Conic sheaves). — (i) We set S n−1 = (Rn \ {0})/R>0 and let q : Rn \ {0} → − S n−1 be the quotient map. We say that F ∈ D(kRn ) is conic ≃ q −1 G. By Proposition I.2.9 if there exists G ∈ D(kS n−1 ) such that F | n R \{0}

F is conic if and only if, for any x ̸= 0 ∈ Rn , we have SS(F ) ∩ Tx∗ Rn ⊂ (Tl∗ Rn )x , where l = R · x is the line generated by x. (ii) In Example I.2.3-(iv) we have given a bound for SS(kλ ) when λ is a closed convex cone in Rn , namely SS(kλ ) ⊂ Rn × λ◦ and SS(kλ ) ∩ T0∗ Rn = λ◦ , where λ◦ is the polar cone of λ. Since kλ is conic, the above discussion also implies SS(kλ ) ∩ Tx∗ Rn ⊂ (Tl∗ Rn )x , where l = R · x, for any x ̸= 0 ∈ Rn . We remark that Int(λ◦ ) = {ξ ∈ (Rn )∗ ; ⟨v, ξ⟩ > 0 for all v ̸= 0 ∈ λ} (if λ contains a line, then Int(λ◦ ) = ∅). In particular, for x ̸= 0 ∈ λ, we have Int(λ◦ ) ∩ (Tl∗ Rn )x = ∅ and we deduce the more precise bound (I.2.4)

SS(kλ ) ∩ T ∗ (Rn \ {0}) ⊂ (Rn \ {0}) × ∂λ◦ ,

where ∂λ◦ = λ◦ \ Int(λ◦ ). Let δM : M → − M 2 be the diagonal embedding. For conic subsets A, B ⊂ T ∗ M we set (I.2.5)

♯ b B = δM A+ (A × B).

b B is the set of (x; ξ) such that there exist two seIn local coordinates A + quences (xn ; ξn ) in A and (yn ; ηn ) in B such that xn , yn → − x, ξn + ηn → − ξ and |xn − yn ||ξn | → − 0 when n → − ∞. ∗ M, Example I.2.12. — If A, B ⊂ T ∗ M are closed conic subsets such that Aa ∩B ⊂ TM b B = A + B. then A +

For the definition of cohomologically constructible we refer to [28, §3.4]. An example of a cohomologically constructible sheaf F is given by the case where F is constructible with respect to a Whitney stratification (that is, the restriction of F to each stratum is locally constant and of finite rank). Theorem I.2.13 (See [28, Cor. 6.4.5])). — Let F, G ∈ D(kM ). Then, using the notation b of (I.2.5) (see also Example I.2.12), + L

b SS(G), SS(F ⊗ G) ⊂ SS(F ) + b SS(G), SS(RHom (F, G)) ⊂ SS(F )a + SS(D′ F ) = SS(F )a . ∗ If we assume that SS(F )∩SS(G) ⊂ TM M and that F is cohomologically constructible, L

then the natural morphism D′ F ⊗ G → − RHom (F, G) is an isomorphism.

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Remark I.2.14. — Let M, N be manifolds and let q1 , q2 be the projections from M × N to M , N . For F ∈ D(kM ) and G ∈ D(kN ) Theorem I.2.13 implies L

SS(RHom (q1−1 F, q2−1 G)) ⊂ SS(F )a × SS(G) and SS(q1−1 F ⊗ q2−1 G) ⊂ SS(F ) × SS(G). b operation we can give a version of Proposition I.2.4 for an open emUsing the + bedding. We only state the case where the boundary of the open subset is smooth (see [28] for the general case). Theorem I.2.15 (See [28, Thm. 6.3.1])). — Let j : U ,→ M be the embedding of an open b SS(kU )a subset with a smooth boundary. Let F ∈ D(kU ). Then SS(Rj∗ F ) ⊂ SS(F ) + b SS(kU ). and SS(Rj! F ) ⊂ SS(F ) + The next result follows immediately from Proposition I.2.4 and Example I.2.3 (i). It is a particular case of a microlocal Morse result (see [28, Cor. 5.4.19]), the classical theory corresponding to the constant sheaf F = kM . Corollary I.2.16. — Let F ∈ D(kM ), let ϕ : M → − R be a function of class C 1 and assume that ϕ is proper on supp(F ). Let a < b in R and assume that dϕ(x) ∈ / SS(F ) for a ≤ ϕ(x) < b. Then the natural morphisms RΓ(ϕ−1 (]−∞, b[); F ) → − RΓ(ϕ−1 (]−∞, a[); F ) and RΓϕ−1 ([b,+∞[) (M ; F ) → − RΓϕ−1 ([a,+∞[) (M ; F ) are isomorphisms. Here is another useful consequence of the properties of the microsupport which appears in [38]. Corollary I.2.17. — Let M be a manifold and I an open interval of R. Let F , G ∈ D(kM ×I ). We assume (i) the projection supp(F ) ∩ supp(G) → − I is proper, (ii) F , G are non-characteristic for all maps it : M × {t} → − M × I, t ∈ I, that is, ∗ ˙ SS(A) ∩ (TM M × Tt∗ I) = ∅ for A = F , G, ˙ ˙ (iii) setting Λt = i♯t (SS(F )) and Λ′t = i♯t (SS(G)), we have Λt ∩ Λ′t = ∅ for all t ∈ I. −1 Then RHom(i−1 t F, it G) is independent of t.

Proof. — We recall that in the non-characteristic case we have i♯t = itd it −1 (see π Example I.2.7). We set H = RHom (F, G). Since Λ := SS(F ) and Λ′ := SS(G) are non∗ characteristic for it and Λt ∩ Λ′t = ∅, we can see that Λ ∩ Λ′ ⊂ TM ×I (M × I) and then a ′ a ′ SS(H) ⊂ Λ + Λ . We can see moreover that Λ + Λ is also non-characteristic for it . −1 ! ! Hence Theorem I.2.8 gives i−1 t G ≃ it G[1] and it H ≃ it H[1]. Then it follows from −1 −1 −1 Proposition I.1.1-(h) that RHom (it F, it G) ≃ it H. By the base change formula we have RΓ(M ; i−1 − I is the projection. Hence t H) ≃ (Rq! H)t , where q : M × I → it is enough to check that Rq! H is locally constant, that is, SS(Rq! H) ⊂ TI∗ I. By Proposition I.2.4 this follows from the fact that SS(H) is non-characteristic for each it .

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I.2.3. CONSTRUCTIBILITY

17

Remark I.2.18. — The particular case F = kM ×I of Corollary I.2.17 gives the following. Let G ∈ D(kM ×I ). We assume that G is non-characteristic for all maps it : M × {t} → − M × I, t ∈ I, and that the map supp(G) → − I is proper. Then the sections RΓ(M ; i−1 t G) are independent of t.

I.2.3. Constructibility We say a few words about the notion of constructibility for sheaves and its relation with the microsupport. F A stratification Σ = {Σi }i∈I of a manifold M is a partition M = i∈I Σi by locally closed subsets such that the closure of each stratum is a union of strata. We will always assume that our stratifications are locally finite (any compact subset meets finitely many strata). We say that F ∈ D(kM ) is weakly constructible with respect to Σ if F | is locally constant for each i ∈ I. If k is a Noetherian ring, we say that F is Σi

constructible with respect to Σ if F ∈ Db (kM ), F is weakly constructible and, for each x ∈ M and j ∈ Z, the stalk H j (F )x is finitely generated over k. We recall an important property of stratifications introduced by Kashiwara and Schapira because it helps to bound the microsupport of sheaves. Definition I.2.19 (Def. 8.3.19 of [28]). — A stratification Σ = {Σi }i∈I of M satisfies b Σ = ΛΣ , where the µ-condition if the strata are locally closed submanifolds and ΛΣ +Λ F ∗ ΛΣ = i∈I TΣi M . In a subanalytic framework, the µ-condition implies Whitney’s conditions (a) and (b) for any two strata (see Exercise VIII.12 of [28]). In Chapter 8 of [28] it is proved that, if M is real analytic, any stratification by subanalytic subsets can be refined into a subanalytic stratification satisfying the µ-condition. The following proposition appears in the same chapter but the proof does not require analyticity. Proposition I.2.20 (Prop. 8.4.1 in [28]). — Let Σ be a stratification of M satisfying the µ-condition. Then F ∈ D(kM ) is weakly constructible with respect to Σ if and only if SS(F ) ⊂ ΛΣ . Proof. — (i) We first assume that SS(F ) ⊂ ΛΣ . Let Σi be a stratum and let U be a neighborhood of Σi such that Σi is closed in U . By Theorem I.2.13 we have b TΣ∗ M ) ∩ T ∗ U . Since ΛΣ + b TΣ∗ M ⊂ ΛΣ + b ΛΣ = ΛΣ and SS(FΣi ) ∩ T ∗ U ⊂ (ΛΣ + i i ∗ ∗ ∗ FΣi vanishes outside Σi we deduce SS(FΣi ) ∩ T U ⊂ TΣi M ∩ T U . It then follows from Example I.2.5 that F | (= FΣi | ) is locally constant. Σi Σi (ii) Now we assume that F is weakly constructible with respect to Σ. Since the problem is local on M we can assume that the stratification is finite, say Σ = {Σ1 , . . . , ΣN }. We Fk also assume that dim(Σi ) ≥ dim(Σi+1 ). Hence Σk = i=1 Σi is an open subset of M . −1 −1 We prove by induction on k that SS(F ) ∩ πM (Σk ) ⊂ ΛΣ ∩ πM (Σk ). The case k = 1 is

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CHAPTER I.2. MICROSUPPORT

clear because F | is locally constant. Let us assume that the claim is proved for k. Σ1 We have the excision distinguished triangle − F| FΣk →

+1

Σk+1

→ − FΣk+1 −−→

over Σk+1 . We recall that FΣk ≃ j! j∗ (F ), where j is the inclusion of Σk in Σk+1 . By Theorem I.2.15, the induction hypothesis and the µ-condition, we have −1 −1 (Σk+1 ) ⊂ ΛΣ ∩ πM (Σk+1 ). Since F | is locally constant and Σk+1 is SS(FΣk ) ∩ πM Σk+1

−1 −1 smooth and closed in Σk+1 we have SS(FΣk+1 ) ∩ πM (Σk+1 ) ⊂ TΣ∗k+1 M ∩ πM (Σk+1 ). Now the result follows from the triangular inequality for the microsupport.

Corollary I.2.21. — Let Σ be a stratification of M satisfying the µ-condition and let F, G ∈ Db (kM ) be given. We assume that F and G are weakly constructible with L

respect to Σ. Then F ⊗ G and RHom (F, G) are also weakly constructible with respect to Σ. L

Proof. — This follows from Proposition I.2.20 and Theorem I.2.13 (the case of F ⊗ G L

also follows from the fact that ⊗ commutes with the inverse image and does not need the µ-condition). Remark I.2.22. — (a) If M is of dimension 1, a (locally finite) stratification Σ is the data of a discrete set of points, say Σ0 , and the connected components of M \ Σ0 . is Then F ∈ Db (kM ) is weakly constructible with respect to Σ if and only if F | 0 M \Σ

locally constant (indeed the condition of being locally constant at the points of Σ0 is L

empty). Since ⊗ and RHom both commute with the restriction to an open subset, Corollary I.2.21 is obvious in the case of dimension 1. (b) More generally, let us assume that the stratification Σ consists of a discrete set of disjoint closed (not only locally closed) submanifolds and the connected components of their complement. We see easily that Σ satisfies the µ-condition. Moreover Proposition I.2.20 follows directly from Proposition I.2.9. If M is a real analytic manifold and Λ ⊂ T˙ ∗ M is an R>0 -conic real analytic Lagrangian submanifold, Corollary 8.3.22 of [28] says that there exists a stratification Σ of M satisfying the µ-condition such that Λ ⊂ ΛΣ . We can deform a C ∞ Lagrangian submanifold into an analytic one and obtain the next proposition. We thank Sylvain Courte for his suggestions about this result. Proposition I.2.23. — Let M be a manifold and Λ ⊂ T˙ ∗ M an R>0 -conic closed Lagrangian submanifold (both M and Λ of class C ∞ ). Then there exists an R>0 -homogeneous Hamiltonian isotopy Ψ : T˙ ∗ M × [0, 1] → − T˙ ∗ M such that Ψ0 = id and Ψ1 (Λ) ⊂ ΛΣ , for some stratification Σ of M satisfying the µ-condition. Moreover Ψ1 (Λ) can be chosen arbitrarily close to Λ. Proof. — We reduce to the real analytic case, following the ideas in [34], and apply results of [28].

ASTÉRISQUE 440

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I.2.3. CONSTRUCTIBILITY

¯ = Λ/R>0 and S ∗ M = T˙ ∗ M/R>0 . Then Λ ¯ is a Legendrian submani(i) We set Λ fold of S ∗ M and the contact version of the Weinstein neighborhood theorem gives a contact embedding j : U → − S ∗ M of class C ∞ , where U is an open neighborhood 1 ¯ ∗¯ ¯ of Λ in J (Λ) = T Λ × R. We denote by ξΛ¯ and ξM the standard contact structures ¯ and S ∗ M and by αΛ¯ and αM the standard contact forms. of J 1 (Λ) ¯ and M compatible We know by [53] that there exist real analytic structures on Λ ∞ ¯ S ∗ M and with their C structures. They induce real analytic structures on J 1 (Λ), r αΛ¯ , αM are analytic. We know also that we can find a C map ȷ˜: U ×[0, 1] → − S ∗ M , for s 0 1 any integer r, such that, setting j = ȷ˜| , we have: j = j, j is an analytic open U ×{s} s embedding and the j are as close as required to j in the compact open C r -topology. On U we thus have a family of contact structures ξ s = j s∗ (ξM ), with 0 ξ = j ∗ (ξM ) = ξΛ¯ | and ξ 1 analytic. For each given s we consider a linear inU terpolation between ξ 0 and ξ s : αts = (1 − t)αΛ¯ + t j s∗ (αM ),

ξts = ker(αts ).

We remark that the αt1 are analytic. Choosing j s C r -close to j, we can assume that j s∗ (αM ) is C r−1 -close to αΛ¯ . By Gray’s theorem, for each s we can find an ¯ as we recall in (ii) (see for isotopy φst , t ∈ [0, 1], such that (φst )∗ (ξts ) = ξ 0 near Λ example Theorem 2.2.2 of [16]). (ii) For a given s ∈ [0, 1], we define the vector field Xts , t ∈ [0, 1], on U uniquely determined by Xts ∈ ξts

(I.2.6) (recall that

dαts | s ξ t

and

Xts ⌟ dαts = −α˙ ts |

is non-degenerate). We remark that

Xt1

ξts

is analytic. We also remark

= j (αM ) − αΛ¯ is close to 0. Hence Xts is as close is close to dαΛ¯ and that to 0 as required and the flow φ of {Xts }t∈[0,1] is defined on a set V × [0, 1] where V is ¯ in U . Moreover φ1 is analytic. some neighborhood of Λ . By (I.2.6) we have Now we check that (φst )∗ (ξts ) = ξ 0 , where φst = φs | V ×{t} s s s s s s Xt ⌟dαt + α˙ t = ft αt for some function ft and d ((φst )∗ (αts ) ∧ αΛ¯ ) = (φst )∗ (LXts (αts ) + α˙ ts ) ∧ αΛ¯ dt = (φst )∗ (fts αts ) ∧ αΛ¯ dαts

α˙ ts s

s∗

= (fts ◦ φst )((φst )∗ (αts ) ∧ αΛ¯ ). Since (φs0 )∗ (αΛ¯ )∧αΛ¯ = 0, we obtain (φst )∗ (αts )∧αΛ¯ = 0 for all t ∈ [0, 1], which implies (φst )∗ (ξts ) = ξ 0 . (iii) By construction the map ψ s = j s ◦ φs1 : V → − S ∗ M is contact. Moreover ψ 1 is s s ¯ ¯ ¯ with Λ ¯ 1 analytic. analytic. Hence Λ = ψ (Λ) defines a Legendrian deformation of Λ 1 ¯ The existence of the required stratification for Λ is given by Corollary 8.3.22 of [28]. Now a Legendrian deformation can be lifted to an ambient contact isotopy, which is the same as an R>0 -homogeneous Hamiltonian isotopy of T˙ ∗ M .

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CHAPTER I.3 SATO’S MICROLOCALIZATION

We quickly review the definition of the specialization and microlocalization functors as introduced in [28]. We first recall the notion of normal deformation. Let M be a manifold and N a closed submanifold of N . The normal deformation of N in M is a fN together with three maps manifold M fN , fN → fN → s : TN M → − M p: M − M, t: M − R, such that  s is an embedding and im(s) = t−1 (0),      p(im(s)) = N and p ◦ s is the projection TN M → − N,   p−1 (M \ N ) ≃ (M \ N ) × (R \ {0}),    t−1 (R \ {0}) ≃ M × (R \ {0}),    −1 p − M is a diffeomorphism, for all u ̸= 0. | −1 : t (u) → t

(u)

fN as an open subset of some blow-up as follows. The blow-up We can define M BN ×{0} (M × R) is set theoretically the union of U = (M × R) \ (N × {0}) and P (TN M × R), the projectivization of the vector bundle TN M × R → − N . We consider (M \ N ) × {0} as a subset of U and P (TN M × {0}) as a subset of P (TN M × R). We set fN = BN ×{0} (M × R) \ (((M \ N ) × {0}) ∪ P (TN M × {0})). M Now BN ×{0} (M ×R) comes with a map to M ×R and this map induces the maps p, t. The difference P (TN M × R) \ P (TN M × {0}) is identified with TN M and this gives s. fN be the inclusion and set p+ = p ◦ j. We set Ω = t−1 (]0, +∞[). We let j : Ω → − M We have the following formula for the cone of a subset A ⊂ M along N : CN (A) = s−1 (p−1 + (A)). The sheaf counterpart of the cone construction is the following specialization functor.

Definition I.3.1 (See [28, Def. 4.2.2]). — With the above notations we define the functor νN : D(kM ) → − D(kTN M ) by νN (F ) = s−1 Rj∗ p−1 + (F ).

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CHAPTER I.3. SATO’S MICROLOCALIZATION

The sheaf νN (F ) is conic, that is, invariant by the multiplicative action of R>0 on the fibers of TN M . We can deduce a sheaf over TN∗ M using the Fourier-Sato transform defined as follows. Definition I.3.2 (See [28, Def. 3.7.8]). — Let qi be the i-th projection from TN M ×N TN∗ M and let P ⊂ TN M ×N TN∗ M be the subset P = {(ν, ξ); ⟨ν, ξ⟩ ≤ 0}. For F ∈ D(kTN M ) we define F ∧ ∈ D(kTN∗ M ) by F ∧ = Rq2 ! (q1−1 F ⊗ kP ). In [28] the Fourier-Sato transform is actually defined for general vector bundles. It is proved that it gives an equivalence between conic sheaves on a vector bundle and conic sheaves on its dual. Definition I.3.3 (See [28, Def. 4.3.1]). — The microlocalization µN : D(kM ) → − D(kTN∗ M ) is defined by µN (F ) = (νN (F ))∧ .

functor

If V ⊂ TN∗ M is a convex open cone, we have, using notation (I.1.6), i (U ; F ), H i (V ; µN (F )) ≃ lim HZ∩U −→ U,Z

where U runs over the open subsets of M containing πM (V ) and Z over the closed subsets of M such that CN (Z) ⊂ V ◦ (recall that V ◦ is the polar cone of V ). In [28] we also find a generalization of Sato’s microlocalization which will be important when we consider the Kashiwara-Schapira stack. Let ∆M be the diagonal ∗ of M × M . Let q1 , q2 : M × M → − M be the projections. We identify T∆ (M × M ) M with T ∗ M through the first projection. Definition I.3.4 (See [28, Def. 4.4.1]). — µhom(F, G) ∈ D(kT ∗ M ) by

For

F, G ∈ D(kM )

we

define

µhom(F, G) = µ∆M (RHom (q2−1 F, q1! G)).

(I.3.1)

− T ∗ M the inclusion, we have For a submanifold N of M and i : TN∗ M → i∗ µN (G) ≃ µhom(kN , G), for any G ∈ D(kM ). The functor µhom is a refinement of the functor RHom in view of the following properties: (I.3.2)

RπM ∗ µhom(F, G) ≃ RHom (F, G),

(I.3.3)

−1 RπM ! µhom(F, G) ≃ δM RHom (q2−1 F, q1−1 G),

where δM : M → − M ×M is the diagonal embedding. For a conic sheaf H on T ∗ M (or on any vector bundle) we have a natural isomorphism RπM ! (H) ≃ RπM ∗ RΓM (H) (where M is here the zero section of T ∗ M ) and the natural morphism RπM ! (H) → − RπM ∗ (H) coincides with the morphism deduced from RΓM (H) → − H. The excision distinguished triangle associated with the inclusion M ⊂ T ∗ M then gives a version of Sato’s distinguished triangle: (I.3.4)

−1 δM RHom (q2−1 F, q1−1 G) → − RHom (F, G) +1

→ − Rπ˙ M ∗ (µhom(F, G)| ˙ ∗ ) −−→ . T M

ASTÉRISQUE 440

23

CHAPTER I.3. SATO’S MICROLOCALIZATION

If F is cohomologically constructible, then the first term of (I.3.4) is isomorphic L

to D′ (F ) ⊗ G by Theorem I.2.13 and we obtain (I.3.5)

L

+1

D′ (F ) ⊗ G → − RHom (F, G) → − Rπ˙ M ∗ (µhom(F, G)| ˙ ∗ ) −−→ . T M

Proposition I.3.5 (Cor. 6.4.3 of [28]). — Let F, G ∈ D(kM ). Then (I.3.6) (I.3.7)

supp(µhom(F, G)) ⊂ SS(F ) ∩ SS(G), SS(µhom(F, G)) ⊂ (H −1 (C(SS(G), SS(F ))))a ,

where H is the Hamiltonian isomorphism. When F = G, the inclusion (I.3.6) is an equality. More precisely, by (I.3.2), idF ∈ Hom(F, F ) gives a global section of µhom, say (I.3.8)

idµF ∈ H 0 (T ∗ M ; µhom(F, F ))

and [28, Cor. 6.1.3] says that (I.3.9)

supp(idµF ) = supp µhom(F, F ) = SS(F ).

An important consequence of (I.3.7) and (I.3.9) is the following involutivity theorem. Theorem I.3.6 (Thm. 6.5.4 of [28]). — Let M be a manifold and F ∈ D(kM ). Then S = SS(F ) is a coisotropic subset of T ∗ M in the sense that Cp (S) contains the symplectic orthogonal of Cp (S, S), for all p ∈ S. When Λ ⊂ T ∗ M is a Lagrangian submanifold we have H −1 (T Λ) = TΛ∗ T ∗ M . Hence (I.3.6), (I.3.7) and Example I.2.5 give the following result. Corollary I.3.7. — Let Λ be a conic Lagrangian submanifold of T˙ ∗ M . Let F, G ∈ D(kM ). We assume that there exists a neighborhood Ω of Λ such that SS(F ) ∩ Ω ⊂ Λ and SS(G) ∩ Ω ⊂ Λ. Then µhom(F, G)|Ω is supported on Λ and is locally constant on Λ. By the following result we can see µhom as a microlocal version of RHom . Let p ∈ T ∗ M be a given point. By the triangular inequality the full subcategory Np of D(kM ) formed by the F such that p ̸∈ SS(F ) is triangulated and we can set D(kM ; p) = D(kM )/Np (see the more general Definition 6.1.1 of [28]). The functor µhom(·, ·)p induces a bifunctor on D(kM ; p) and we have Theorem I.3.8 (Theorem 6.1.2 of [28]). — For all F, G ∈ D(kM ), the morphism HomD(kM ;p) (F, G) → − H 0 (µhom(F, G))p is an isomorphism. We also give the following useful consequence of Theorem I.3.6. Corollary I.3.9. — Let Λ be a conic connected Lagrangian submanifold of T˙ ∗ M . Let ˙ ˙ ˙ F ∈ D(kM ) be such that SS(F ) ̸= ∅ and SS(F ) ⊂ Λ. Then SS(F ) = Λ. ˙ Proof. — Arguing by contradiction we assume that U = Λ \ SS(F ) is non empty. The set U is open in Λ with a non empty boundary ∂U . We choose a chart V in Λ around a point of ∂U and we choose a point p0 ∈ V ∩ U . Let B be the open ball in V with

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CHAPTER I.3. SATO’S MICROLOCALIZATION

˙ ˙ center p0 and maximal radius such that B ∩ SS(F ) = ∅. Then ∂B ∩ SS(F ) is non empty and we let p be any of its points. ˙ Since SS(F ) ⊂ Λ and Λ is smooth, we have Cp (SS(F ), SS(F )) ⊂ Cp (Λ, Λ) = Tp Λ. Since Λ is Lagrangian, it follows that the symplectic orthogonal of Cp (SS(F ), SS(F )) contains Tp Λ. On the other hand Cp (SS(F )) is contained in Cp (V \ B) which is a half space of Tp Λ. Hence Cp (SS(F )) does not contain Tp Λ and this contradicts Theorem I.3.6.

ASTÉRISQUE 440

CHAPTER I.4 SIMPLE SHEAVES

Let Λ be a closed conic Lagrangian submanifold of T˙ ∗ M . We recall the definition of simple and pure sheaves along Λ and give some of their properties. We first recall some notations from [28]. For a function φ : M → − R of class C ∞ we define (I.4.1)

Λφ = {(x; dφ(x)); x ∈ M }.

We notice that Λφ is a closed Lagrangian submanifold of T ∗ M . For a given point p = (x; ξ) ∈ Λ ∩ Λφ we have the following Lagrangian subspaces of Tp (T ∗ M ) (I.4.2)

λ0 (p) = Tp (Tx∗ M ),

λΛ (p) = Tp Λ,

λφ (p) = Tp Λφ .

We recall the definition of the inertia index (see for example §A.3 in [28]). Let (E, σ) be a symplectic vector space and let λ1 , λ2 , λ3 be three Lagrangian subspaces of E. We define a quadratic form q on λ1 ⊕ λ2 ⊕ λ3 by q(x1 , x2 , x3 ) = σ(x1 , x2 ) + σ(x2 , x3 ) + σ(x3 , x1 ) and (I.4.3)

τE (λ1 , λ2 , λ3 ) = sgn(q),

where sgn(q) is the signature of q, that is, p+ − p− , where p± is the number of ±1 in a diagonal form of q. We set τφ = τp,φ = τTp T ∗ M (λ0 (p), λΛ (p), λφ (p)). Proposition I.4.1 (Proposition 7.5.3 of [28]). — Let φ0 , φ1 : M → − R be functions of class C ∞ , let p = (x; ξ) ∈ Λ and let F ∈ D(Λ) (kM ) (see Notation I.2.2). We assume that Λ and Λφi intersect transversely at p, for i = 0, 1. Then there exists an isomorphism (RΓ{φ1 ≥φ1 (x)} (F ))x ≃ (RΓ{φ0 ≥φ0 (x)} (F ))x [ 21 (τφ0 − τφ1 )]. Definition I.4.2 (Definition 7.5.4 of [28]). — In the situation of Proposition I.4.1 we say that F is pure at p if (RΓ{φ0 ≥φ0 (x)} (F ))x is concentrated in a single degree and free, that is, (RΓ{φ0 ≥φ0 (x)} (F ))x ≃ L[d], for some free module L ∈ Mod(k) and d ∈ Z. The half integer (I.4.4)

d + 21 dM + 21 τφ0

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CHAPTER I.4. SIMPLE SHEAVES

is called the shift of F . If L ≃ k, we say that F is simple at p. If F is pure (resp. simple) at all points of Λ we say that it is pure (resp. simple) along Λ. We denote by Ds,f [Λ] (kM ) the full subcategory of D[Λ] (kM ) formed by the F such ˙ that F is simple along SS(F ) and the stalks of F at the points of M \ π˙ M (Λ) are finitely generated. Proposition I.4.3 (Cor. 7.5.7 in [28]). — We assume that Λ is connected and F ∈ D(Λ) (kM ) is pure at some p ∈ Λ. Then F is pure along Λ. Moreover the L ∈ Mod(k) in the above definition is the same at every point. Remark I.4.4. — In Proposition X.5.3 below we will give a more precise result than Propositions I.4.1 and I.4.3. If k is a field, we know that F is pure along Λ if and only if µhom(F, F )| ˙ ∗ is T M concentrated in degree 0 and F is simple along Λ if and only if the natural morphism kΛ → − µhom(F, F )| ˙ ∗ induced by (I.3.2) and the section idF of RHom (F, F ) is T M an isomorphism: ∼ (I.4.5) kΛ −− → µhom(F, F ). For coefficients in a field the property (I.4.5) could be a definition of a simple sheaf. Example I.4.5. — (1) Let N be a submanifold of M of codimension d. Then kN is simple with shift 21 d. To see that kN is simple we can assume M = Rn , N = Rn−d ×{0} and p = (x0 ; ξ0 ) with x0 = 0, ξ0 = (0, . . . , 0, 1). Pn−1 We choose the function φ0 (x) = xn + i=1 x2i . Then RΓ{φ0 ≥0} (kN ) ≃ kN and (RΓ{φ0 ≥0} (kN ))x0 ≃ k. The computation of the shift is not difficult but lengthy. We refer to Example 7.5.5 of [28]. (2) In Example I.2.3 (iii) the sheaves kU and kU are simple; kU has shift −1/2 and kU has shift 1/2. In fact both compare to k∂U , which has shift 1/2 by (1), through +1 ˙ U ) we the distinguished triangle kU → − k → − k∂U −−→. Choosing φ0 with dφ0 ∈ SS(k U

deduce from the triangle RΓ{φ0 ≥φ0 (0)} (kU ) ≃ RΓ{φ0 ≥φ0 (0)} (k∂U )[−1]. The argument is similar for kU . We also refer to Example 7.5.5 of [28]. (3) For i ∈ N we let Λi ⊂ Λ be the set of points such that the rank of dπM |Λ is (dim M − 1 − i). For a generic closed conic Lagrangian connected submanifold Λ in T˙ ∗ M , Λ0 is an open dense subset of Λ and, for a given simple sheaf F ∈ D(Λ) (kM ), the shift of F at p is locally constant on Λ0 and changes by 1 when p crosses Λ1 . These properties correspond exactly to the definition of a Maslov potential for Λ. We recall that the Maslov class of Λ (an element of H 1 (Λ; ZΛ ) is the obstruction to the existence of a Maslov potential for Λ. Hence, if there exists a simple sheaf F ∈ D(Λ) (kM ), then the Maslov class of Λ vanishes and a Maslov potential is given by the shift of F .

ASTÉRISQUE 440

CHAPTER I.4. SIMPLE SHEAVES

27

We can also compute the stalks of µhom for sheaves in D(Λ) (kM ). Let p = (x; ξ) ∈ Λ and φ0 : M → − R be as in Proposition I.4.1. For F, G ∈ D(Λ) (kM ), we have (I.4.6)

µhom(F, G)p ≃ RHom((RΓZ0 (F ))x , (RΓZ0 (G))x ),

where Z0 = {φ0 ≥ φ0 (x)}. Now we prove that a simple sheaf belongs to Ds,f [Λ] (kM ) as soon as its stalk at some given point is finitely generated. We set ZΛ = {x ∈ π˙ M (Λ); there exist a neighborhood W of x and a smooth hypersurface S ⊂ W such that Λ ∩ T ∗ W ⊂ TS∗ W }. The transversality theorem implies the following result. Lemma I.4.6. — Let x, y ∈ M \ π˙ M (Λ). Let I be an open interval containing 0 and 1. Then there exists a C ∞ embedding c : I → − M such that c(0) = x, c(1) = y and c([0, 1]) only meets π˙ M (Λ) at points of ZΛ , with a transverse intersection. Lemma I.4.7. — Let F ∈ D[Λ] (kM ) be a simple sheaf along Λ. We set U = M \ π˙ M (Λ). We that M is connected and that there exists x0 ∈ U such thatL the k-module L assume i i H F is finitely generated. Then, for any x ∈ U , the k-module x0 i∈Z i∈Z H Fx is s,f finitely generated. In other words F belongs to D[Λ] (kM ). Proof. — Let x ∈ U and let I be an open interval containing 0 and 1. By Lemma I.4.6 we can choose a C ∞ path γ : I → − M such that γ(0) = x0 , γ(1) = x and γ([0, 1]) meets π˙ M (Λ) at finitely many points, all contained in ZΛ and with a transverse intersection. We denote these points by γ(ti ), where 0 < t1 < · · · < tk < 1. Since F is locally constant on U , the stalk Fγ(t) is constant for t ∈ ]ti , ti+1 [. Near a point xi = γ(ti ) we have a hypersurface S of M such that Λ ⊂ TS∗ M . Let us first assume that Λ is one half of TS∗ M . Using Example I.2.10 and the fact that F is simple, there exist d ∈ Z, E ′ ∈ D(k) and a distinguished triangle, in some neighborhood of xi , +1

′ EM → − kZ [d] → − F −−→,

where Z is one of the closed half-spaces bounded by S. It follows that the stalks Fγ(ti −ε) and Fγ(ti +ε) differ by k[d] or k[d + 1] for ε > 0 small enough. Now we assume that Λ = TS∗ M . The argument in Example I.2.10 works in the same way to reduce the situation to Example I.2.3-(v), which also works in the same way and gives a +1 distinguished triangle F ′ → − kZ [d] → − F −−→. Now F ′ is no longer constant, but ′ ˙ ˙ ˙ Z [d]) is half of T ∗ M . So we are back to the previous case. SS(F ) = SS(F ) \ SS(k S

SOCIÉTÉ MATHÉMATIQUE DE FRANCE 2023

CHAPTER I.5 COMPOSITION OF SHEAVES

We will use several times a usual operation associated with sheaves called “composition” or “convolution”. We refer to [28, §3.6] for more details, or [21, §1.6, §1.10]. Let Mi , i = 1, 2, 3, be three manifolds. We denote by qij the projection from M1 × M2 × M3 to Mi × Mj . For K1 ∈ D(kM1 ×M2 ) and K2 ∈ D(kM2 ×M3 ) we denote by K1 ◦ K2 ∈ D(kM1 ×M3 ) the composition of K1 and K2 : (I.5.1)

L

−1 −1 K1 ◦ K2 = Rq13 ! (q12 K1 ⊗ q23 K2 ).

The Fourier-Sato transform of Definition I.3.2 is an example of composition of sheaves. Using the base change formula we see that the composition product is associative in the sense that, for another manifold M4 and K3 ∈ D(kM3 ×M4 ), we have a natural isomorphism (K1 ◦ K2 ) ◦ K3 ≃ K1 ◦ (K2 ◦ K3 ). If M1 = M2 , the constant sheaf on the diagonal K1 = k∆M1 is a left unit for this product: we have k∆M1 ◦ K2 ≃ K2 for any K2 ∈ D(kM2 ×M3 ). Similarly, if M2 = M3 , the sheaf k∆M2 is a right unit. The base change formula gives a useful expression for the stalks of the composition. For (x, z) ∈ M1 × M3 we have (I.5.2)

(K1 ◦ K2 )(x,z) ≃ RΓc (M2 ; (K1 |

L

{x}×M2

) ⊗ (K2 |

M2 ×{z}

)).

For K1 ∈ D(kM1 ×M2 ) we have a natural candidate for an inverse, denoted K1−1 and defined as follows. ∼ Let q2 : M1 × M2 → − M2 be the projection and v : M1 × M2 −− → M2 × M1 the swap ! isomorphism. We recall that q2 (kM2 ) ≃ ωM1 ⊠ kM2 . We define (I.5.3)

K1−1 = v −1 RHom (K1 , q2! (kM2 )) ∈ D(kM2 ×M1 ).

Let δ2 : M2 → − M2 × M2 and δ2′ : M1 × M2 → − M2 × M1 × M2 be the diagonal −1 embeddings. The base change formula δ2−1 ◦ Rq23 ! ≃ Rq2 ! ◦ δ2′ implies L

δ2−1 (K1−1 ◦ K1 ) ≃ Rq2 ! (K1 ⊗ RHom (K1 , q2! (kM2 ))). L

Using the contraction K1 ⊗ RHom (K1 , L) → − L and the adjunction morphisms for (Rq2 ! , q2! ) and (δ2−1 , Rδ2∗ ) we deduce the first morphism in (I.5.4) below; the

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CHAPTER I.5. COMPOSITION OF SHEAVES

second morphism is obtained in the same way. K1−1 ◦ K1 → − k∆M2 ,

(I.5.4)

K1 ◦ K1−1 → − k ∆ M1 .

Using the bounds given by Proposition I.2.4 and Theorem I.2.8, I.2.13 for the behavior of the microsupport under sheaves operations we obtain the following result. We denote by pij the projections from T ∗ (M1 × M2 × M3 ) similar to the qij . We also define a2 on T ∗ (M1 × M2 ) by a2 (x, y; ξ, η) = (x, y; ξ, −η). Lemma I.5.1. — Let K1 ∈ D(kM1 ×M2 ) and K2 ∈ D(kM2 ×M3 ) be given. We assume −1 −1 that q13 is proper on q12 supp(K1 ) ∩ q23 supp(K2 ) and −1 −1 ∗ ∗ ∗ p−1 12 a2 SS(K1 ) ∩ p23 SS(K2 ) ∩ (TM1 M1 × T M2 × TM3 M3 )

is contained in the zero-section of T ∗ (M1 × M2 × M3 ). Then SS(K1 ◦ K2 ) ⊂ SS(K1 ) ◦a SS(K2 ),

(I.5.5)

where the operation ◦a for A1 ⊂ T ∗ (M1 × M2 ) and A2 ⊂ T ∗ (M2 × M3 ) is defined −1 −1 by A1 ◦a A2 = p13 (p−1 12 a2 (A1 ) ∩ p23 (A2 )). We will also use a relative version of the composition. For a manifold I we denote by qijI the projections from M1 × M2 × M3 × I to Mi × Mj × I similar to the qij . For K1 ∈ D(kM1 ×M2 ×I ) and K2 ∈ D(kM2 ×M3 ×I ) we set L

−1 −1 K1 ◦ | K2 = Rq13I ! (q12I K1 ⊗ q23I K2 ).

(I.5.6)

I

The definition is chosen so that (K1 ◦ | K2 )| ≃ K1,t ◦ K2,t for all t ∈ I, I M1 ×M3 ×{t} where K1,t = K1 | , K2,t = K2 | . M1 ×M2 ×{t}

M2 ×M3 ×{t}

The previous results generalize to the relative setting (see [21]). In particular we can define K1−1 = v −1 RHom (K1 , q2! (kM2 ×I )) (an object of D(kM2 ×M1 ×I )) and we have natural morphisms K1−1 ◦ | K1 → − k∆M2 ×I ,

(I.5.7) We also have

I

K1−1 | M1 ×M2 ×{t}

≃

K1 ◦ | K1−1 → − k∆M1 ×I . I

−1 K1,t .

Remark I.5.2. — For K1 ∈ D(kM1 ×M2 ) the composition ΦK1 : F 7→ K ◦ F is a functor D(kM2 ) → − D(kM1 ) and we have ΦK1 ◦K2 ≃ ΦK1 ◦ ΦK2 . When M1 = M2 = M , Φk∆M is the identity functor. When M1 = M2 = M3 = M , if K1 ◦ K2 ≃ k∆M , then K2 ◦ K1 ≃ k∆M and ΦK1 , ΦK2 are mutually inverse equivalences of categories.

ASTÉRISQUE 440

PART II SHEAVES ASSOCIATED WITH HAMILTONIAN ISOTOPIES

In this part we recall the main result of [21], which, following ideas of Tamarkin in [45], gives a sheaf version of the Chekanov-Sikorav theorem about generating functions (see [10] and [44]). Let Ψ : J 1 (N ) × I → − J 1 (N ) be a contact isotopy of the 1-jet bundle of some manifold N . The Chekanov-Sikorav theorem says that, if a Legendrian submanifold L of J 1 (N ) has a generating function, so does Ψs (L), for any s ∈ I. The sheaf version is more functorial. We can associate a sheaf KΨ on N 2 × I with Ψ. This sheaf acts by composition on D(kN ) and induces equivalences of categories, F 7→ (KΨ | 2 ) ◦ F . We state this result with homogeneous Hamiltonian isotopies N ×{s} ∗ ˙ of T N (which is the same thing as contact isotopies of the sphere bundle of T ∗ N ). We recall how it implies the non homogeneous case and give some complementary remarks.

CHAPTER II.1 HOMOGENEOUS CASE

Let N be a manifold and I an open ball of Rd containing 0 (in general I will be an open interval of R containing 0). We consider a homogeneous Hamiltonian isotopy Ψ : T˙ ∗ N × I → − T˙ ∗ N of class C ∞ . For s ∈ I, p ∈ T˙ ∗ N we set Ψs (p) = Ψ(p, s). Hence Ψ0 = idT˙ ∗ N and, for each s ∈ I, Ψs is a symplectic diffeomorphism such that Ψs (x; λξ) = λ · Ψs (x; ξ), for all (x; ξ) ∈ T˙ ∗ N and λ > 0. We let ΛΨs ⊂ T˙ ∗ N 2 be the twisted graph of Ψs , that is, (II.1.1) ΛΨ = {(Ψ(x, ξ, s), (x; −ξ)); (x; ξ) ∈ T˙ ∗ N }. s

We can see that there exists a unique conic Lagrangian submanifold ΛΨ ⊂ T˙ ∗ (N 2 ×I), described in (II.1.2) below, such that ΛΨs = i♯s (ΛΨ ), for all s ∈ I, where is is the embedding N 2 × {s} → − N 2 × I. Our homogeneous Hamiltonian isotopy Ψ is the flow of some h : T˙ ∗ N × I → − R and there is a unique such h which is homogeneous of degree 1 in the variable ξ. Then we have (II.1.2) ΛΨ = {(Ψ(x, ξ, s), (x; −ξ), (s; −h(Ψ(x, ξ, s), s))); (x; ξ) ∈ T˙ ∗ N, s ∈ I}. We also remark that ΛΨ is non-characteristic for the inclusion is , for any s ∈ I. We recall that Dlb (kM ) is the subcategory of D(kM ) of locally bounded complexes. Theorem II.1.1 (Theorem. 3.7 and Remark 3.9 of [21] – see also Proposition 3.16 ˙ of [45]). — There exists a unique KΨ ∈ Dlb (kN 2 ×I ) such that SS(K Ψ ) ⊂ ΛΨ and ˙ KΨ | 2 ≃ k∆N . Moreover we have SS(K ) = Λ , K is simple along ΛΨ , Ψ Ψ Ψ N ×{0} both projections supp(KΨ ) → − N × I are proper and the morphisms (I.5.7) are isomorphisms: −1 −1 ∼ ∼ KΨ ◦ | KΨ −− → k∆N ×I , KΨ ◦ | KΨ −−→ k∆N ×I . I I ˙ ˙ Remarks II.1.2. — (1) The equivalence between SS(K Ψ ) ⊂ ΛΨ and SS(KΨ ) = ΛΨ follows from Corollary I.3.9. (2-a) The fact that KΨ is simple along ΛΨ is not explicitly stated in [21] (although it is used in the proof of the Arnol’d conjecture about the intersection of the zero section of a cotangent bundle which its image under a Hamiltonian isotopy). However it is easily deduced from the construction of KΨ which is obtained as a composition of

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CHAPTER II.1. HOMOGENEOUS CASE

constant sheaves over open subsets with smooth boundaries. Such sheaves are simple and a composition of simple sheaves is simple by [28, Thm. 7.5.11]. (2-b) We can also check the simplicity without going back to the construction ˙ of KΨ and using only the properties SS(K ≃ k∆N . Indeed, Ψ ) = ΛΨ and KΨ | 2 N ×{0} by Proposition I.4.3 it is enough to check the simplicity at one point of ΛΨ . The map is,π : T ∗ N 2 × Ts∗ I → − T ∗ (N 2 × I) is transverse to ΛΨ , for any s ∈ I, and the ∗ 2 projection is,d : ΛΨ ∩(T N ×Ts∗ I) → − ΛΨs is a bijection. By [28, Cor. 7.5.13] it follows that the type of KΨ | 2 at a point p ∈ ΛΨs is the same as the type of KΨ at the N ×{s}

point p′ ∈ ΛΨ such that is,d (p′ ) = p. For s = 0 we deduce from KΨ |

N 2 ×{0}

∗

≃ k ∆N

2

that KΨ is simple at any point of ΛΨ ∩ (T N × {0}). (3) We can define the isotopy Ψ′ : T˙ ∗ N × I → − T˙ ∗ N by Ψ′s = Ψ−1 for all s ∗ 2 ˙ ′ s ∈ I. Using Lemma I.5.1 we see that SS(K ◦ K ) ⊂ T (N × I). Since Ψ |I Ψ ∆N ×I ′ ′ (KΨ ◦ | KΨ )| 2 ≃ k∆N , Proposition I.2.9 gives KΨ ◦ | KΨ ≃ k∆N ×I . The I

N ×{0}

I

−1 uniqueness of the inverse then implies that KΨ ≃ KΨ′ .

(4) By Remark I.5.2 the composition with KΨ,s = KΨ |

N ×{s}

gives an equivalence of

categories D(kN ) → − D(kN ), F 7→ KΨ,s ◦ F . Its inverse is given by G 7→ (KΨ,s )−1 ◦ G; with the notations of (3) we have (KΨ,s )−1 ≃ KΨ′ ,s . Remark II.1.3. — The link with the Chekanov-Sikorav theorem is as follows. Let M be a manifold and let Λ ⊂ J 1 (M ) be a Legendrian submanifold which admits a generating function f : M ×Rd → − R quadratic at infinity. We define the epigraph of f , + d Γ+ ⊂ M × R × R, by Γ = {(x, v, t); t ≥ f (x, v)}. We let q : M × Rd × R → − M × R be f f ˙ the projection and set Ff = Rq! (kΓ+ ). Then we can check that SS(Ff ) = Λ, where we f identify Λ with a conic Lagrangian submanifold of T˙ ∗ (M × R) (this follows directly from Proposition I.2.4 if q is proper on Γ+ f ; in general we have to check that q |Γ+ has f a good behavior at infinity and we need some hypotheses on f —quadratic at infinity for example). Now we assume to be given a Legendrian isotopy Ψ of J 1 (M ), that we identify with a homogeneous Hamiltonian isotopy of T˙ ∗ (M × R). The Chekanov-Sikorav theorem says that, up to adding extra variables, we can modify f into fs , such that fs is a generating function of Ψs (Λ). Then we have Ffs ≃ KΨ,s ◦ Ff .

Example II.1.4. — The easiest illustration of Theorem II.1.1 is the sheaf associated with the (normalized) geodesic flow of T˙ ∗ Rn . It is Example 3.10 of [21] and we recall ξ it briefly. We set I = R and define Ψ : T˙ ∗ Rn × I → − T˙ ∗ Rn , by Ψs (x; ξ) = (x + s ∥ξ∥ ; ξ). It is the Hamiltonian flow of h(x; ξ) = ∥ξ∥ (it is −∥ξ∥ in [21], so our example is the same up to inverting time). For s > 0 we define Us = {(x, y) ∈ R2n ; ∥x − y∥ < s}. ˙ U ). The Formula (I.5.2) Using Example I.2.3 (iii) we see directly that ΛΨs = SS(k s gives (kUs ◦ kUt )(x,z) ≃ RΓc (Rn ; kB(x,s) ⊗ kB(z,t) ) ≃ RΓc (Rn ; kB(x,s)∩B(z,t) ),

ASTÉRISQUE 440

CHAPTER II.1. HOMOGENEOUS CASE

35

where B(x, s) is the open ball of radius s centered at x. The intersection B(x, s) ∩ B(z, t) is empty or homeomorphic to an n-ball and we deduce (kUs ◦ kUt )(x,z) ≃ k[−n] if ∥x − z∥ < t + s. We cannot deduce a sheaf from its stalks. However we can make the same computation for a small ball instead of the stalks (it is a similar computation and the sets involved are cylinders instead of balls). Alternatively we have a bound for the microsupport of a composition (Lemma I.5.1) which says in our case that kUs ◦ kUt is locally constant on Us+t , hence constant since Us+t is contractible. It follows kUs ◦ kUt ≃ kUs+t [−n]. Defining Ks = kUs [n] we thus have Ks ◦ Kt ≃ Ks+t and we can expect that the sheaf associated with Ψ is given on {s > 0} by K = kU [n], where U = {(x, y, s) ∈ R2n+1 ; s > 0, ∥x − y∥ < s}. What about negative times? For s > 0 we have Ks−1 ≃ v −1 RHom (Ks , q2! (kRn )) ≃ v −1 RHom (Ks , kR2n [n]) ≃ kUs (see (I.5.3) for the definition of Ks−1 and use (I.1.8)). The composition Ks−1 ◦ K is a sheaf on R2n × R>0 . We shift it by s to the left and get a sheaf, say L(s), on R2n × ]−s, +∞[, whose restriction to R2n × R>0 is K. We can compute its restriction to R2n × ]−s, 0] and find L(s)| 2n ≃ ki(Us ) where i(x, y, t) = (x, y, −t). The R

×]−s,0]

microsupport of L(s) is again given by Lemma I.5.1 and thus is the graph of Ψ. Since L(s)| 2n ≃ k∆Rn we deduce that L(s) ≃ KΨ . We can take s arbitrarily big and R ×{0} obtain ( KΨ | 2n ≃ kU [n], R ×]0,+∞[ (II.1.3) KΨ | 2n ≃ kZ , R

×]−∞,0]

2n+1

where U = {(x, y, s) ∈ R ; s > 0, ∥x − y∥ < s} and Z = {(x, y, s) ∈ R2n+1 ; s ≤ 0, ∥x − y∥ ≤ −s}. We remark that we have a distinguished triangle (II.1.4)

u

kU [n] → − KΨ → − kZ − → kU [n + 1]

which is not split, that is, KΨ ̸≃ kZ ⊕ kU [n]. Indeed this would imply SS(KΨ ) = SS(kU ) ∪ SS(kZ ). But Example I.2.3 (iv) says that SS(kZ ) is bigger than the graph of Ψ at the points T ∗ R2n+1 above ∆Rn × {0}. In particular the morphism u in (II.1.4) is non zero. The computation RHom (kZ , kU ) ≃ RHom (kZ , D′ (kU )) ≃ RHom (kZ ⊗ kU , kR2n+1 ) ≃ RHom (k∆Rn ×{0} , kR2n+1 ) ≃ k∆Rn ×{0} [−n − 1] shows that Hom(kZ , kU [n + 1]) ≃ k. Hence, if k is a field, we have only one non trivial distinguished triangle like (II.1.4) up to isomorphism (see Lemma IV.4.4 below).

For a conic subset A of T˙ ∗ N we let D[A] (kN ) be the full subcategory of D(kN ) ˙ formed by the F with SS(F ) ⊂ A (see Notation I.2.2). Using the notation ◦a of (I.5.5) ′ ∗ ˙ we define A ⊂ T (N × I) by A′ = ΛΨ ◦a A. More explicitly we have (II.1.5)

A′ = {(Ψ(x, ξ, s), (s; −h(Ψ(x, ξ, s), s))); (x; ξ) ∈ A, s ∈ I}.

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CHAPTER II.1. HOMOGENEOUS CASE

We see that A′ is non-characteristic for the inclusion is , for any s ∈ I. Moreover i♯s (A′ ) = Ψs (A). We obtain an inverse image (II.1.6)

i−1 − D[Ψs (A)] (kN ). s : D[A′ ] (kN ×I ) →

We can deduce from Theorem II.1.1 that it is an equivalence (see [21, §3.4]), with an inverse induced by KΨ (however the categories involved in (II.1.6) and the functor ′ i−1 s only depend on the sets Ψs (A), A , not on Ψ itself): Corollary II.1.5. — (i) For any s ∈ I the composition F 7→ KΨ,s ◦ F induces an ∼ equivalence of categories D[A] (kN ) −− → D[Ψs (A)] (kN ), where KΨ,s = KΨ | . N ×{s}

(ii) For any s ∈ I the inverse image functor (II.1.6) is an equivalence of categories, −1 with an inverse given by F 7→ KΨ ◦KΨ,s ◦F . In particular, for any F, G ∈ D(kN ), we have isomorphisms ∼ Hom(F, G) ← −− Hom(KΨ ◦ F, KΨ ◦ G) (II.1.7) ∼ −−→ Hom(KΨ,s ◦ F, KΨ,s ◦ G) ∼ ∼ and RΓ(N ; G) ← −− RΓ(N × I; KΨ ◦ G) −− → RΓ(N ; KΨ,s ◦ G). Remark II.1.6. — The functor (II.1.6) and Corollary II.1.5 do not depend on the whole isotopy Ψ but only on the deformation Ψs (A), s ∈ I, of A. It is only required that this deformation is given by some Hamiltonian isotopy. Remark II.1.7. — Instead of composing with KΨ on the left in Corollary II.1.5 we can compose on the right. We can also add parameters, that is, consider F, G on a product N × P or P × N for some other manifold P . We only quote the following formulas for later use: let F, G ∈ D(kN ×P ), F ′ , G′ ∈ D(kP ×N ) be given; then the restriction at time 0 ∈ I gives isomorphisms ∼ (II.1.8) HomD(k − → HomD(k ) (KΨ ◦ F, KΨ ◦ G) − ) (F, G), N ×P ×I

(II.1.9)

ASTÉRISQUE 440

N ×P

∼ HomD(kP ×N ×I ) (F ′ ◦ KΨ , G′ ◦ KΨ ) −− → HomD(kP ×N ) (F ′ , G′ ).

CHAPTER II.2 LOCAL BEHAVIOR

We check here that the restriction of KΨ,s ◦ F to an open subset V of N only depends on F | for some bigger open subset U . U

Lemma II.2.1. — Let U , V be two open subsets of N and let s ∈ I be given. We ˙∗ ˙∗ assume that Ψ−1 t (T V ) ⊂ T U for all t ∈ [0, s]. Then, for any F ∈ D(kN ), the morphism FU → − F induces an isomorphism ∼ (KΨ ◦ FU )| −− → (KΨ ◦ F )| . V ×[0,s] V ×[0,s] ∼ In particular (KΨ,s ◦ FU )| −− → (KΨ,s ◦ F )| . V V Proof. — Let us set G = KΨ ◦ FN \U ∈ D(kN ×I ) and Gt = G| . We have a N ×{t} +1

distinguished triangle KΨ ◦ FU → − KΨ ◦ F → − G −−→ and the assertion of the lemma is equivalent to G| ≃ 0. Since G0 | ≃ 0 it is enough to see that G| is V ×[0,s] V V ×[0,s] locally constant. We first remark that Gt | is locally constant for each t ∈ [0, s]. Indeed V ˙ ˙ N \U )) ⊂ Ψt (T˙ ∗ N \ T˙ ∗ U ) and the hypothesis T˙ ∗ V ⊂ Ψt (T˙ ∗ U ) SS(Gt ) = Ψt (SS(F ˙ ˙∗ implies SS(G t ) ∩ T V = ∅. It follows that G is locally on V × [0, s] of the form G ≃ q −1 G′ for some G′ ∈ D(kI ), where q : N × I → − I is the projection. ∗ ∗ Hence SS(G| ) ⊂ TN N × T I. On the other hand G is non-characteristic for V ×[0,s] ˙ the inclusion of N × {t} in N × I, for any t ∈ I. Hence SS(G) = ∅ as required. Proposition II.2.2. — Let F, G ∈ D(kN ). The restriction morphism i−1 − RHom (F, G) 0 RHom (KΨ ◦ F, KΨ ◦ G) → is an isomorphism. Using the bound A′ of (II.1.5) with A = T˙ ∗ N we could argue as in the proof of b A′ is non-characteristic Corollary I.2.17. The main step is then to check that (A′ )a + for i0 ; here is another proof avoiding this computation. Proof. — We set F ′ = KΨ ◦ F , G′ = KΨ ◦ G. It is enough to check that the restriction morphism induces an isomorphism in each degree of the stalks at any point x ∈ N .

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CHAPTER II.2. LOCAL BEHAVIOR

We recall that H 0 (U ; RHom (F, G)) ≃ Hom(F | , G| ) ≃ Hom(FU , G). Hence U U (H k RHom (F, G))x ≃ lim Hom(F | , G[k]| ), U U −→ x∈U

(II.2.1)

(H k RHom (F ′ , G′ ))(x,0) ≃

lim −→

Hom(F ′ |

(x,0)∈U ×J

U ×J

, G′ [k]|

U ×J

),

where U runs over the open neighborhoods of x ∈ N and J over the neighborhoods of 0 ∈ I. For such U and J, with J contractible, we have ∼ Hom(KΨ ◦ FU | , G′ [k]| ) −− → Hom(FU , G[k]) N ×J N ×J by (II.1.7). ˙∗ ˙∗ If V ⊂ U satisfies Ψ−1 t (T V ) ⊂ T U for all t ∈ J, then KΨ ◦ FU |V ×J ≃ KΨ ◦ F |V ×J by Lemma II.2.1. We deduce a natural morphism Hom(FU , G[k]) → − Hom(F ′ |

V ×J

, G′ [k]|

V ×J

)

which gives a commutative diagram Hom(F ′ |

U ×J

, G′ [k]|

)

Hom(F | , G[k]| )

Hom(F ′ |

V ×J

, G′ [k]|

)

Hom(F | , G[k]| ).

U ×J

V ×J

U

V

U

V

It follows that the two limits in (II.2.1) are isomorphic, which proves the proposition.

Let A be a conic subset of T˙ ∗ N and let A′ ⊂ T ∗ (N × I) be as in (II.1.5). Let U be an open subset of N . We would like to find a neighborhood V of U × {0} in N × I such that the inverse image by the inclusion of U in V is an equivalence of categories ∼ D[A′ ∩T ∗ V ] (kV ) −− → D[A∩T ∗ U ] (kU ). We do not know if such a V always exists and give a weaker statement which will be sufficient for our purposes. Proposition II.2.3. — Let A ⊂ T˙ ∗ N and A′ ⊂ T ∗ (N × I) be as in (II.1.5). Let j : U → − N be the inclusion of an open subset. (i) Let F ∈ D[A∩T ∗ U ] (kU ). Then there exist a neighborhood V of U × {0} in N × I and G ∈ D[A′ ∩T ∗ V ] (kV ) such that F ≃ G| . U ×{0}

(ii) Let V be a neighborhood of U × {0} in N × I and G ∈ D[A′ ∩T ∗ V ] (kV ). Then there exists a smaller neighborhood V ′ of U × {0} such that G| ′ ≃ (KΨ ◦ F )| ′ , V V where F = Rj! (G| ). In particular, for G, G′ ∈ D[A′ ∩T ∗ V ] (kV ) the inverse U ×{0} image by the inclusion gives an isomorphism ∼ RHom (G, G′ )| −− → RHom (G| , G′ | ) U ×{0} U ×{0} U ×{0} and, if G| ≃ G′ | , then there exists a smaller neighborhood V ′′ U ×{0} U ×{0} of U × {0} such that G| ′′ ≃ G′ | ′′ . V V

ASTÉRISQUE 440

39

CHAPTER II.2. LOCAL BEHAVIOR

Proof. — (i) We set F ′ = Rj! F and G′ = KΨ ◦ F ′ . We have the rough bounds F −1 −1 ′ ′ ˙ ˙ SS(F ) ⊂ A ∪ πN (∂U ) and SS(G ) ⊂ A′ ∪ π˙ N ×R (Z) where Z = s∈I Zs × {s} and −1 Zs = πN (Ψs (π˙ N ×R (∂U ))). Then V = N × R \ Z is a neighborhood of U × {0} and G = G′ | has the required property. V

−1 (ii) We let k : V → − N × I be the inclusion and set G1 = Rk! G, H = KΨ ◦ | G1 . Then I −1 H ∈ D(kN ×I ) and Hs ≃ KΨ,s ◦ (G1 | ). Using the notation ΛΨ ◦a − (see (II.1.5)) N ×{s}

we have SS(H) ⊂ ΛΨ−1 ◦a | SS(G1 ). Then I ˙ ˙ SS(H) ⊂ ΛΨ−1 ◦a | SS(G 1) I −1 ∗ ⊂ ΛΨ−1 ◦a | (A′ ∪ πN ×I (∂V )) ⊂ (A × TI I) ∪ B, I

a

−1 |I πN ×I (∂V

where B = ΛΨ−1 ◦ ). We remark that W = (N ×I)\π(B) is a neighborhood of U × {0}. We let V1 ⊂ V ∩ W ∩ (U × I) be a smaller neighborhood such that the ∗ ˙ fibers of the projection p : V1 → − U are intervals. Then SS(H | ) ⊂ A × TI I and V1

Proposition I.2.9 implies that H |

V1

≃ p−1 (F ) for some F ∈ D(kU ).

−1 Hence KΨ ◦ | Rk! G is isomorphic to Rj! F ⊠kI on some neighborhood of U in N ×I. I It follows easily that Rk! G is isomorphic to KΨ ◦ | (Rj! F ⊠ kI ) ≃ KΨ ◦ Rj! F on some I smaller neighborhood V ′ . In the same way G′ can be written G′ | ′ ≃ (KΨ ◦ F ′ )| ′ , maybe up to shrinkV V ing V ′ . Now the last assertions of the proposition follow from Proposition II.2.2 and Corollary II.1.5.

SOCIÉTÉ MATHÉMATIQUE DE FRANCE 2023

CHAPTER II.3 NON HOMOGENEOUS CASE

Now we consider the case of a non homogeneous Hamiltonian isotopy on the cotangent bundle T ∗ M of some manifold M . We reduce this case to the homogeneous framework by a common trick of adding one variable. Let (x; ξ) and (t; τ ) be the coordinates on T ∗ M and T ∗ R. We define ρM : T ∗ M × T˙ ∗ R → − T ∗ M by (II.3.1)

ρM (x, t; ξ, τ ) = (x; ξ/τ ).

The fibers of ρM have dimension 2. They are stable by the translation in the t variable and are conic, that is, stable by the action of R>0 on the fibers of T˙ ∗ (M × R). Let Φ : T ∗M × I → − T ∗ M be a Hamiltonian isotopy of class C ∞ . We assume that Φ has compact support, that is, there exists a compact subset C ⊂ T ∗ M such that Φ(p, s) = p for all p ∈ T ∗ M \ C and all s ∈ I. Then Φ is the Hamiltonian flow of a function h : T ∗ M × I → − R such that h is locally constant outside C × I. In particular, if M does not have a connected component diffeomorphic to the circle S1 , then any Hamiltonian isotopy on T ∗ M with compact support can be defined by a Hamiltonian function h with compact support. Proposition II.3.1 (See Prop. A.6 of [21]). — Let Φ : T ∗ M ×I → − T ∗ M be a Hamiltonian ∗ isotopy with compact support and let h : T M ×I → − R be a function with Hamiltonian flow Φ. (i) Let h′ : T˙ ∗ (M × R) × I → − R be the Hamiltonian function given by h′ ((x, t; ξ, τ ), s) = τ h((x; ξ/τ ), s). Then the flow Φ′ of h′ is a homogeneous Hamiltonian isotopy Φ′ : T˙ ∗ (M × R) × I → − T˙ ∗ (M × R) whose restriction ∗ ∗ ˙ to T M × T R × I gives the commutative diagram T ∗ M × T˙ ∗ R × I (II.3.2)

Φ′

ρM

ρM ×idI

T ∗M × I

T ∗ M × T˙ ∗ R

Φ

T ∗M .

(ii) The isotopy Φ′ preserves the subset {τ > 0} of T˙ ∗ (M × R) and commutes with the vertical translations Tc of T˙ ∗ (M × R) given by Tc (x, t; ξ, τ ) = (x, t + c; ξ, τ ), for all c ∈ R.

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CHAPTER II.3. NON HOMOGENEOUS CASE

Defining q : (M × R)2 × I → − M 2 × R × I, (x, t, x′ , t′ , s) 7→ (x, x′ , t − t′ , s), the graph ΛΦ′ defined in (II.1.2) satisfies ΛΦ′ = qd qπ−1 (Λ′h ), where ′ ∗ 2 ˙ Λh ⊂ T (M × R × I) is given by Λ′h = qπ qd−1 (ΛΦ′ ). Proof. — (i) is Prop. A.6 of [21]. (ii) Since h′ does not depend on t, the isotopy Φ′ commutes with the Hamiltonian flow of τ which is Tc . The flow Φ′ also preserves the variable τ , that is, ΛΦ′ is contained in Σ:={τ +τ ′ = 0}. Then Σ = im qd and the quotient map to the symplectic reduction of Σ is qπ . Hence we can write ΛΦ′ = qd qπ−1 (Λ′h ), where Λ′h = qπ qd−1 (ΛΦ′ ). Corollary II.3.2. — Let Φ : T ∗ M × I → − T ∗ M be a Hamiltonian isotopy with compact ∗ support and let h : T M ×I → − R be a function with Hamiltonian flow Φ. Then there ex˙ ists a unique K ∈ Dlb (kM 2 ×R×I ) such that SS(K) = Λ′h and K | 2 ≃ k∆M ×{0} , M ×R×{0}

where Λ′h is defined in Proposition II.3.1. Proof. — We use the notations of Proposition II.3.1. By Proposition I.2.9 the inverse ′ ˙ image functor q −1 gives an equivalence between {K ′ ∈ Dlb (kM 2 ×R×I ); SS(K ) = Λ′h } ˙ and {K ∈ Dlb (k(M ×R)2 ×I ); SS(K) = ΛΦ′ }. Then the existence and uniqueness of K follows from Theorem II.1.1.

ASTÉRISQUE 440

PART III CUT-OFF LEMMAS

In this part we recall several results of [28] which are called “(dual) (refined) cut-off lemmas”. We also give another version which removes some convexity hypotheses. We apply these results to decompose a sheaf with respect to a partition of its microsupport (see Proposition III.3.2 below). The starting point is the following problem: for a given sheaf F ∈ D(kV ), with V = Rn , and an open cone C ⊂ T0∗ V = V ∗ , find a sheaf G such that SS(G) = SS(F ) \ (SS(F ) ∩ C) and F ≃ G in the quotient category of D(kV ) by {H; (SS(H) ∩ T0∗ V ) ⊂ C}. As we have seen in Corollary I.3.9 the involutivity Theorem I.3.6 says this problem has no solution in general. However, for a given neighborhood Ω of SS(F ) ∩ ∂C in T ∗ V , it is possible to find G with SS(G) = (SS(F ) \ (SS(F ) ∩ C)) ∪ Ω near T0∗ V (see Proposition III.3.1 below). We first discuss a weaker version of this problem where we don’t ask that SS(G) = SS(F ) \ (SS(F ) ∩ C) but we only ask for a “cut-off” functor F 7→ P (F ) so that SS(P (F )) is contained in V × (V ∗ \ C). There are several ways to define such a P and we give four variations on the cut-off functors of [28] in (III.1.1). If we want P to be a projector (see Remark III.1.9) these are in fact the only possible choices (see §III.5 for a quick discussion). One version of the cut-off lemma says that the category of sheaves on V with microsupport in some convex cone γ is equivalent to the category of sheaves on V endowed with another topology. We use it to prove that SS(H i (F )) is contained in the convex hull of SS(F ) (see Corollary III.4.3 below – this will be used to construct a graph selector in Part V). In the last paragraph we consider a relative version of the cut-off functor and recall some its properties already considered by Tamarkin which we will use to recover non-squeezing results in §VI.2.

CHAPTER III.1 GLOBAL CUT-OFF

Let V be a vector space of dimension n and let γ ⊂ V be a closed convex cone (with vertex at 0). We denote by γ a = −γ its opposite cone and by γ ◦ ⊂ V ∗ its polar cone (see (I.2.2)). We also define γ e = {(x, y) ∈ V 2 ; x − y ∈ γ}. Let qi : V 2 → − V , i = 1, 2, be the projection to the i-th factor and let ∆V be the diagonal of V 2 . The following functors are introduced in [28]:

(III.1.1)

Pγ : D(kV ) → − D(kV ),

F 7→ Rq2 ∗ (kγe ⊗ q1−1 F ),

Qγ : D(kV ) → − D(kV ),

F 7→ Rq2 ! (RHom (kγea , q1! F )),

Pγ′ : D(kV ) → − D(kV ),

F 7→ Rq2 ! (RHom (kγea \∆V [1], q1! F )),

Q′γ : D(kV ) → − D(kV ),

F 7→ Rq2 ∗ (kγe\∆V [1] ⊗ q1−1 F ).

We will mainly use the first three functors and introduce Q′γ because these functors come in pairs (adjoint pairs or pairs of projectors—see §III.5). We will see the effect of these functors on the microsupport in (III.1.8) and Propositions III.1.7, III.1.8 g = ∆V and P{0} (F ) ≃ Q{0} (F ) ≃ F . and III.1.10 below. For γ = {0}, we have {0} +1

Using the distinguished triangle kγe → − k∆V → − kγe\∆V [1] −−→ (and the same with γ a ) we obtain morphisms of functors u γ : Pγ → − id, vγ : id → − Qγ , (III.1.2) ′ ′ ′ u γ : Pγ → − id, vγ : id → − Q′γ and, for any F ∈ D(kV ), the distinguished triangles (III.1.3)

uγ (F )

vγ′ (F )

+1

u′γ (F )

vγ (F )

+1

Pγ (F ) −−−−−→ F −−−−→ Q′γ (F ) −−→, Pγ′ (F ) −−−−−→ F −−−−→ Qγ (F ) −−→ .

The functors Pγ , Qγ , Pγ′ , Q′γ have similarities with the composition operation (I.5.1). Let us recall the composition: for three manifolds X, Y, Z and K ∈ D(kX×Y ), L ∈ D(kY ×Z ) −1 −1 K ◦ L = Rq13 ! (q12 K ⊗ q23 L),

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CHAPTER III.1. GLOBAL CUT-OFF

where qij is the projection from X × Y × Z to the i × j-factor. Taking X = {pt} and Y = Z = V , we have, with the notations of (III.1.1), F ◦ kγe = Rq2 ! (kγe ⊗ q1−1 F ), which is almost Pγ (F ) up to the change from Rq2 ! to Rq2 ∗ . The following lemma gives hypothesis on F so that both functors are the same. Lemma III.1.1. — Let F ∈ D(kV ). We assume that supp(F ) is compact or that there exists a linear function ξ : V → − R such that γ \ {0} ⊂ ξ −1 (]−∞, 0[) and −1 supp(F ) ⊂ ξ ([a, ∞[), for some a ∈ R. Then Pγ (F ) ≃ F ◦ kγe and supp(Pγ (F )) ⊂ ξ −1 ([a, ∞[). Proof. — As was already remarked, the difference between Pγ (F ) and F ◦ kγe is the change from Rq2 ∗ to Rq2 ! in (III.1.1). Hence it is enough to prove that the projection q2 is proper on S = supp(kγe ⊗ q1−1 F ) to deduce the isomorphism. This is clear when supp(F ) is compact. We assume the other hypothesis. Let ∥ · ∥ be some Euclidean norm on V . Since γ is a closed cone, we actually have γ ⊂ {v ∈ V ; ξ(v) ≤ −c∥v∥} for some c > 0. Hence S ⊂ {(v1 , v2 ); ξ(v1 − v2 ) ≤ −c∥v1 − v2 ∥, ξ(v1 ) ≥ a}. For a given v2 ∈ V we see that S ∩ q2−1 ({v2 }) ⊂ {v1 ; ∥v1 − v2 ∥ ≤ c−1 (ξ(v2 ) − a)} is compact. Hence Pγ (F ) ≃ F ◦ kγe . Moreover, if (Pγ (F ))v2 ̸= 0, we must have S ∩ q2−1 ({v2 }) ̸= ∅, hence ξ(v2 ) − a ≥ 0. It follows that supp(Pγ (F )) ⊂ ξ −1 ([a, ∞[). ea = d−1 (γ a ) with We can also reformulate Qγ and Pγ′ with the composition. Since γ d(x, y) = x − y, we have kγea ≃ d−1 (kγ a ) and Theorem I.2.8 gives SS(kγea ) ⊂ {(x, y; ξ, −ξ)}. We also have SS(q1! F )) ⊂ {(x, y; ξ, 0)}. Hence the microsupports of kγea and q1! F do not intersect outside the zero section and Theorem I.2.13 implies L ∼ D′ (kγea ) ⊗ q1! F −− → RHom (kγea , q1! F ).

(III.1.4)

Choosing an orientation for V we have q1! F ≃ q1−1 F [n] and we deduce the first isomorphism below; the second one is proved in the same way, using γ ea \∆V = d−1 (γ a \{0}), Qγ (F ) ≃ F ◦ (D′ (kγea )[n]),

(III.1.5)

Pγ′ (F ) ≃ F ◦ (D′ (kγea \∆V )[n − 1]).

We deduce the following adjunction properties: Lemma III.1.2. — For any F, G ∈ D(kV ) we have Hom(Qγ (F ), G) ≃ Hom(F, Pγ (G)) and Hom(Pγ′ (F ), G) ≃ Hom(F, Q′γ (G)). Proof. — We only prove the first isomorphism, the second one being similar. Using (III.1.5) we see Qγ as a composition of left adjoint functors and we have L

Hom(Qγ (F ), G) ≃ Hom(Rq2 ! ((D′ (kγea )[n]) ⊗ q1−1 (F ), G) ≃ Hom(F, Rq1 ∗ (RHom ((D′ (kγea )[n], q2! G))).

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By a microsupport argument similar to the proof of (III.1.4) we obtain L ∼ D′ (D′ (kγea )[n]) ⊗ q ! G −− → RHom ((D′ (kγea )[n], q ! G). Since kγea is cohomologically 2

constructible, D′ (D′ (kγea )) ≃ kγea and we obtain

2

Hom(Qγ (F ), G) ≃ Hom(F, Rq1 ∗ (kγea ⊗ q2−1 G)). We conclude with the remark that the automorphism (x, y) 7→ (y, x) of V 2 interchanges q1 , q2 and γ ea , γ e. Examples III.1.3. — (i) The easiest example is the image of the skyscraper sheaf k{x} for a given x ∈ V . Using Lemma III.1.1 and the Formula (I.5.2) for the stalks of a composition, we find Pγ (k{x} ) ≃ kx−γ . If Int(γ) ̸= ∅, we have D′ (kγea ) ≃ kInt(eγ a ) by (I.1.8). Hence (III.1.5) gives Qγ (F ) ≃ F ◦kInt(eγ a ) [n]. By (I.5.2) again Qγ (k{x} ) ≃ kInt(x+γ) [n]. We illustrate these results in Fig. III.1.1 (for a sheaf of the type LZ , Z locally closed, we draw the set Z and label it with L).

k γ

Qγ (k{x} )

x•

Pγ (k{x} ) k[n]

Figure III.1.1.

The morphism vγ (k{x} ) is given by the inverse image of 1 in the sequence of isomorphisms: Hom(k{x} , kInt(x+γ) [n]) ≃ Hom(k{x} , D(kx+γ )) n ≃ Hom(k{x} ⊗ kx+γ , kV [n]) ≃ H{x} (kV ) ≃ k.

(ii) Here are other easy examples in dimension 1. We set γ = ]−∞, 0]. F

Pγ (F )

Qγ (F )

Pγ′ (F )

Q′γ (F )

k[0,1] k[0,1[

k[0,+∞[ k[0,1[

(k]−∞,0[ )[1] k[0,1[

k]−∞,1] 0

(k]1,+∞[ )[1] 0

k]0,1]

0

0

k]0,1]

k]0,1]

k]0,1[

(k[1,+∞[ )[−1]

k]−∞,1[

(k]−∞,0] )[−1]

k]0,+∞[

It is easy to check here the properties stated in Proposition III.1.10: if (t; τ ) are the coordinates on T ∗ R we have ˙ γ (F )) = SS(F ˙ ˙ γ′ (F )) = SS(F ˙ SS(P ) ∩ {τ > 0}, SS(P ) ∩ {τ < 0}

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and F is decomposed “up to a constant sheaf” as Pγ (F ) ⊕ Pγ′ (F ) (triangle (III.1.14)); for example we have the triangle +1

kR → − k[0,+∞[ ⊕ k]−∞,1] → − k[0,1] −−→ . We use the identifications Tx∗ V = V ∗ for any x ∈ V and T ∗ V = V × V ∗ . Lemma III.1.4. — Let F ∈ D(kV ). We assume that F has a compact support. Then, for any x ∈ V , (III.1.6)

SS(Pγ (F )) ∩ Tx∗ V ⊂ p2 (SS(F ) ∩ SS(kx+γ )a ),

(III.1.7)

SS(Qγ (F )) ∩ Tx∗ V ⊂ p2 (SS(F ) ∩ SS(kx+γ a )),

where p2 : T ∗ V → − Tx∗ V is the projection. Proof. — We only prove the first inclusion, the proof of the second one being similar. We set G = kγe ⊗ q1−1 F , hence Pγ (F ) = Rq2 ∗ (G). Using γ e = d−1 (γ) with d(x, y) = x − y, we see by Theorems I.2.8 and I.2.13: SS(kγe ) = {(x, y; ξ, −ξ); (x − y; ξ) ∈ SS(kγ )}, SS(G) ⊂ SS(kγe ) + (SS(F ) × TV∗ V ). Since supp(F ) is compact, Proposition I.2.4 implies that SS(Pγ (F )) is bounded − T ∗ V is the second projection. by p′2 (SS(G) ∩ (TV∗ V × T ∗ V )) where p′2 : T ∗ V 2 → ∗ ∗ A point (x1 , x2 ; 0, η) of TV V × T V belongs to SS(G) if and only if there exist (x1 , x2 ; ξ, −ξ) ∈ SS(kγe ) and (x1 ; ξ1 ) ∈ SS(F ) such that (0, η) = (ξ, −ξ) + (ξ1 , 0). In other words (x1 − x2 ; −η) ∈ SS(kγ ) and (x1 ; η) ∈ SS(F ). Now, if x2 = x is fixed, we obtain (x1 ; η) ∈ SS(kx+γ )a ∩ SS(F ) and the result follows. By Example I.2.3 SS(kx+γ a ) ⊂ V × γ ◦a and Lemma III.1.4 gives SS(Pγ (F )) ∪ SS(Qγ (F )) ⊂ V × γ ◦a if F has compact support; but this result holds without the compactness hypothesis: Lemma III.1.5. — For any F ∈ D(kV ) we have (III.1.8)

SS(Pγ (F )) ∪ SS(Qγ (F )) ⊂ V × γ ◦a .

Proof. — We prove the result for Pγ (F ), the case of Qγ (F ) being similar. We set G = kγe ⊗ q1−1 F as in the proof of Lemma III.1.4. We have already seen SS(G) ⊂ SS(kγe ) + (SS(F ) × TV∗ V ) (this didn’t require the compactness of supp(F )) and we deduce the rough bound SS(G) ⊂ T ∗ V × γ ◦a . If q2 were proper on supp(G) we could conclude with Proposition I.2.4. In our case we can use a compactification ∼ of V . We choose a diffeomorphism R −− → ]0, 1[ and deduce V = Rn ≃ ]0, 1[n and 2 ∼ n then φ : V −−→ ]0, 1[ ×V . We let j : ]0, 1[n ×V ,→ Rn × V be the inclusion. Since φ is proper, we can use Proposition I.2.4 and we obtain SS(Rφ∗ (G)) ⊂ T ∗ ]0, 1[n ×γ ◦a . Then Theorem I.2.15 gives SS(Rj∗ Rφ∗ (G)) ⊂ T ∗ V × γ ◦a . Since q2 is proper on [0, 1]n × V , we can apply Proposition I.2.4 again and, using Rq2 ∗ Rj∗ Rφ∗ (G) ≃ Rq2 ∗ (G), we obtain the result.

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CHAPTER III.1. GLOBAL CUT-OFF

49

The first cut-off results say roughly that the bound (III.1.8) characterizes sheaves of the type Pγ (F ) or Qγ (F ) (see Propositions III.1.7 and III.1.8). For a given subset Ω of T ∗ V , a morphism a : F → − G in D(kV ) is said to be an isomorphism on Ω if SS(C(a)) ∩ Ω = ∅, where C(a) is given by the distinguished a +1 triangle F − →G→ − C(a) −−→. This implies that SS(F ) ∩ Ω = SS(G) ∩ Ω. We will need the following composition result: Lemma III.1.6. — If a : F → − G, b : G → − H are isomorphisms on Ω, then so is b ◦ a. Proof. — The octahedron axiom implies that we have a distinguished triangle C(a) → − C(b ◦ a) → − C(b) → − C(a)[1], where C(a), C(b), C(b ◦ a) are the cones of a, b, b ◦ a. Now the result follows from the triangular inequality for the microsupport. The next two propositions are taken from [28] where they are stated using the γ-topology (see §III.4). The link between the γ-topology and the functors Pγ , Qγ is also explained in Proposition 3.5.4 of [28]. Proposition III.1.7 (See [28] Prop. 5.2.3). — For F ∈ D(kV ) we let uγ (F ) : Pγ (F ) → − F be the morphism in (III.1.2). Then (i) uγ (F ) is an isomorphism on V × Int(γ ◦a ), (ii) uγ (F ) is an isomorphism if and only if SS(F ) ⊂ V × γ ◦a . Proposition III.1.8 (See [28] Lem. 6.1.5). — We assume that γ is proper (that is, γ contains no line) and Int(γ) ̸= ∅. Let F ∈ D(kV ). We assume that F has compact support. Then the morphism vγ (F ) : F → − Qγ (F ) in (III.1.2) is an isomorphism on V × Int(γ ◦a ). Remark III.1.9. — (1) The bound (III.1.8) and Proposition III.1.7-(ii) show that uγ (Pγ (F )) : Pγ (Pγ (F )) → − Pγ (F ) is an isomorphism. In other words the ∼ morphism of functors uγ : Pγ → − idD(kV ) induces an isomorphism Pγ ◦ Pγ −− → Pγ . The functor Pγ is then called a projector (see for example [29, §4.1]). More precisely it is the projector to the subcategory DV ×γ ◦a (kV ) of D(kV ) formed by the F such that SS(F ) ⊂ V × γ ◦a (Notation I.2.2). By the dual statements of [29, Prop. 4.1.3, 4.1.4] the functor R : D(kV ) → − DV ×γ ◦a (kV ) induced by Pγ is both right adjoint and left inverse to the inclusion ι of DV ×γ ◦a (kV ) and we have Pγ ≃ ι ◦ R. We will see this again in a slightly different form in §III.4 where DV ×γ ◦a (kV ) is identified with the category of sheaves on Vγ , which is the space V with the “γ-topology” (see Proposition III.4.1—then ι, R correspond to ϕ−1 γ , Rϕγ ∗ ). (2) Since Qγ is left adjoint to Pγ , it is also a projector. More precisely, Qγ ◦Qγ is adjoint to Pγ ◦ Pγ ≃ Pγ and by uniqueness of adjoints we have Qγ ◦ Qγ ≃ Qγ . Instead of a morphism Pγ → − id we have here a morphism id → − Qγ . By [29, Prop. 4.1.3, 4.1.4] Qγ is then a projector to some full subcategory, say D, of D(kV ). Using the decomposition ∼ Pγ ≃ ι◦R in (1) and R◦ι ≃ id we can prove Hom(Pγ (F ), Pγ (G)) −− → Hom(Pγ (F ), G), ∼ for any F, G ∈ D(kV ). We deduce Hom(Qγ (Pγ (F )), G) −−→ Hom(Pγ (F ), G), proving that Qγ ◦ Pγ ≃ Pγ . This implies that DV ×γ ◦a (kV ) is stable by Qγ . Hence we have

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DV ×γ ◦a (kV ) ⊂ D. In the same way we have Pγ ◦ Qγ ≃ Qγ and we obtain finally DV ×γ ◦a (kV ) = D. Hence Qγ is also a projector to DV ×γ ◦a (kV ). The difference with Pγ is that we could write Qγ ≃ ι ◦ L where L is left adjoint to the embedding ι. (3) Since, by (2), Qγ is a projector to DV ×γ ◦a (kV ), we have as in Proposition III.1.7(ii): vγ (F ) is an isomorphism if and only if SS(F ) ⊂ V × γ ◦a , for any F ∈ D(kV ) and without the additional hypothesis that Int(γ) ̸= ∅. However we will only use the statement of Proposition III.1.8. A consequence of Proposition III.1.7-(i) is the bound (III.1.9)

SS(Pγ (F )) ∩ (V × Int(γ ◦a )) = SS(F ) ∩ (V × Int(γ ◦a )).

However SS(Pγ (F )) ∩ (V × ∂γ ◦a ) could be bigger than SS(F ) ∩ (V × ∂γ ◦a ). If we ˙ assume that SS(F ) does not meet V × ∂γ ◦a and supp(F ) is compact, then SS(Pγ (F )) does not meet V × ∂γ ◦a by (III.1.6). Hence (III.1.8) and (III.1.9) actually imply SS(Pγ (F )) = SS(F ) ∩ (V × Int(γ ◦a )). In Proposition III.1.10 below we see the similar ˙ ˙ γ′ (F )) = SS(F ) ∩ (V × (V \ γ ◦a )) and that F is split as Pγ (F ) ⊕ Pγ′ (F ) up bound SS(P to a constant sheaf. Actually, in Prop. III.1.10 we give a weaker hypothesis than SS(F ) ∩ (V × ∂γ ◦a ) = ∅. We want to split F only on some open subset W of V and we can replace V × ∂γ ◦a S by some smaller on W , namely x∈W Sxγ where Sxγ is defined below S set depending (remark that x∈V Sxγ = V × ∂γ ◦a ). This more precise result is used in the proof of Proposition III.2.3. The motivation to introduce Sxγ is the following. We first want to ensure ˙ γ (F )) ∩ T ∗ V coincides with SS(F ˙ that SS(P ) ∩ (V × Int(γ ◦a )) ∩ Tx∗ V . By (III.1.8) and x ˙ γ (F )) ∩ Tx∗ V ∩ ({x} × (∂γ ◦a )) = ∅, where Proposition III.1.7 it is enough that SS(P ◦a ◦a ◦a ˙ ∂γ = γ \ Int γ . Using (III.1.6) this holds if we assume that SS(F ) does not meet ◦a a γ ˙ (V × ∂γ ) ∩ SS(kx+γ ) , which is half of Sx . The other half is introduced to bound ∗ ˙ SS(Q γ (F )) ∩ Tx V in the same way, using (III.1.7). Let γ ⊂ V be a closed convex proper cone. For x ∈ V we define Sxγ ⊂ T ∗ V by ˙ x+γ )a ∪ SS(k ˙ x+γ a )) \ ({x} × Int(γ ◦a )) Sxγ = (SS(k (III.1.10) ˙ x+γ )a ∪ SS(k ˙ x+γ a )) ∩ (V × ∂γ ◦a ), = (SS(k where the second equality follows from (I.2.4). ˙ x+γ )a , SS(k ˙ x+γ a ) and Sxγ in dimension 2 In Fig. III.1.2 we have pictured SS(k ∗ 2 2 ˙ ˙ x+γ a ) contain (we identify T R with T R ; we note that SS(kx+γ )a and SS(k ◦a γ {x} × Int(γ )). We can give an easy description of Sx when γ has a smooth boundary away from 0, as in the following example (this is in fact the only case we need). For n > 1 we write the coordinates in Rn as x = (x′ , xn ). We define γ = {(x′ , xn ) ∈ V ; xn ≤ −∥x′ ∥} and C = {(x′ , xn ) ∈ V ; xn = ±∥x′ ∥}. Then (x0 + γ) ∪ (x0 + γ a ) has boundary Cx0 = x0 + C, which is a smooth hypersurface except at x0 . We set Cx′ 0 = Cx0 \ {x0 } and define Λ = TC∗x′ Rn ∩ T˙ ∗ Rn . Then 0

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x

γ

x

˙ x+γ )a SS(k

x

Sxγ

˙ x+γ a ) SS(k

Figure III.1.2.

Λ is a smooth closed conic Lagrangian submanifold of T˙ ∗ Rn with two connected components and Sxγ0 is the component with ξn > 0. We deduce from (III.1.6), (III.1.7) and Example I.2.3-(iv) that, if F ∈ D(kV ) has compact support, then, for G = Pγ (F ) or G = Qγ (F ), ˙ ˙ (III.1.11) SS(G) ∩ (Tx∗ V \ Int(γ ◦a )) ⊂ p2 (SS(F ) ∩ Sxγ ). Since uγ (F ) and vγ (F ) are isomorphisms on V × Int(γ ◦a ), we also have (III.1.12)

SS(G) ∩ (V × Int(γ ◦a )) = SS(F ) ∩ (V × Int(γ ◦a )).

Proposition III.1.10. — Let F ∈ D(kV ) be such that supp(F ) is compact and let W ⊂ V be an open subset such that ˙ (III.1.13) Sxγ ∩ SS(F )=∅ for any x ∈ W . Then vγ (F ) ◦ uγ (F )| : Pγ (F )| → − Qγ (F )| is an isomorphism on T˙ ∗ W and we W W W have a distinguished triangle in D(kW ) (III.1.14)

Pγ (F )|

W

⊕ Pγ′ (F )|

(uγ (F ),u′γ (F ))

W

−−−−−−−−−−→ F |

W

+1

→ − L −−→,

where L ∈ D(kW ) is locally constant and ˙ γ (F )| ) = SS(F ˙ SS(P ) ∩ (W × Int(γ ◦a )), W

′ ◦a ˙ ˙ SS(P γ (F )| ) = SS(F ) ∩ (W × (V \ γ )). W

Proof. — (i) Let G be Pγ (F )| or Qγ (F )| . By (III.1.11) and (III.1.13) we have W W ˙ SS(G) ⊂ W × Int(γ ◦a ). By (III.1.12) we deduce ˙ ˙ SS(G) = SS(F ) ∩ (W × Int(γ ◦a )). (ii) We define L ∈ D(kW ) by the distinguished triangle (III.1.15)

Pγ (F )|

W

vγ (F )◦uγ (F )

−−−−−−−−→ Qγ (F )|

a

W

+1

− → L −−→ .

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˙ By (i) we have SS(L) ⊂ W × Int(γ ◦a ). On the other hand vγ (F ) ◦ uγ (F ) is an isomorphism on W × Int(γ ◦a ) by Propositions III.1.7 and III.1.8 and Lemma III.1.6. Hence ˙ SS(L) ∩ (W × Int(γ ◦a )) = ∅ and we find SS(L) = ∅. This proves the first assertion. (iii) The distinguished triangle of the proposition follows from the triangles (III.1.3), (III.1.15) and Lemma III.1.11 applied with A = Pγ (F )| , B = F | , C = Pγ′ (F )| , W W W C ′ = Qγ (F )| , D = L. W Since vγ (F ) is an isomorphism on V ×Int(γ ◦a ), the second triangle in (III.1.3) gives ˙ γ′ (F )| ) ∩ (W × Int(γ ◦a )) = ∅. It follows that SS(P ˙ γ′ (F )| ) and SS(P ˙ γ (F )| ) SS(P W W W are disjoint. Hence (III.1.14) implies ˙ ˙ γ′ (F )| ) ⊔ SS(P ˙ γ (F )| ). SS(F | ) = SS(P W

W

W

˙ Since (III.1.13) implies in particular SS(F ) ∩ (W × ∂(γ ◦a )) = ∅, the triangular inequality for the microsupport implies the last equalities of the proposition. Lemma III.1.11. — We assume to be given a morphism a : A → − B in a triangulated category and two distinguished triangles c

c′

+1

C→ − B− → C ′ −−→,

c′ ◦a

+1

A −−→ C ′ → − D −−→ .

Then there exists a distinguished triangle (a,c)

+1

A ⊕ C −−−→ B → − D −−→ . (a,c)

+1

Proof. — We define E by the distinguished triangle A⊕C −−−→ B → − E −−→. We also u v +1 have the canonical triangle C − → A⊕C − → A −−→ where u = (0, idC ) and v = (idA , 0). Since (a, c) ◦ u = c : C → − B, the octahedron axiom applied to this last two triangles and the first one in the statement gives a distinguished triangle w

+1

A− → C′ → − E −−→ such that w ◦ v = c′ ◦ (a, c). This implies w = c′ ◦ a. Hence E ≃ D and the lemma follows.

ASTÉRISQUE 440

CHAPTER III.2 LOCAL CUT-OFF—SPECIAL CASE

In this section we want to give local versions of Propositions III.1.7 and III.1.10. For example the conclusion Pγ (F ) ≃ F requires a bound for SS(F ) on V ; in the same way the Hypothesis (III.1.13) requires a knowledge of SS(F ) on some unbounded set. Here we only assume that we know SS(F ) over some open subset U of V and we will give results similar to the conclusions of the mentioned propositions over some smaller open subset W ⊂ U . In fact we will apply Propositions III.1.7 and III.1.10 to a sheaf FZ for some locally closed subset Z ⊂ U such that Z ⊂ U (FZ is considered as a sheaf on V by extension by 0). Then SS(FZ ) coincides with SS(F ) over Int(Z) and is empty over V \ Z. b SS(kZ ). We However, over Z \ Int(Z), we only have the bound SS(FZ ) ⊂ SS(F ) + check in this section that we can choose Z so that FZ satisfies the hypotheses of Propositions III.1.7 and III.1.10. This is done in [28, Prop. 6.1.4] for a convex cone γ. In Proposition III.3.2 we will generalize this result to a more general cone, using Theorem II.1.1 to reduce the situation to the case of a convex cone. Since we can as well reduce to a very special cone, we only explain a particular case of [28, Prop. 6.1.4] (the proof is not different from that of [28] but the exposition is easier). We write V = V ′ ×R, where V ′ = Rn−1 . We take coordinates x = (x′ , xn ) on V and we endow V ′ and V with the natural Euclidean structure. For c > 0 we let γc ⊂ V be the cone (III.2.1)

γc = {(x′ , xn ) ∈ V ; xn ≤ −c ∥x′ ∥}.

Precised cut-off. — We begin with an analog of Propositions III.1.7, III.1.8: we give a functor R similar to Pγ or Qγ which cuts the microsupport along {ξn ≤ 0}, but locally defined (on D(kU ) instead of D(kV ), with U open in V ) and with a bound for SS(R(F )) up to the boundary {ξn = 0}. However R(F ) is only defined on a smaller subset W ⊂ U and we don’t have the ideal bound SS(R(F )) ⊂ SS(F ) ∩ {ξn ≤ 0}, but only SS(R(F )) ⊂ (SS(F ) ∩ {ξn ≤ 0}) ∪ B ′ , where B ′ is an arbitrary neighborhood of SS(F ) ∩ {ξn = 0} chosen in advance. Proposition III.2.1. — Let B ⊂ V ∗ be a closed conic subset and let B ′ ⊂ V ∗ be a conic neighborhood of (B \ {0}) ∩ {ξn = 0}. Let U ⊂ V be a neighborhood of 0. Then there

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exist an open neighborhood W of 0 in U and a functor R : D(kU ) → − D(kW ), together with a morphism of functors (·)| → − R, such that, for any F ∈ D(kU ), W

(a) if SS(F ) ⊂ U × B, then SS(R(F )) ⊂ W × ((B \ {ξn ≥ 0}) ∪ B ′ ), ∼ (b) if SS(F ) ⊂ U × {ξn ≤ 0}, then F | −− → R(F ). W Since the proof is a bit technical we explain the idea. We first cut F by some open set Y with Y ⊂ U (so that FY can be consider as a sheaf on V which is 0 outside Y ) and apply Q′γ to FY . Using (III.1.6) we have SS(Q′γ (FY ))∩Tx∗ V ⊂ p2 (SS(FY )∩SS(kx+γ )a ) b and we know that SS(FY ) ⊂ SS(F ) +SS(k Y ). Of course this last subset could be much bigger than SS(F ), but it turns out that, if SS(kY ) is close enough to SS(kx+γ ) (at the points x′ where both SS(kY ) ∩ Tx∗′ V and SS(kx+γ ) ∩ Tx∗′ V are non trivial), then SS(Q′γ (FY )) is not too much bigger than SS(F ). This relies on Lemma III.2.2 because, at the interesting points x′ , SS(kY ) ∩ Tx∗′ V and SS(kx+γ ) ∩ Tx∗′ V are half-lines. The examples we choose for Y and γ with the required property are for Y a vertical cylinder (III.2.2)

Yrh = {(x′ , xn ) ∈ V ; ∥x′ ∥ < r, |xn | < h}

and for γ a cone γc with big slope c such that ∂γc meets ∂Yrh along its vertical part, which means that h > rc (as in Fig. III.2.1).

Figure III.2.1.

Lemma III.2.2. — Let B, B ′ be as in Proposition III.2.1. Let S be the unit sphere of V ∗ . Then there exists ε > 0 such that, for any (p1 , p2 ) ∈ (S ∩ {ξn = 0}) × S with ∥p1 − p2 ∥ < ε, we have (B + R≥0 · (−p1 )) ∩ (R>0 · p2 ) ⊂ B ′ . (In other words, if the intersection is non empty, then p2 ∈ B ′ .)

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55

This lemma is an elaboration on the following remark: for p ∈ V ∗ \ B we have p ̸∈ (B + R≥0 (−p)) and hence (B + R≥0 · (−p)) ∩ (R>0 · p) = ∅. It follows that, for any p ∈ (S ∩ {ξn = 0}), we have (B + R≥0 · (−p)) ∩ (R≥0 · p) ⊂ (B ∩ {ξn = 0}). In the lemma (p1 , p2 ) is near the diagonal and B ∩ {ξn = 0} is replaced by its neighborhood B ′ . Proof. — For a subset C of the sphere S, we set CS,ε = {v ∈ S; there exists v ′ ∈ C, ∥v − v ′ ∥ < ε}. For a conic subset D of V ∗ we set Dε = R>0 · (D ∩ S)S,ε . We remark that Dε = {av1 + bv2 ; v1 ∈ D ∩ S, v2 ∈ S, ∥v1 − v2 ∥ < ε, a, b ≥ 0} \ {0}. A point p′ ∈ (B + R>0 · (−p1 )) ∩ (R>0 · p2 ) is written p′ = p − ap1 = bp2 , with 1 p ∈ B, a, b ≥ 0. Then p = ap1 + bp2 belongs to B ∩ {ξn = 0}ε . The point q = ∥p∥ p belongs to the arc p1 p2 on S, hence ∥q − p2 ∥ < ε and then p2 ∈ (B ∩ {ξn = 0}ε )ε . Now, for ε small enough this last set is contained in B ′ . Proof of Proposition III.2.1. — (i) We will use the functors Pγ , Q′γ for a cone γ = γc +1

to be chosen in (iii) and the distinguished triangle (III.1.2) Pγ (F ) → − F → − Q′γ (F ) −−→. We have the bound, with the same proof as (III.1.6), SS(Q′γ (F )) ∩ Tx∗ V ⊂ p2 (SS(F ) ∩ Sx′γ ), − Tx∗ V ≃ V ∗ is the projection. where Sx′γ = (SS(k(x+γ)\{x} ))a and p2 : T ∗ V → (ii) We prove that Sx′γ ⊂ V ×(V ∗ \Int(γ ◦a )). More precisely Sx′γ = (SS(kx+γ ))a outside Tx∗ V (which is obvious from the definition of Sx′γ ) and Sx′γ ∩ Tx∗ V = V ∗ \ Int(γ ◦a ). Indeed this can be computed directly from the definition of the microsupport, or we can use [28, Lem. 3.7.10, Thm. 5.5.5] as follows. The Fourier-Sato transform of a conic sheaf G ∈ D(kV ) is G∧ ∈ D(kV ∗ ) and, identifying T ∗ V = V × V ∗ = T ∗ V ∗ , we have SS(G∧ ) = SS(G). In particular SS(G) ∩ T0∗ V = supp(G∧ ). We also have (kλ )∧ ≃ kInt(λ◦ ) for any proper closed convex cone λ, in particular λ = γ and λ = {0}. Hence the exact sequence 0 → − kγ\{0} → − kγ → − k{0} → − 0 gives (kγ\{0} )∧ ≃ kV ∗ \Int(γ ◦ ) and the result follows. (iii) We will choose c, r, h such that h > rc as explained after (III.2.2). Hence ∂γc meets ∂Yrh along its vertical part and the intersection, say I, is the sphere {(x′ , xn ) ∈ V ; ∥x′ ∥ = r, xn = −cr}. Let y ∈ I, p1 , p2 ∈ Ty∗ V be the unit length vectors in Ty∗ V ∩ SS(kYrh ), Ty∗ V ∩ S0′γc . Then ∥p1 − p2 ∥ ≤ c−1 . Let ε > 0 such that the conclusion of Lemma III.2.2 holds. We choose c > 2ε−1 and h, r small enough so that Yrh ⊂ U . We define R0 (F ) = Q′γc (FYrh ). We have a natural morphism F → − R0 (F ). Let W ⊂ Yrh be the set of x such that ∂(x + γc ) meets ∂Yrh along its vertical part and, for any y ∈ ∂(x + γc ) ∩ ∂Yrh , the unit length vectors p1 ∈ Ty∗ V ∩ SS(kYrh ), p2 ∈ Ty∗ V ∩ Sx′γc satisfy ∥p1 − p2 ∥ < ε. The set W is open and contains 0 by construction. b SS(kY h ). We can use + instead By Theorem I.2.13 we have SS(FYrh ) ⊂ (V × B) + r b in the last expression because the term V × B is a product. By (i) and (ii) we of +

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have SS(R0 (F )) ∩ Tx∗ V ⊂ p2 (((V × B) + SS(kYrh )) ∩ Sx′γc ) ˙ Y h )) ∩ Sx′γc ). ⊂ p2 ((V × B) ∩ (V × (V ∗ \ Int(γc◦a ))) ∪ p2 (((V × B) + SS(k r By Lemma III.2.2 and by our definition of W we have ˙ Y h )) ∩ Sx′γc ) ⊂ B ′ p2 (((V × B) + SS(k r

if x ∈ W . Hence SS(R0 (F )) ∩ Tx∗ V ⊂ (B ∩ (V ∗ \ Int(γc◦a )) ∪ B ′ . Since V ∗ \ Int(γc◦a ) = {ξn ≥ c−1 ∥ξ ′ ∥}, we have B ∩ (V ∗ \ Int(γc◦a )) ⊂ ((B \ {ξn ≥ 0}) ∪ B ′ ) for c big enough. Now we set R(F ) = R0 (F )| and we obtain (a). W

(iv) We prove that Pγc (FYrh )| ≃ 0, which implies (b) by the triangle given in (i). W By (I.5.2) we have (Pγc (FYrh ))y ≃ RΓ(V ; FEy ) where Ey = Yrh ∩ (y + γc ). If we restrict to the open subset {xn < h}, the microsupports of F , kYrh and ky+γc , for y ∈ V , are all contained in {ξn ≤ 0}. For y ∈ W we have Ey ⊂ {xn < h}, hence SS(FEy ) ⊂ {ξn ≤ 0} by Theorem I.2.13. Since supp(FEy ) ⊂ Yrh is compact, we deduce by Corollary I.2.16 that RΓ(V ; FEy ) ≃ 0 (taking ϕ(x) = xn in the corollary we obtain ∼ RΓ(ϕ−1 (]−∞, 2h[); FEy ) −− → RΓ(ϕ−1 (]−∞, −2h[); FEy ) ≃ 0). Hence Pγc (FYrh )| ≃ 0 W as claimed. Local splitting. — Recall the cone γc of (III.2.1) and the subset Sxγc ⊂ T ∗ V of (III.1.10). We have πV (S0γc ) = {(x′ , xn ; ξ ′ , ξn ); |xn | = c ∥x′ ∥} and, for a non zero y = (y ′ , yn ) ∈ πV (S0γc ) we have S0γc ∩ Ty∗ V = {(λ˜ y ′ , −c−1 λ˜ yn ); λyn > 0},

(III.2.3)

where (˜ y ′ , y˜n ) ∈ V ∗ corresponds to y through the identification V ′ ≃ (V ′ )∗ given by the Euclidean product. We will use a cylinder similar to Yrh Zrh = {(x′ , xn ) ∈ V ; ∥x′ ∥ ≤ r, −h ≤ xn < 0}

(III.2.4)

⊔ {(x′ , xn ) ∈ V ; ∥x′ ∥ < r, 0 ≤ xn < h}

which is locally closed (and not open) and such that SS(kZrh ) is close to (S0γc )a along ∂Zrh ∩ πV (S0γc ) for c big. Indeed over the vertical boundary of Zrh , that is, for y = (y ′ , yn ) with ∥y ′ ∥ = r and 0 < |yn | < h we have, by Example I.2.3-(iii), SS(kZrh ) ∩ Ty∗ V = {(λ˜ y ′ , 0); λyn > 0},

(III.2.5)

where y˜′ ∈ (V ′ )∗ is as in the description of S0γc (in Fig. III.2.2 we have pictured SS(kZrh ) and S0γc ). Proposition III.2.3. — Let U ⊂ V be an open subset containing 0 and let c > 0 be given. Then there exist an open neighborhood W of 0 in U and two functors P, P ′ : D(kU ) → − D(kW ) together with morphisms of functors u : P → − (·)| , W ′ ′ − (·)| , such that, for any F ∈ D(kU ) satisfying u:P → W

(III.2.6)

ASTÉRISQUE 440

SS(F ) ⊂ U × (γc◦ ∪ γc◦a ),

CHAPTER III.2. LOCAL CUT-OFF—SPECIAL CASE

57

Figure III.2.2.

we have ◦ ˙ ˙ SS(P (F )) = SS(F | ) ∩ (W × γc ), W

′ ◦a ˙ ˙ SS(P (F )) = SS(F | ) ∩ (W × γc ) W

and the object L ∈ D(kW ) given by the distinguished triangle (u(F ), u′ (F ))

P (F ) ⊕ P ′ (F ) −−−−−−−−−→ F |

+1

W

→ − L −−→

˙ satisfies SS(L) = ∅. Proof. — (i) As in the proof of Proposition III.2.1 we assume to be given d, r, h > 0 such that h > rd. Hence πV (S0γd ) meets ∂Zrh along its vertical part, where Zrh is defined in (III.2.4), and the intersection is the union of the two spheres {(x′ , xn ) ∈ V ; ∥x′ ∥ = r, xn = ±dr}. Along these two spheres S0γd and SS(kZrh ) are the two half lines described in (III.2.3) and (III.2.5) (so they make an angle arctan(d−1 )). Using Lemma III.2.2 with (III.2.7)

B = γc◦ ∪ γc◦a = {(ξ ′ , ξn ); c |ξn | ≤ ∥ξ ′ ∥}

and B ′ = ∅ we obtain (III.2.8)

S0γd ∩ (SS(kZrh ) + V × (γc◦ ∪ γc◦a )) = ∅

for d big enough, whatever r, h. We fix such a d and then choose r, h such that h > rd (as already assumed) and small enough so that Zrh ⊂ U . (ii) We define W as the set of x ∈ Int(Zrh ) such that πV (Sxγd ) meets ∂Zrh along its vertical part and (III.2.8) holds with S0γd replaced by Sxγd . Hence W is an open neighborhood of 0. For F ∈ D(kU ) we can extend F ⊗ kZrh by 0 as an object of D(kV ). We define P, P ′ by P (F ) = Pγd (F ⊗ kZrh )| and P ′ (F ) = Pγ′ d (F ⊗ kZrh )| . The functors u, u′ W

W

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are induced by uγd , u′γd (we remark that W ⊂ Zrh , hence (F ⊗ kZrh )| = F | ). If W W F satisfies (III.2.6), then (III.2.9)

SS(F ⊗ kZrh ) ⊂ (SS(kZrh ) + V × (γc◦ ∪ γc◦a ))

by Theorem I.2.13. Hence F ⊗ kZrh satisfies the Hypothesis (III.1.13) of Proposition III.1.10 by (III.2.8) (with S0γd replaced by Sxγd ). Now the result follows from Proposition III.1.10

ASTÉRISQUE 440

CHAPTER III.3 LOCAL CUT-OFF—GENERAL CASE

Here we extend the results of §III.2 replacing the cones {ξn ≤ 0} or γc◦ by more general ones. We deduce Propositions III.3.1, III.3.2 from Propositions III.2.1, III.2.3. The process of reduction to the results of §III.2 is the same for both propositions and we only prove the second one. (Moreover Proposition III.3.1 is only a precised version of [28, Prop. 6.1.4] but Proposition III.3.2 is new.) Proposition III.3.1. — Let U be an open subset of V = Rn and let A ⊂ V ∗ \ {0} be an open cone. We assume that there exists a homotopy ψ : (V ∗ \ {0}) × [0, 1] → − V ∗ \ {0} 1 of class C and homogeneous of degree 1 such that ψ1 (A) = {ξn > 0}. Let B ⊂ V ∗ be a closed conic subset and B ′ ⊂ V ∗ be a conic neighborhood of B ∩ ∂A. Let x0 ∈ U be given. Then there exist a neighborhood W of x0 and a functor R : D(kU ) → − D(kW ), of the form R(F ) = K ◦ F for some K ∈ D(kW ×U ), together with a morphism of functors (·)| → − R, such that, for any F ∈ D(kU ), W (i) if SS(F ) ⊂ U × B, then SS(R(F )) ⊂ W × ((B \ A) ∪ B ′ ), ∼ ˙ (ii) if SS(F ) ∩ (U × A) = ∅, then F | −− → R(F ). W Proposition III.3.2. — Let U be an open subset of V = Rn and let A, A′ ⊂ V ∗ \ {0} be two disjoint closed conic subsets. We assume that there exists a homotopy ψ : (V ∗ \ {0}) × [0, 1] → − V ∗ \ {0} of class C 1 and homogeneous of degree 1 such that ψ1 (A) and ψ1 (A′ ) are separated by some hyperplane of V ∗ . Let x0 ∈ U be given. Then there exist a neighborhood W of x0 in U and two functors P, P ′ : D(kU ) → − D(kW ) together with morphisms of functors u : P → − (·)| , W ′ ′ u:P → − (·)| such that, for any F ∈ D(kU ) satisfying W

(III.3.1)

˙ SS(F ) ⊂ U × (A ⊔ A′ ),

we have ˙ ˙ SS(P (F )) = SS(F | ) ∩ (W × A), W

′ ′ ˙ ˙ SS(P (F )) = SS(F | ) ∩ (W × A ) W

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and the object L ∈ D(kW ) given by the distinguished triangle (u(F ),u′ (F ))

P (F ) ⊕ P ′ (F ) −−−−−−−−→ F |

+1

W

→ − L −−→

˙ satisfies SS(L) = ∅. Proof. — (i) Let c > 0 be given. We recall the notation γc of (III.2.1). Up to a linear change of coordinates in Rn we can assume that x0 = 0, that the homotopy ψ in the hypothesis is defined on some open interval I containing [0, 1] and that ψ1 (A) ⊂ γc◦ , ψ1 (A′ ) ⊂ γc◦a . We first extend ψ to a homogeneous Hamiltonian isotopy Φ : T˙ ∗ U × I → − T˙ ∗ U such that Φt (T˙0∗ U ) = T˙0∗ U for all t ∈ I and Φ1 ({0} × A) ⊂ {0} × γc◦ ,

Φ1 ({0} × A′ ) ⊂ {0} × γc◦a .

To see that such a Φ exists we choose local coordinates (x; ξ) around 0 and write Pn ∂ψ ∗ i=1 ai (ξ, t)∂ξi (for (ξ, t) ∈ V × I). Then we choose a Hamiltonian func∂t (ξ, t) = − R homogeneous of degree 1 in ξ and with supportPin C × V ∗ for tion h : T˙ ∗ U × I → n some compact subset C of U such that, near 0, we have h(x; ξ) = − i=1 ai (ξ, t)xi . Then the Hamiltonian flow Φ of h satisfies the above relations. We let RΦ : D(kU ) → − D(kU ), F 7→ KΦ,1 ◦ F , be the equivalence of categories given ˙ ˙ by Corollary II.1.5. We have in particular SS(R Φ (F )) = Φ1 (SS(F )) for all F ∈ D(kU ). (ii) We can find a neighborhood U1 of 0 such that ◦ ◦a Φ1 (U × A) ∩ T˙ ∗ U1 ⊂ U1 × γ2c , Φ1 (U × A′ ) ∩ T˙ ∗ U1 ⊂ U1 × γ2c . Applying Proposition III.2.3 (with U1 and 2c instead of U and c) we find a neighborhood of 0, say W1 , and functors P1 , P1′ : D(kU1 ) → − D(kW1 ) together with morphisms of functors u1 : P1 → − (·)| , u′1 : P1′ → − (·)| satisfying the conclusion of ProposiW1 W1 tion III.2.3. (iii) We let W be an open neighborhood of 0 such that Φt (T˙ ∗ W ) ⊂ T˙ ∗ W1 for all t ∈ [0, 1]. Let j denote the inclusion of W1 in U . We define the functor P by −1 P (F ) = (RΦ j! P1 (RΦ (F )| ))| W U1

and P ′ from P1′ by the same formula. We let u, u′ be the morphisms of functors induced by u1 , u′1 . By Proposition III.2.3 we have a distinguished triangle on W1 − RΦ (F )| P1 (RΦ (F )| ) ⊕ P1′ (RΦ (F )| ) → U1

U1

+1

W1

→ − L −−→,

−1 where L is locally constant on W1 . Since Φ1 (T˙ ∗ W ) ⊂ T˙ ∗ W1 we see that RΦ (j! (L))| W −1 is locally constant. Applying (RΦ (j! (−)))| to the distinguished triangle, we find W −1 that G = (RΦ ((RΦ (F ))W1 ))| is isomorphic to P (F )⊕P ′ (F ) up to a locally constant W sheaf on W . It only remains to check that G is isomorphic to F | . Lemma II.2.1 applied with W −1 −1 Ψt = Φ−1 t , U = W1 and V = W gives RΦ (HW1 )|W ≃ RΦ (H)|W for any H. Hence −1 G ≃ (RΦ (RΦ (F )))| ≃ F | , as required. W

ASTÉRISQUE 440

W

CHAPTER III.4 CUT-OFF AND γ-TOPOLOGY

In [28] the Proposition III.1.7 has another formulation in terms of the γ-topology. It gives directly a decomposition of the projector Pγ as in Remark III.1.9-(1). As in the previous paragraphs, let γ ⊂ V be a closed convex cone. The γ-topology is defined in [28] as follows. We say that an open subset Ω of V is γ-stable if x + y ∈ Ω for all (x, y) ∈ Ω × γ. The γ-stable open subsets define a topology on V and we denote by Vγ this topological space. The identity map induces a continuous map ϕγ : V → − Vγ . Then Propositions 3.5.3, 3.5.4 and 5.2.3 of [28] give Proposition III.4.1. — (i) For any G ∈ D(kVγ ) the adjunction morphism G→ − Rϕγ ∗ ϕ−1 γ G is an isomorphism. (ii) For F ∈ D(kV ) the adjunction morphism ϕ−1 − F is an isomorphism if γ Rϕγ ∗ F → and only if SS(F ) ⊂ V × γ ◦a . ◦a In particular for any G ∈ D(kVγ ) we have SS(ϕ−1 γ G) ⊂ V × γ . −1 (iii) There exists an isomorphism of functors Pγ ≃ ϕγ ◦ Rϕγ ∗ and the morphism uγ of (III.1.2) corresponds to the adjunction morphism. In other words, using the notation DV ×γ ◦a (kV ) of Notation I.2.2, the functors and Rϕγ ∗ give mutually inverse equivalences of categories between D(kVγ ) and DV ×γ ◦a (kV ). In Proposition III.4.1 the conditions on the microsupport are global on V . However we can also consider local situations using the following lemma. ϕ−1 γ

Lemma III.4.2. — Let U be an open subset of V and F ∈ D(kU ). We assume that SS(F ) ⊂ U × γ ◦a . Let x0 ∈ U . Then there exist a neighborhood U ′ of x0 in U and G ∈ D(kV ) such that G| ′ ≃ F | ′ and SS(G) ⊂ V × γ ◦a . U

U

Proof. — For x ∈ V , ξ ∈ γ ◦a and t > 0 we define the truncated cone C = Cx,ξ,t = (x − γ) ∩ {y; ⟨ξ, y − x⟩ < t} with vertex at x. We have SS(kC ) ⊂ V × γ ◦a by Example I.2.3 and Theorem I.2.13. We can find x, ξ, t such that x0 ∈ Int(C) and C ⊂ U . By Theorem I.2.13 the sheaf G = FC satisfies SS(G) ⊂ U × γ ◦a . Since G has a compact support we can extend

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it by 0 on V and we still have SS(G) ⊂ V × γ ◦a . Setting U ′ = Int(C), we also have G| ′ ≃ F | ′ . U U We thank Pierre Schapira for the following useful result. Corollary III.4.3. — (i) Let U be an open subset of V and F ∈ D(kU ). We assume that SS(F ) ⊂ U × γ ◦a . Then SS(H i F ) ⊂ U × γ ◦a for all i ∈ Z. (ii) Let M be a manifold and F ∈ D(kM ). Let C ⊂ T ∗ M be the convex hull of SS(F ) in the sense that C is the intersection of all closed conic subsets S of T ∗ M which contain SS(F ) and are fiberwise convex (S ∩ Tx∗ M is convex for any x ∈ M ). We assume that C does not contain any line. Then SS(H i F ) ⊂ C for all i ∈ Z. Proof. — (i) The statement is local on U . Let x0 ∈ U be given and let U ′ , G be given by Lemma III.4.2. By Proposition III.4.1 there exists G′ ∈ D(kVγ ) such ′ ′ −1 i ′ that G ≃ ϕ−1 is exact we have H i G ≃ ϕ−1 γ G (G = Rϕγ ∗ G). Since ϕγ γ H G . By Proposition III.4.1 again we deduce SS(H i G) ⊂ V × γ ◦a . Since G| ′ ≃ F | ′ , this U U gives the required bound for SS(H i F ) near x0 . (ii) The statement is local on M and we can assume that M is open in a vector space V . Then T ∗ M = M × V ∗ . For a given x ∈ M we can find a closed convex cone δ in V ∗ , contained in an arbitrarily small neighborhood of C ∩ ({x} × V ∗ ), and a neighborhood U of x such that SS(F | ) ⊂ U × δ. Then the result follows from (i). U

Remark III.4.4. — 1) Let U be an open subset of V and F ∈ Mod(kU ). We assume that SS(F ) ⊂ U × γ ◦a . Then, for any s ∈ F (U ), supp(s) is “locally γ-closed”, that is, for any x0 ∈ U , there exists a neighborhood B of x0 (for the usual topology) such that B ∩ supp(s) = B ∩ Z for some Z ⊂ V which is closed for the γ-topology. Indeed, we consider U ′ , G given by Lemma III.4.2. Then s| ′ is identified with a U ′ section of G. By Proposition III.4.1 there exists G′ ∈ Mod(kVγ ) such that G ≃ ϕ−1 γ G . ′ ′ ′ Since Gx0 ≃ Gx0 , there exist a section s of G (over some γ-open set containing x0 ) and a neighborhood U ′′ of x0 in V such that s| ′′ is the section of G induced by s′ . U In particular sx ≃ s′x for all x ∈ U ′′ and we get supp(s) ∩ U ′′ = supp(s′ ) ∩ U ′′ . Since supp(s′ ) is closed in Vγ we are done. 2) If a subset Z of V is locally γ-closed, then, for any x0 ∈ Z, there exists a neighborhood B of 0 in V such that x0 − y ∈ Z for all y ∈ B ∩ γ.

ASTÉRISQUE 440

CHAPTER III.5 REMARKS ON PROJECTORS—TAMARKIN PROJECTOR

We have seen in Remark III.1.9 that the functors Pγ , Qγ : D(kV ) → − D(kV ) are projectors to the subcategory DV ×γ ◦a (kV ) of D(kV ) and induce adjoints to the embedding ι : DV ×γ ◦a (kV ) → − D(kV ); namely Pγ induces the right adjoint to ι and Qγ the left adjoint. In this paragraph we give a similar interpretation to Pγ′ , Q′γ and give a relative version of Pγ′ . It turns out that our projectors come in pairs. Using [29, Ex. 10.15] or [22, Prop. 4.21], we can deduce from the first triangle in (III.1.3) that Q′γ is a projector to the right orthogonal of DV ×γ ◦a (kV ): (III.5.1)

D⊥,r V ×γ ◦a (kV ) = {F ∈ D(kV ); Hom(G, F ) ≃ 0 for all G ∈ DV ×γ ◦a (kV )}.

Indeed, applying Pγ to (III.1.3) and using Pγ ◦ Pγ ≃ Pγ we obtain Pγ ◦ Q′γ ≃ 0. Then, ∼ → Q′γ ◦ Q′γ . This proves that Q′γ is putting F = Q′γ (G) in (III.1.3) we have Q′γ −− ∼ a projector. Using Hom(Pγ (F ), Pγ (G)) −−→ Hom(Pγ (F ), G), (see Remark III.1.9) we deduce Hom(Pγ (F ), Q′γ (G)) ≃ 0 and we can deduce that the image of Q′γ is ′ indeed D⊥,r V ×γ ◦a (kV ). More precisely Qγ induces a left adjoint to the embedding of D⊥,r V ×γ ◦a (kV ). In the same way, we can see that Pγ′ is also a projector, with image the left orthogonal of DV ×γ ◦a (kV ): (III.5.2)

D⊥,l V ×γ ◦a (kV ) = {F ∈ D(kV ); Hom(F, G) ≃ 0 for all G ∈ DV ×γ ◦a (kV )}

and that it induces a right adjoint to the embedding of D⊥,l V ×γ ◦a (kV ). The relation between Pγ′ , Q′γ is not the same as the relation between Pγ , Qγ . In Remark III.1.9 we deduced that Qγ was a projector from the fact that its was adjoint to Pγ . To deduce that they have the same image we used the more precise fact: Pγ induces the right adjoint to ι and Pγ is the right adjoint to Qγ . Now Pγ′ induces the right adjoint to the embedding of its image but Pγ′ is left adjoint to Q′γ ; so we cannot conclude that Pγ′ and Q′γ have the same image. The fact that the embedding ι has both a left and a right adjoint comes from the property that DV ×γ ◦a (kV ) has arbitrary small sums and products and ι commutes

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CHAPTER III.5. REMARKS ON PROJECTORS—TAMARKIN PROJECTOR

with sums and products (this is the Brown representability theorem—see for example [29, §14.2]). However D⊥,l V ×γ ◦a (kV ) is not stable by infinite products even for V = R, γ = [0, +∞[ (see Example III.5.1) and we cannot apply Brown theorem to find a new projector. Example III.5.1. — In the case V = R, γ = [0, +∞[, we have, forQ any x ∈ R, Pγ′ (k[x,+∞[ ) ≃ k[x,+∞[ . Let us check that Pγ′ (F ) ̸≃ F , where F = i∈N k[−i,+∞[ , thus proving that D⊥,l V ×γ ◦a (kV ) is not stable by infinite products. We compute the stalks (Pγ′ (F ))x using (III.1.5) and (I.5.2). We find (Pγ′ (F ))x ≃ RΓc (R; FIx ), with Ix = ]−∞, x]. Using the exact sequence Q 0→ − G→ − kN − F → − 0, with G = ≃ 0 for any R → i∈N k]−∞,−i[ , and RΓc (R; LIx ) L constant sheaf L, we obtain (Pγ′ (F ))x ≃ RΓc (R; GIx )[1]. Now G ≃ i∈N k]−∞,−i[ because this sum is locally finite. The functors RΓc (R; ·) and (·)Ix commute with direct sums. Hence RΓc (R; GIx ) ≃ k(N) for x > 0. On the other hand we have Fx ≃ kN for x > 0. Hence Pγ′ (F ) ̸≃ F . It should be noted that the cut-off functor Pγ is defined in [28] in a relative situation where V is replaced by M × V for some manifold M . In this general setting Pγ is defined on D(kM ×V ) and projects onto the subcategory DT ∗ M ×(V ×γ) (kM ×V ) formed by the F such that SS(F ) ⊂ T ∗ M × (V × γ ◦a ) (Notation I.2.2). We have Pγ (F ) = Rq2 ∗ (kM ×eγ ⊗ q1−1 F ), where q1 , q2 are now the projections M × V 2 → − M × V and γ e is defined as in (III.1.1). For later use we quote some properties of Pγ already considered by Tamarkin in [45] in the case V = R, γ = [0, +∞[ or γ = ]−∞, 0]. We introduce coordinates (t; τ ) on T ∗ R and local coordinates (x, t; ξ, τ ) on T ∗ (M × R). We denote for short by {τ ≥ 0} or {τ > 0} the subsets of T ∗ (M ×R) defined by the corresponding conditions on τ . In this relative setting the projector P]−∞,0] : D(kM ×R ) → − D{τ ≥0} (kM ×R ) can be rewritten as (III.5.3)

P]−∞,0] (F ) = Rs∗ (q1−1 (F ) ⊗ q2−1 (k[0,+∞[ )),

where s, q1 : M × R2 → − M × R and q2 : M × R2 → − R are defined by s(x, t1 , t2 ) = (x, t1 + t2 ), q1 (x, t1 , t2 ) = (x, t1 ) and q2 (x, t1 , t2 ) = t2 . ′ In [45] Tamarkin was rather interested in the projector P[0,+∞[ , defined on the category D(kM ×R ) with image (III.5.4) D⊥,l {τ ≤0} (kM ×R ) = {F ∈ D(kM ×R ); Hom(F, G) ≃ 0 for all G ∈ D{τ ≤0} (kM ×R )} as noted in (III.5.2). Using (III.1.5) and noticing that D′ (k]−∞,0]a \{0} ) ≃ D′ (k]−∞,0[ ) ≃ k]−∞,0] , we see ′ that P]−∞,0] and P[0,+∞[ have very similar expressions. With the notations of (III.5.3) we can rewrite ′ P[0,+∞[ (F ) = Rs! (q1−1 (F ) ⊗ q2−1 (k[0,+∞[ )).

ASTÉRISQUE 440

CHAPTER III.5. REMARKS ON PROJECTORS—TAMARKIN PROJECTOR

65

′ Using this formula we find the bound SS(P[0,+∞[ (F )) ⊂ {τ ≥ 0}, like SS(P]−∞,0] (F ))

in (III.1.6). This proves that D⊥,l {τ ≤0} (kM ×R ) is in fact contained in D{τ ≥0} (kM ×R ). ′ An important remark of Tamarkin is that the functors P]−∞,0] or P[0,+∞[ have a natural morphism not only to the identity functor but also to the direct image functor Tc∗ for c ≥ 0, where Tc denotes the translation along R Tc : M × R → − M × R,

(x, t) 7→ (x, t + c).

In other words the objects F of D{τ ≥0} (kM ×R ) or D⊥,l {τ ≤0} (kM ×R ) come with a natural morphism τc (F ) : F → − Tc∗ (F ) for c ≥ 0. To define τc we first remark that we have an isomorphism of functors Tc∗ ◦ P]−∞,0] ≃ P]−∞,0] ◦ Tc∗ and that (III.5.5)

Tc∗ ◦ P]−∞,0] (F ) ≃ Rs∗ (q1−1 (F ) ⊗ q2−1 (k[c,+∞[ )).

For c ≥ 0 the natural morphism k[0,+∞[ → − k[c,+∞[ induces the morphism of functors (III.5.6) and hence the morphism (III.5.7) for sheaves in the image of P]−∞,0] : (III.5.6) (III.5.7)

τc : P]−∞,0] → − Tc∗ ◦ P]−∞,0] , τc (F ) : F → − Tc∗ (F ),

for F ∈ D{τ ≥0} (kM ×R ).

Tamarkin then used this morphism τc to give non-displaceability conditions on subsets of T ∗ M . We will consider a variant of Tamarkin criterion by looking at the minimal c such that τc (F ) = 0 (see §VI.2). Remark III.5.2. — Assuming γ ⊂ V is a closed convex cone which contains no line, we can also easily build projectors which cut the microsupport by the complement of Int(γ ◦ ). Let us set Zγ = V × (V ∗ \ Int(γ ◦ )) and let DZγ (kV ) be the subcategory of D(kV ) of sheaves with microsupport contained in Zγ . We can generalize Tamarkin’s construction in higher dimension and define projectors from D(kV ) to itself with image the subcategory DZγ (kV ) or its left or right orthogonal. For example it is proved in [22, Prop. 4.21] that Lγ : F 7→ F ◦ kγea is a projector on D(kV ) with image D⊥,l Zγ (kV ). We also know that D⊥,l Zγ (kV ) is contained in DV ×γ ◦ (kV ), which is the image of Pγ a (we can prove that SS(Lγ (F )) ⊂ V × γ ◦ like (III.1.8), or we refer to loc. cit.). We recall that Pγ a and Lγ often coincide (see Lemma III.1.1). Of course, when V = R we have Zγ = V × γ ◦a , Lγ = Pγ′ and we recover Tamarkin’s projector (see around (III.5.4)).

SOCIÉTÉ MATHÉMATIQUE DE FRANCE 2023

PART IV CONSTRUCTIBLE SHEAVES IN DIMENSION 1

In this part we apply Gabriel’s theorem to describe the constructible sheaves on the real line and the circle with coefficients in a field k.

CHAPTER IV.1 GABRIEL’S THEOREM

We give a quick reminder of a part of Gabriel’s theorem that we will use in this part. We follow the presentation of Brion’s lecture [9] on the subject and we refer the reader to [9] for further details. In this section k is a field. A quiver is a finite directed graph, that is, a quadruple Q = (Q0 , Q1 , s, t) where Q0 , Q1 are finite sets (the set of vertices, resp. arrows) and s, t : Q1 → − Q0 are maps assigning to each arrow its source, resp. target. A representation of a quiver Q consists of a family of k-vector spaces Vi indexed by the vertices i ∈ Q0 , together with a family of linear maps fα : Vs(α) → − Vt(α) indexed by the arrows α ∈ Q1 . For a representation Q0 ({Vi }, fα ) the dimension vector is (dim Vi )i∈Q0 . The Pspace 2R Pof dimension vectors is endowed with the Tits form defined by qQ (d) = i∈Q0 di − α∈Q1 ds(α) dt(α) . A quiver is of finite orbit type if it has only finitely many isomorphism classes of representations of any prescribed dimension vector. Gabriel’s theorem describes the quivers of finite orbit type. It says that they are the quivers with a positive defined Tits form and also says that this is equivalent to be of type A, D, E. Another part of Gabriel’s theorem gives the structure of the representations of the quivers of finite type. A representation is said indecomposable if it cannot be split as the sum of two non zero representations. A representation V is Schur if Hom(V, V ) ≃ k · idV . Theorem IV.1.1 (See Theorem 2.4.3 in [9]). — Assume that the Tits form qQ is positive definite. Then: (i) Every indecomposable representation is Schur and has no non-zero selfextensions. (ii) The dimension vectors of the indecomposable representations are exactly those d ∈ NQ0 such that qQ (d) = 1. (iii) Every indecomposable representation is uniquely determined by its dimension vector, up to isomorphism. Remark IV.1.2. — We are actually only interested in quivers of type Am , that is, quivers whose underlying graph is the linear graph with m vertices and m − 1 edges.

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In this case we have Q0 = {0, . . . , m − 1} and we find qQ (d) =

m−1 1 2 X (d0 + (di − di−1 )2 + d2m−1 ). 2 i=1

Hence a dimension vector d satisfies the condition qQ (d) = 1 if and only if there exist i ≤ j ∈ {0, . . . , m − 1} such that dk = 1 if i ≤ k ≤ j and dk = 0 else.

ASTÉRISQUE 440

CHAPTER IV.2 CONSTRUCTIBLE SHEAVES ON THE REAL LINE

We apply the results of Section IV.1 to sheaves on R with coefficients in a field k. Let x = {x1 < · · · < xn } be a finite family of points in R. We denote by Modx (kR ) the category of constructible sheaves on R with respect to the stratification induced by x. Setting x0 = −∞ and xn+1 = +∞, a sheaf F belongs to Modx (kR ) if and only if the stalks Fy are finite dimensional for all y ∈ R and the restrictions F | are ]xk ,xk+1 [ constant for k = 0, . . . , n. If I is an interval of R and x1 , . . . , xn ∈ I we define in the same way Modx (kI ). We say that a sheaf F on R is constructible if, for any n > 0, F| ∈ Modx (k]−n,n[ ) for some finite family x = {x1 < · · · < xk } of ]−n, n[. We ]−n,n[ denote by Modc (kR ) the category of constructible sheaves on R. A sheaf F ∈ Modx (kR ) is determined by the data of the spaces of sections ∼ V2i+1 = F (]xi , xi+2 [) −− → Fxi+1 , for i = 0, . . . , n − 1, (IV.2.1) V2i = F (]xi , xi+1 [), for i = 0, . . . , n, together with the restriction maps V2i+1 → − V2i and V2i+1 → − V2i+2 for i = 0, . . . , n − 1. Conversely, any such family of vector spaces {Vi }2n and linear maps defines a sheaf i=0 in Modx (kR ). Hence (IV.2.1) gives an equivalence between Modx (kR ) and the category of representations of the quiver Q = (Q0 , Q1 , s, t) of type A2n+1 where Q0 = {0, . . . , 2n} and there is exactly one arrow in Q1 from 2i − 1 to 2i − 2 and from 2i − 1 to 2i, for i = 1, . . . , n. Since Q is of type A2n+1 , we can apply Gabriel’s theorem and Remark IV.1.2. Hence the indecomposable representations of Q are in bijection with the dimension vectors d such that dk = 1 if i ≤ k ≤ j and dk = 0 else, for some i ≤ j ∈ {0, . . . , 2n}. Through the equivalence (IV.2.1) the corresponding sheaves in Modx (kR ) are the constant sheaves kI on the intervals I with ends −∞, x1 , . . . , xn or +∞ (the intervals can be open, closed or half-closed). Gabriel’s theorem gives the following decomposition result for constructible sheaves with compact support. Extensions of Gabriel’s theorem gives the general case (see for example Theorem 1.1 in [12]). We can also deduce the general case from the case of compact support; this is done in [30, Thm. 1.17] and we reproduce the proof below.

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Corollary IV.2.1. — We recall that k is a field. Let F ∈ Modc (kR ). Then there exist a locally finite family of distinct intervals {Ia }a∈A and integers {na }a∈A such that M (IV.2.2) F ≃ knIaa . a∈A

Moreover this decomposition is unique in the following sense. If we have another L b ∼ decomposition F ≃ b∈B km − →B Jb like (IV.2.2), then there exists a bijection σ : A − such that Jσ(a) = Ia and mσ(a) = na for all a ∈ A. Proof. — (i) For an integer n ≥ 1 we set Un = ]−n, n[. Identifying Un with R, the restriction F | belongs to Modx (kR ) for some finite set of points x. Through the Un equivalence (IV.2.1) (and coming back to Un ) Gabriel’s theorem gives a decomposition L F | ≃ c∈Cn kIcn , where Cn is a finite set of intervals of Un . Each factor kIcn has Un multiplicity 1, but we may have Icn = Idn for c, d ∈ Cn . This decomposition is unique by (i) of Theorem IV.1.1 and by the Krull-Schmidt theorem. The uniqueness implies that we can find an injective map γn : Cn → − Cn+1 , for for any c ∈ C . We set C = lim C . For c ∈ C, any n ≥ 1, such that Icn = Un ∩ Iγn+1 n n (c) −→ n S represented by c˜ ∈ Cn for some n, we set Ic = m≥n Iγmm (˜c) , where γnm = γm−1 ◦· · ·◦γn . n We then have Ic ∩Um = Iγmm (˜c) . We remark that the obvious map Cn → − C is injective, n for any n. (ii) Let c ∈ C be given. We claim that we can find ic : kIc → − F and pc : F → − kIc such that pc ◦ ic = idkIc . For n ≥ 1 we define En = Hom(kIc |

Un

,F|

Un

) × Hom(F |

Un

En′ = {(i, p) ∈ En ; p ◦ i = idkIc |

, k Ic |

Un

),

}.

Un

− En for m ≥ n. We clearly have The restriction morphisms induce enm : Em → ′ enm (Em ) ⊂ En′ . We remark that En′ ̸= ∅ because kIc ∩Un is a direct summand of F | . Un ′ Let us prove that enm (Em ) = enm (Em ) ∩ En′ , for any m ≥ n. This is clear when Ic ∩ Un = ∅ (in this case En = En′ = {(0, 0)}). If Ic ∩ Un ̸= ∅, then we have k ≃ Hom(kIc |

Um

, kIc |

Um

r

)− → Hom(kIc |

Un

, kIc |

Un

) ≃ k,

and the restriction map r is an isomorphism. In particular r(u) = idkIc |

Un

implies u = idkIc |

Um

′ and we get enm (Em ) = enm (Em ) ∩ En′ . Since F is con-

structible, the spaces En are all finite dimensional. Hence, for a given n, the ′ sequence {enm (E stabilizes and it follows that {enm (Em )}m≥n also stabilizes. Tm )}m≥n ′∞ n ′ n ′∞ We set En = m≥n em (Em ). The maps em induce surjective maps Em → − En′∞ , for ′ ), for some m big all m ≥ n. Finally we have En′∞ ̸= ∅ for all n; indeed En′∞ = enm (Em ′ ′∞ enough, and we have Em ̸= ∅. Hence {En )}n≥1 is a projective system of non empty sets, with surjective structural maps. It follows that limn En′∞ ̸= ∅ and any element ←− in this limit is a sequence of compatible pairs of morphisms (in , pn ) ∈ En′ which glue into a pair (ic , pc ) as claimed.

ASTÉRISQUE 440

CHAPTER IV.2. CONSTRUCTIBLE SHEAVES ON THE REAL LINE

73

L (iii) Let n ≥ 1 be an integer. We claim that we can write F ≃ F1 ⊕ a∈A1 kIa , where F1 | ≃ 0 and A1 is some finite family of intervals of R. Un If F | ≃ 0, the claim is trivial. If not, we pick c ∈ C such that Ic ∩ Un ̸= ∅. Using Un (ic , pc ) found in (ii) we write F ≃ kIc ⊕ F 1 . Then F 1 is constructible. If F 1 ≃ 0 we are done. If not, we apply the same argument to F 1 and write F 1 | ≃ kI1 ⊕ F 2 Un for some interval I1 meeting Un (the interval I1 is in the family C associated with F but we do not need to know that). We go on with F 2 and write inductively F ≃ kIc ⊕ kI1 ⊕ · · · ⊕ kIk ⊕ F k+1 , where the Ij ’s meet Un , as long as F k | ̸≃ 0. Un Since F is constructible, the space Hom(F | , F | ) is finite dimensional. Since the Un Un Ij ’s meet Un , the elements idkIj give a free family in Hom(F | , F | ). Hence the Un Un process will stop after finitely many steps and the claim is proved. (iv) Using (iii) we can write inductively for k ≥ 1, (Dk )

F ≃ Fk ⊕

k M

Gi ,

i=1

L where Fk | ≃ 0, Gi = a∈Ai kIa , where Ai is a finite family of intervals of R and Uk Gi | ≃ 0 for i ≥ 2. Indeed (iii) applied with F and n = 1 gives the first step Ui−1

and (iii) applied with Fk and n = k + 1 gives the (k + 1)-th step. This inductive construction also makes the decompositions (Dk ) compatible in the sense that the projection ui : F → − Gi deduced from (Dk ) for k ≥ i is actually independent of k. Pk Lk Since Fk | ≃ 0, the sum i=1 ui | : F | → − i=1 Gi |Uk is an isomorphism. We U U U k k k L∞ P∞ set G = i=1 Gi and define u = i=1 ui : F → − G. This last sum makes sense because, over each interval Uk , only the terms ui for i = 1, . . . , k are non zero. Since u| is Uk an isomorphism for each k, u itself is an isomorphism. Putting together the intervals which appear several times (only finitely many times by constructibility) we obtain a decomposition of F as stated in the corollary. (v) To prove the uniqueness statement we first consider a bounded interval Ia , for a ∈ A. We choose an interval Un such that Ia ⊂ Un . Then kIa appears in the decomposition of F | with the same multiplicity as in F . By the uniqueness of the Un decomposition of F | we deduce that Ia = Jb for some b ∈ B and mb = na . Defining Un A′ = {a ∈ A; Ia is bounded} and B ′ similarly, we thus have a bijection between A′ and B ′ with respects the multiplicities.L L mb na We are left with an isomorphism a∈A\A′ kIa ≃ b∈B\B ′ kJb . Now, an unbounded interval I with one end x is determined by I ∩ Un , as soon as x ∈ Un . Defining A′′ = {a ∈ A; Ia is unbounded and not equal to R} and B ′′ similarly, we can then identify A′′ with B ′′ as in the bounded case. Now we are left with the summand m n kR , indexed by, say a0 ∈ A, b0 ∈ B, and we have kRa0 ≃ kR b0 . Hence na0 = mb0 and this concludes the proof.

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We also remark that (i) of Theorem IV.1.1 is given in our case by the following easy result, which we quote for later use, and the fact that Ext1 (kI , kI ) ≃ 0. Lemma IV.2.2. — Let I, J be two intervals of R. Then ( k if I ∩ J ̸= ∅ and I ∩ J is closed in I and open in J, Hom(kI , kJ ) ≃ 0 otherwise. In particular, if I and J are distinct, then we have Hom(kI , kJ ) ≃ 0 or Hom(kJ , kI ) ≃ 0.

ASTÉRISQUE 440

CHAPTER IV.3 CONSTRUCTIBLE SHEAVES ON THE CIRCLE

In this section we extend Corollary IV.2.1 to the circle. Like in the case of R the result is also a particular case of quiver representation theory and Auslander-Reiten theory (see for example [41] §3.6 p.153 and Theorem 5 p.158). However it is quicker to prove the facts we need than recall these general results. We denote by S1 the circle and we let e : R → − S1 ≃ R/2πZ be the quotient map. We 1 use the coordinate θ on S defined up to a multiple of 2π. We also denote by T : R → − R the translation T (x) = x+2π. We recall that k is a field. We say that F ∈ Mod(kS1 ) is constructible if F | is constructible in the sense of §IV.2 for any arc I ⊂ S1 . We denote I by Modc (kS1 ) the category of such sheaves. Since e is a covering map, we have an isomorphism of functors e! ≃ e−1 , hence an adjunction (e! , e−1 ). For any n ∈ Z we have e ◦ T n = e, hence natural isomorphisms of functors (T n )−1 e−1 ≃ e−1 and e! T!n ≃ e! . For G ∈ Mod(kR ) the isomorphism ∼ e! (T!n (G)) −− → e! (G) gives by adjunction in (G) : T!n (G) → − e−1 e! (G). For x ∈ R we L −1 n have (e e! (G))x ≃ (e! (G))e(x) ≃ n∈Z GT (x) . We deduce that the sum of the in (G) gives an isomorphism M ∼ (IV.3.1) T n (G) −− → e−1 e! (G). ∗

n∈Z

∼ Let I be a bounded interval of R. We have e! (kI ) −− → e∗ (kI ). Let AI be the algebra AI = Hom(e∗ (kI ), e∗ (kI )). The adjunction (e−1 , e∗ ) gives a morphism e−1 e∗ (kI ) → − kI . Using also the adjunction (e! , e−1 ) we obtain a natural morphism (IV.3.2)

εI : AI ≃ Hom(kI , e−1 e∗ (kI )) → − Hom(kI , kI ) ≃ k.

Lemma IV.3.1. — Let I be a bounded interval of R and let AI be the algebra AI = Hom(e∗ (kI ), e∗ (kI )). Then the morphism εI defined in (IV.3.2) is an algebra morphism and ker(εI ) is a nilpotent ideal of AI . Moreover, a morphism u ∈ AI is an isomorphism if and only if εI (u) ̸= 0, More precisely, if I is closed or open, then εI is an isomorphism. If I is half-closed, say I = [a, x[ or I = ]x, a] and we set Ea = I ∩ e−1 (e(a)), then the identification (e∗ (kI ))e(a) ≃ kE(a) induces a morphism AI → − Hom(kE(a) , kE(a) ),

φ 7→ φe(a)

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which identifies AI with the subalgebra of matrices generated by the standard nilpotent matrix of order |E(a)|. L n Proof. — Using e−1 e∗ (kI ) ≃ n∈Z T∗ (kI ) and Lemma IV.2.2, the cases I closed or open are obvious. If I is half-closed of length l, we find AI ≃ k|E(a)| as a vector space. Assuming I = [a, x[ (the case ]x, a] is similar), a basis of AI is given by the morphisms e∗ (un ), for n = 0, . . . , |E(a)| − 1, where un : kI → − kT n (I) is the natural morphism and we use the natural identification ϕn : e∗ (kT n (I) ) ≃ e∗ (kI ). At the level of stalks ϕn identifies the summands (kI )a+2πk of (e∗ (kI ))e(a) and (kT n (I) )a+2π(k+n) of (e∗ (kT n (I) ))e(a) . We obtain that (e∗ (un )) acts on kE(a) by (s1 , s2 , . . .) 7→ (s1+n , s2+n , . . .). We deduce that the image of AI in End(kE(a) ) is as claimed in the lemma. The characterization of the isomorphisms then follows from the structure of AI . Lemma IV.3.2. — Let F ∈ Mod(kS1 ). Let I be a bounded interval of R such that kI is a direct summand of e−1 (F ). Then e∗ (kI ) is a direct summand of F . Proof. — We let i0 : kI → − e−1 (F ) ≃ e! (F ) and p0 : e−1 (F ) → − kI be morphisms such that p0 ◦ i0 = idkI and we denote by i′0 : e∗ (kI ) ≃ e! (kI ) → − F and p′0 : F → − e∗ (kI ) their adjoint morphisms. In general p′0 ◦ i′0 ̸= ide∗ (kI ) but it is enough to see that p′0 ◦ i′0 is an isomorphism. For this we use Lemma IV.3.1. Let us compute εI (p′0 ◦ i′0 ). Let a : e−1 e∗ (kI ) → − kI and b : kI → − e−1 e! (kI ) be the adjunction morphisms. Then εI (p′0 ◦ i′0 ) = a ◦ e−1 (p′0 ◦ i′0 ) ◦ b = p0 ◦ i0 = idkI and we deduce that p′0 ◦ i′0 is an isomorphism. Lemma IV.3.3. — Let F ∈ Modc (kS1 ). We assume that F is not locally constant. Then there exists a bounded interval I of R such that kI is a direct summand of e−1 (F ). Proof. — By Corollary IV.2.1 there exist a locally finite family of intervals {Ia }a∈A L and integers {na }a∈A such that e−1 (F ) ≃ a∈A knIaa . Since F is not locally constant, one of these intervals, say I, is not R. Let T be the translation T (x) = x + 2π. Since T −1 e−1 (F ) ≃ e−1 (F ), the intervals T n (I) also appear in the decomposition, for all n ∈ Z. If I were not bounded, this would contradict the constructibility of F . Proposition IV.3.4. — Let F ∈ Modc (kS1 ). Then there exist a finite family {(Ia , na )}a∈A , of bounded intervals and integers, and a locally constant sheaf of finite rank L ∈ Mod(kS1 ) such that M (IV.3.3) F ≃L⊕ e∗ (knIaa ). a∈A

Proof. — (i) We choose a finite stratification {Σi }i∈I of S1 such that F is constructible with P respect to {Σi }. We choose one point θi ∈ Σi for each i ∈ I and set r(F ) = i∈I dim(Fθi ). We prove the proposition by induction on r(F ). If r(F ) = 0, then F ≃ 0 and the result is clear.

ASTÉRISQUE 440

CHAPTER IV.3. CONSTRUCTIBLE SHEAVES ON THE CIRCLE

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(ii) We assume r(F ) ̸= 0. If F is locally constant, the result is clear. Else, by Lemmas IV.3.3 and IV.3.2 there exists a bounded interval I of R such that F ≃ e∗ (kI )⊕F ′ for some F ′ ∈ Modc (kS1 ). Then r(F ′ ) < r(F ) and the induction proceeds.

SOCIÉTÉ MATHÉMATIQUE DE FRANCE 2023

CHAPTER IV.4 COHOMOLOGICAL DIMENSION 1

The decomposition results for sheaves in dimension 1 extend to the derived category. For M = R or M = S1 we denote by Dbc (kM ) the full subcategory of Db (kM ) formed by the F such that H i F ∈ Modc (kM ) for all i ∈ Z. We first recall a well-known decomposition result (see for example [29, Ex. 13.22]). Lemma IV.4.1. — Let C be an abelian category and X ∈ Db (C ) a complex such that Extk (H i X, H j X) ≃ 0 for all i,L j ∈ Z and all k ≥ 2. Then X is split, that is, there exists an isomorphism X ≃ i∈Z H i X[−i] in Db (C ). This applies in particular to constructible sheaves in dimension 1. Indeed, if M = R or M = S1 and k is a field, we have Extk (F, G) ≃ 0 for all F, G ∈ Modc (kM ) and for all k ≥ 2. We deduce: Lemma IV.4.2. — Let M L = R or M = S1 and let k be a field. Then, for all b F ∈ Dc (kM ) we have F ≃ i∈Z H i F [−i] in Dbc (kM ). Using Corollary IV.2.1 and Proposition IV.3.4 we obtain immediately: Corollary IV.4.3. — let k be a field. Let M be R or S1 and let F ∈ Db (kM ) be a constructible object. (i) If M = R, then there exist a locally finite family of intervals {Ia }a∈A and integers {na }a∈A , {da }a∈A such that M (IV.4.1) F ≃ knIaa [da ]. a∈A 1

(ii) If M = S , then there exist a finite family of bounded intervals {Ia }a∈A , integers {na }a∈A , {da }a∈A and L ∈ Db (kS1 ) with locally constant cohomology sheaves of finite rank such that M (IV.4.2) F ≃L⊕ e∗ (knIaa )[da ], a∈A 1

where e : R → − S ≃ R/2πZ is the quotient map.

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CHAPTER IV.4. COHOMOLOGICAL DIMENSION 1

The next lemma is related with the results of this section and was already used in Example II.1.4. This is a classical result (see [29, Ex. 13.20, 13.21]). Let C be an Abelian category. For A ∈ D(C ) we let Aut(A) ⊂ Hom(A, A) be the isomorphism group of A. For another B ∈ D(C ) the product Aut(A) × Aut(B) acts on Hom(A, B) by composition. We recall that we have truncation functors τ≤n , τ≥n : Db (C ) → − Db (C ) b together with morphisms of functors cn which yield for any X ∈ D (C ) and n ∈ Z a distinguished triangle (IV.4.3)

cn (X)

τ≤n (X) → − X→ − τ≥n+1 (X) −−−−→ τ≤n (X)[1].

n For A, B ∈ C and n ≥ 1, we let EA,B ⊂ Db (C ) be the full subcategory of objects n X such that H 0 (X) ≃ B, H n (X) ≃ A and H i (X) ≃ 0 for i ̸= 0, n. For X ∈ EA,B ∼ ∼ and isomorphisms a : A[−n] −−→ τ≥1 (X), b : B −−→ τ≤0 (X), we define ϕa,b (X) = (b[1])−1 ◦ c0 (X) ◦ a ∈ Hom(A[−n], B[1]) ≃ Hom(A, B[n + 1]).

Lemma IV.4.4. — Let C be an abelian category and let A, B ∈ C and n ≥ 1 be ¯ n be the set of isomorphism classes in E n . For a given X ∈ E n given. Let E A,B A,B A,B ∼ ∼ and isomorphisms a : A[−n] −− → τ≥1 (X), b : B −− → τ≤0 (X), the image of ϕa,b (X) in Hom(A, B[n + 1])/Aut(A) × Aut(B) is independent of the choice of a and b and ¯ n and Hom(A, B[n + 1])/Aut(A) × Aut(B). yields a bijection between E A,B Proof. — (i) Changing a and b modifies (b[1])−1 ◦c0 (X)◦a by the action of an element in Aut(A) × Aut(B). This proves that the image of (b[1])−1 ◦ c0 (X) ◦ a in the quotient only depends on X. n , then τ≥1 (u) and τ≤0 (u)[1] are iso(ii) If u : X → − Y is an isomorphism in EA,B morphisms and make a commutative square with c0 (X) and c0 (Y ). Hence c0 (X) is conjugate to c0 (Y ) through Aut(A) × Aut(B). ¯n → This defines a map c¯0 : E A,B − Hom(A, B[n + 1])/Aut(A) × Aut(B). n be given. If c¯0 (X) = c¯0 (Y ), then there exists (α, β) such that (iii) Let X, Y ∈ EA,B the square (S) below commutes:

τ≤0 (X)

X

≀ β[−1]

τ≤0 (Y )

τ≥1 (X) ≀ α

Y

τ≥1 (Y )

c0 (X)

(S) c0 (Y )

τ≤0 (X)[1] ≀ β

τ≤0 (Y )[1].

By the axioms of triangulated categories, we deduce that X ≃ Y . Hence c¯0 is injective. (iv) For ϕ ∈ Hom(A, B[n + 1]) we define Xϕ such that Xϕ [1] is the cone of ϕ. Then c¯0 (Xϕ ) = [ϕ] in Hom(A, B[n + 1])/Aut(A) × Aut(B). Hence c¯0 is surjective.

ASTÉRISQUE 440

PART V GRAPH SELECTORS Let M be a manifold and let Λ ⊂ J 1 (M ) be a closed Legendrian submanifold. We let Γ be the projection of Λ on M × R. A graph selector for Λ is a continuous function φ : M → − R whose graph is contained in Γ. If Λ is generic, then Γ is a union of transverse immersed hypersurfaces outside a set of codimension 1. In this case φ is differentiable on a dense open set where the graph of dφ is contained in the Lagrangian projection of Λ in T ∗ M . We see Λ as a closed conic Lagrangian submanifold of T˙ ∗ (M × R) as follows. We choose coordinates (t; τ ) on T ∗ R and denote by Tτ∗>0 (M × R) the open set {τ > 0} in T ∗ (M × R). Then the quotient by the multiplicative R>0 -action in the fibers gives an identification J 1 (M ) ≃ (Tτ∗>0 (M × R))/R>0 and we abusively write Λ for its inverse image by the quotient map; hence Λ ⊂ Tτ∗>0 (M × R). In this short section we prove that Λ has a graph selector as soon as it is the microsupport of a sheaf F satisfying some conditions at infinity; the graph of φ is then the boundary of the support of a section of F . Using Theorem XIII.5.1 below, we recover Theorem 1.2 of [5] which says that a compact exact Lagrangian submanifold of a cotangent bundle has a graph selector. For a map φ : M → − R we set Γφ = {t = φ(x)} and Γ+ φ = {t ≥ φ(x)}. We assume in this section that M is connected. For F ∈ D(kM ×R ) we consider the following condition: there exists A > 0 such that supp(F ) ⊂ M × [−A, +∞[ and (V.0.1) ˙ SS(F ) ⊂ Tτ∗>0 (M × [−A, A]). We recall that the category of sheaves Mod(kM ×R ) is embedded in its derived category D(kM ×R ) as the subcategory of complexes concentrated in degree 0. We remark the notion of support doesn’t make sense for a class s ∈ H i (M × R; F ) for a general F ∈ D(kM ×R ), but makes sense for s ∈ H 0 (M × R; F ) = F (M × R) when F ∈ Mod(kM ×R ). Proposition V.0.1. — Let F ∈ Mod(kM ×R ) and let s ∈ F (M ×R) be a non-zero section. We assume that F satisfies (V.0.1). Then there exists a unique map φ : M → − R such that supp(s) = Γ+ and this map is continuous. More precisely, for a given chart U φ

82

in M , which is identified with a ball of Rn with coordinates (x; ξ) on T ∗ U , if we have Pn 1 ˙ a bound SS(F ) ∩ T ∗ (U × R) ⊂ {τ ≥ C∥ξ∥} for some C > 0, where ∥ξ∥ = ( i=1 ξi2 ) 2 , then the map φ is C −1 -Lipschitz on U , that is, |φ(x) − φ(y)| ≤ C −1 ∥x − y∥, for any x, y ∈ U , where ∥ · ∥ is again the norm induced by the coordinates. The link between the Lipschitz condition and the assumption on the microsupport can also be found in [49] and [25]. Proof. — (i) To prove that supp(s) is of the form Γ+ φ it is enough to check that supp(s) ∩ ({x} × R) is an interval of the form [a, +∞[ for any x ∈ M . Let z = (x0 , t0 ) ∈ M × R such that sz ̸= 0. We set i : R → − M × R, t 7→ (x0 , t) and G = i−1 F . Then SS(G) ⊂ {τ ≥ 0} by the hypotheses on F and by Theorem I.2.8. The section s induces a section s′ of G over R and we have s′t = si(t) . Hence supp(s′ ) = i−1 (supp(s)) and this is non empty since sz ̸= 0. By Corollary I.2.16 the restriction map H 0 (]a, c[; G) → − H 0 (]b, c[; G) is an isomor′ phism for any a ≤ b < c. It follows that st = 0 implies s′t1 = 0 for all t1 ≤ t. Indeed, we can find a, b, c such that a < t1 , t1 < b < t < c and s′ | = 0. Then s′ | = 0 and ]b,c[ ]a,c[ ′ ′ in particular st1 = 0. Since supp(s ) is closed, non empty and contained in [A, +∞[, this proves that it must be an interval of the form [a, +∞[, as required. It remains to check that, for any x ∈ M , there exists t such that s(x,t) ̸= 0. The hypothesis on SS(F ) implies that F | , where V = M × ]A, +∞[, is a locally constant V sheaf. Since M is connected, the support of any global section of F | is either empty V or V . We have seen that supp(s) contains {x0 } × [a, +∞[ for some a. Hence it must − R. contain V . Finally we obtain that supp(s) = Γ+ φ for some function φ : M → (ii) Now we assume that we are given a chart U and C > 0 as in the second part of ˙ the statement; hence SS(F ) ∩ T ∗ (U × R) ⊂ (U × R) × {τ ≥ C∥ξ∥}. The chart U is identified with an open ball in Rn and we let γ ⊂ Rn+1 be the closed convex cone {t ≤ −C −1 ∥x∥}. We have γ ◦a = {τ ≥ C∥ξ∥}. By Remark III.4.4 Γ+ φ = supp(s) is locally γ-closed, that is, for any z ∈ U × R, there exists a neighborhood B of z (for n+1 which is closed for the usual topology) such that B ∩ Γ+ φ = B ∩ Z for some Z ⊂ R n+1 n+1 the γ-topology. By definition this means that R \ Z = (R \ Z) + γ; it implies Z = Z + γa. Let x ∈ U be given and z = (x, φ(x)). Hence there exists a neighborhood B a of z such that B ∩ Γ+ φ contains B1 = B ∩ (z + γ ) and is contained in B \ B2 , where B2 = B∩(z+γ). Setting W = p(B1 )∩p(B2 ) we thus have |φ(x) − φ(y)| ≤ C −1 ∥x − y∥ for all y ∈ W and we can see that W is a neighborhood of x in U . Since this holds for any x ∈ U , we can deduce that φ is C −1 -Lipschitz on U . Indeed for given x, x′ ∈ U , the segment [x, x′ ] is contained in U (U is a ball) and we can find finitely many points SN xi ∈ [x, x′ ] with neighborhoods Wi as above, i = 1, . . . , N , such that [x, x′ ] ⊂ i=1 Wi . Then the result follows from the triangular inequality. ˙ (iii) Since SS(F ) is conic and closed in T˙ ∗ (M × R) and contained in {τ > 0}, for any compact subset K of some coordinate chart of M × R we can find C > 0 such

ASTÉRISQUE 440

83

−1 ˙ that SS(F ) ∩ πM ×R (K) ⊂ K × {τ ≥ C∥ξ∥}. By (ii) it follows that φ is continuous everywhere.

In order to apply Proposition V.0.1 in the situation of Corollary V.0.4 we have to replace a complex of sheaves F ∈ D(kM ×R ) by a sheaf in Mod(kM ×R )—we will take H 0 F —and ensure that our sheaf has a section. The next lemma implies that H 0 F still satisfies the hypothesis of the proposition and Lemma V.0.3 deals with the sections. Lemma V.0.2. — Let F ∈ D(kM ×R ) which satisfies (V.0.1). Then, for any i ∈ Z, the i ˙ ˙ )). sheaf H i F also satisfies (V.0.1) and moreover we have πM ×R (SS(H F )) ⊂ πM ×R (SS(F Proof. — (i) Let us check (V.0.1) for H i F . We have the general inclusion i ˙ supp(H i F ) ⊂ supp(F ). Corollary III.4.3 gives SS(H F )) ⊂ Tτ∗>0 (M × R). Since ˙ SS(F ) is contained in T ∗ (M × [−A, A]), F is locally constant outside M × [−A, A] i ˙ and so is H i F . Hence SS(H F ) is contained in T ∗ (M × [−A, A]) and we have (V.0.1). (ii) Now we prove the last assertion. We recall that, for any G ∈ D(kM ×R ), a point x is ˙ not in πM ×R (SS(G)) if and only if G is constant in a neighborhood of x (see Exam˙ ple I.2.3-(i)). Now, if F is constant, so is H i F . Hence, if x ̸∈ πM ×R (SS(F )), we have i ˙ x ̸∈ πM ×R (SS(H F )), as required. Lemma V.0.3. — Let F ∈ D(kM ×R ) which satisfies (V.0.1). Then the restriction morphism RΓ(M × R; F ) → − RΓ(M × ]B, +∞[; F ) is an isomorphism, for any B ∈ R. We remark that if M were compact, this would follow directly from Corollary I.2.16. Proof. — We set Z = M × ]−∞, B] and U = M × ]B, +∞[. We have to prove that r : RΓ(M × R; F ) → − RΓ(M × R; RΓU (F )) is an isomorphism. The cone of r is RΓ(M × R; RΓZ (F )). Let p : M × R → − M be the projection. It is enough to prove Rp∗ RΓZ (F ) ≃ 0. We remark that p is proper on supp(RΓZ (F )). For a given x ∈ M and ix : R → − M × R, t 7→ (x, t), we set G = i−1 x RΓZ (F ). Then the base change formula gives (Rp∗ RΓZ (F ))x ≃ RΓ(R; G). By Theorems I.2.13 and I.2.8 we have ∼→ RΓ(]a, +∞[; G) SS(G) ⊂ {τ ≥ 0}. Since G has compact support, we have RΓ(R; G) −− ∼ for some a ∈ R. By Corollary I.2.16 we also have RΓ(]a, +∞[; G) −−→ RΓ(]b, +∞[; G) for a ≤ b and this vanishes for b ≫ 0 since supp(G) is compact. Corollary V.0.4. — Let Λ ⊂ Tτ∗>0 (M × R) be a closed conic Lagrangian submanifold. We assume that Λ ⊂ Tτ∗>0 (M × [−A, A]) for some A and that there ˙ exists F ∈ D(kM ×R ) such that SS(F ) = Λ, supp(F ) ⊂ M × [−A, +∞[ and F| ≃ kM ×]A,+∞[ . Then Λ has a graph selector: there exists a continuous M ×]A,+∞[ map φ : M → − R such that Γφ ⊂ πM ×R (Λ). Moreover, as in Proposition V.0.1, for a given chart U in M , which is identified with a ball of Rn with coordinates (x; ξ) on T ∗ U , if we have a bound Λ ∩ T ∗ (U × R) ⊂ {τ > C∥ξ∥} for some C > 0, then φ is C −1 -Lipschitz on U .

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Proof. — (i) The sheaf F | ≃ kM ×]A,+∞[ ≃ H 0 F | has a M ×]A,+∞[ M ×]A,+∞[ section over M × ]A, +∞[, say s, corresponding to 1 ∈ k. By Lemma V.0.2 H 0 F satisfies (V.0.1) and by Lemma V.0.3 the section s can be extended to s ∈ H 0 (M × R; H 0 F ). Proposition V.0.1 associates a continuous function φ to s and we have Γφ ⊂ πM ×R (Λ) by Lemma V.0.2 again. (ii) Let Ξ ⊂ T ∗ (M ×R) be the convex hull of Λ in the sense that Ξ is the intersection of all closed conic subsets S of T ∗ M which contain Λ and are fiberwise convex. For a local chart as in the statement we also have Ξ ∩ T ∗ U ⊂ {τ > C∥ξ∥}. By Corollary III.4.3 0 ˙ we have SS(H F ) ⊂ Ξ. Hence the Lipschitz constant given in Proposition V.0.1 0 for H F is the same as the one given in the current corollary. An example of a sheaf satisfying the hypotheses of Corollary V.0.4 is given by ∗ Corollary II.1.5 as follows. We start with Λ0 = TM ×{0} (M × R) which is the microsupport of F 0 = kM ×[0,+∞[ . Let Φt , t ∈ I, where I is an open interval containing [0, 1], be a homogeneous Hamiltonian isotopy of T˙ ∗ (M × R) which preserves T˙τ∗>0 (M × R) (for example the homogeneous lift of a Hamiltonian isotopy of T ∗ M , whose support is proper over M , or the lift of a contact isotopy of J 1 M ). Corol˙ t ) = Φt (Λ0 ), where lary II.1.5 gives a sheaf F on M × R × I with F0 = F 0 and SS(F ˙ Ft = F | . We remark that SS(F ) ∩ T ∗ N = ∅ for N = M × ]B, +∞[ × [0, 1] and M ×R×{t} B ≫ 0. Hence F | is locally constant. Since F0 | ≃ kM ×]B,+∞[ , we deduce N M ×]B,+∞[ that F1 | ≃ kM ×]B,+∞[ . So Corollary V.0.4 implies that Φ1 (Λ0 ) has a graph M ×]B,+∞[ selector. More generally, using Corollary V.0.4 and Theorem XIII.5.1 below, we recover Theorem 1.2 of [5]: Corollary V.0.5. — Let Λ ⊂ Tτ∗>0 (M × R) be a closed conic Lagrangian submanifold which is the conification of a compact exact Lagrangian submanifold of T ∗ M . Then Λ has a graph selector.

ASTÉRISQUE 440

PART VI THE GROMOV NONSQUEEZING THEOREM

In this part we use the microlocal theory of sheaves to give a proof of the famous Gromov nonsqueezing theorem (see [18]), which says that there is no symplectomorphism of R2n which sends the ball of radius R into a cylinder Dr × R2n−2 , where Dr is the disk of radius r, if r < R. There is already a proof with generating functions by Viterbo [50] and it is no wonder that we can also find a proof with sheaves. The proof we give is inspired by the papers of Chiu [11], to define a projector associated with a square in §VI.1, and Tamarkin [45] to define a displacement energy (see Definition VI.2.1—this displacement energy is also used in [7]). It is in fact a baby case of the main result of [11] which is the nonsqueezing in the contact setting (see also [54] for a survey of Tamarkin’s and Chiu’s results and [14] for another proof). We also give a non squeezing result for a Lagrangian submanifold of the ball; this is related with a result of Théret in [46].

CHAPTER VI.1 CUT-OFF IN FIBER AND SPACE DIRECTIONS

In this section we prove Corollary VI.1.6 which says that if a sheaf F on Rn has its microsupport contained in some prescribed cone, then the natural morphism τc (F ) of (III.5.7) vanishes for c big enough. This will be used in the next section to recover classical non squeezing results. The idea is to construct explicitly a sheaf K∞ on R2n ∼ such that F ◦ K∞ −− → F for F as above and check that τc (K∞ ) vanishes. In Part III we have seen several functors on the category of sheaves whose effect is to change the microsupport and make it avoid some given set. For a vector space V and a closed convex cone γ ⊂ V , we have seen Pγ : D(kV ) → − D(kV ) which is a projector, in the sense that Pγ ◦ Pγ ≃ Pγ , and satisfies SS(Pγ (F )) ⊂ V × γ ◦a . More usual functors reduce the support of a sheaf, for example D(kM ) → − D(kM ), F 7→ FZ , for a locally closed set Z ⊂ M . This functor is also a projector and satisfies SS(FZ ) ⊂ Z × V ∗ . If Z is closed, a sheaf satisfying SS(F ) ⊂ Z × γ ◦a is stable by both functors Pγ and (·)Z , hence by the composition Q : F 7→ Pγ (FZ ). This Q is not a projector but we will see in a special case that Q◦i converges when i → − ∞ and gives a projector (in fact we will work with a variant of Q). The construction we give in this section is in fact a baby case of a construction by Chiu in [11] who defines a projector corresponding to a subset C of T ∗ Rn+1 which is the cone over a ball in T ∗ Rn ; here we do the case where C is the cone over a square [−1, 1]2 in T ∗ R = R × R∗ . We will use this projector to recover nonsqueezing results in the symplectic case. It will be more convenient to use a composition functor F 7→ F ◦ kγe rather than Pγ because the composition is associative. This is similar to the Tamarkin projector of §III.5: see Remark III.5.2 where this functor is denoted Lγ a . We recall that, for a manifold M and a conic subset A ⊂ T ∗ M , DA (kM ) is the full subcategory of D(kM ) formed by the F with SS(F ) ⊂ A (see Notation I.2.2). The composition − ◦ kγe is still a projector and its image is D⊥,l Aγ (kV ), the left orthogonal of DAγ (kV ), where Aγ = V × (V ∗ \ Int(γ ◦a )). We have D⊥,l Aγ (kV ) ⊂ DV ×γ ◦a (kV ). (See Remark III.5.2 and [22, Prop. 4.10].) However we don’t use the results of [22]; the microsupport estimates will rely on the properties of Pγ and Lemma III.1.1.

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CHAPTER VI.1. CUT-OFF IN FIBER AND SPACE DIRECTIONS

We introduce some notations. Let n ≥ 2 be given. We consider the vector spaces V ′ = Rn−2 , V = R2 × V ′ and put coordinates (x; ξ) on T ∗ V . We let γ ⊂ V be the cone γ = {(x) ∈ Rn ; x2 ≤ −|x1 |, x3 = · · · = xn = 0}. ◦ ∗ Hence γ ⊂ V is given by γ ◦ = {(ξ) ∈ Rn ; ξ2 ≤ −|ξ1 |}. We recall the notation γ e = {(x, y) ∈ V 2 ; x − y ∈ γ} of §III.1. We let Z ⊂ V be the open strip Z = ]−1, 1[ × Rn−1 . Let Rγ , RZ : D(kV ) → − D(kV ) be the functors Rγ (F ) = F ◦ kγe and RZ (F ) = FZ . They come with natural morphisms Rγ (F ) → − F and RZ (F ) → − F . We remark that RZ can also be written as a composition RZ (F ) ≃ F ◦ kδV (Z) , where δV is the diagonal embedding. We define R = RZ ◦ Rγ ◦ RZ and we find R(F ) ≃ F ◦ kW ,

(VI.1.1) where W =γ e ∩ (Z × Z)

= {(x, y) ∈ R2n ; y2 − x2 ≥ |x1 − y1 |, |x1 | < 1, |y1 | < 1, xi = yi , i = 3, . . . , n}. Since W ∩∆V is closed in W and open in ∆V we have a natural morphism kW → − k ∆V ∼ which gives a morphism of functors R → − id. If a sheaf satisfies Rγ (F ) −− → F and ∼ ∼ ∼ FZ −− → F , then R(F ) −− → F and R◦i (F ) −− → F for all i ∈ N. We now compute R◦i ; of course this is the composition with Ki = kW ◦ · · · ◦ kW

(i factors kW ).

For c ∈ R we define f, s, Tc : V → − V by f (y) = (−y1 , y2 + 2, y3 , . . . , yn ) s(y) = (y1 , −y2 , y3 , . . . , yn ) Tc (y) = (y1 , y2 + c, y3 , . . . , yn ). 2

Let Z± ⊂ R be the closed half planes Z± = {y2 ≥ ±y1 }. We have natural morphisms u± : kR2 → − kZ± and we define L ∈ D(kR2 ) by (VI.1.2)

L=0→ − kR2 → − kZ+ ⊕ kZ− → − 0,

where kR2 is in degree 0. We have H 0 L ≃ kR2 \(Z+ ∪Z− ) , H 1 L ≃ kZ+ ∩Z− and a distinguished triangle kR2 \(Z+ ∪Z− ) → − L→ − kZ+ ∩Z− [−1] → − kR2 \(Z+ ∪Z− ) [1]. (In fact we have already met L in (II.1.4): we have L ≃ KΨ | in the case n = 1.) 2 {0}×R

Lemma VI.1.1. — We set C1 = W ∩ (idV × f )((idV × s)(Int(W ))) and W2 = (idV × f )(W ). The sheaf K2 = kW ◦ kW appears in a distinguished triangle (VI.1.3)

ASTÉRISQUE 440

u

kC1 − → K2 → − kW2 [−1] → − kC1 [1]

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89

and, in a small enough neighborhood of {y1 = −x1 , y2 = x2 + 2} × ∆V ′ , we have K2 ≃ p−1 (L) ⊠ k∆V ′ , with p : R4 → − R2 , (x1 , x2 , y1 , y2 ) 7→ (x1 + y1 , y2 − x2 − 2). ∼ Moreover kC1 ◦ kW −−→ kC1 . Proof. — (i) Let us forget the variables xi , yi for i ≥ 3. Let q : R6 → − R4 be the 2 projection q(x, y, z) = (x, z). Then K2 = Rq! (kA ) where A = (W × R ) ∩ (R2 × W ). For a subset E ⊂ R6 and (x, z) ∈ R4 we set E(x,z) = E ∩ q −1 (x, z). If x ̸∈ Z or z ̸∈ Z ∼ we have A(x,z) = ∅. This proves that (K2 )Z×Z −− → K2 . Now we assume (x, z) ∈ Z 2 2 and we find A(x,z) = {y ∈ R ; y2 − x2 ≥ |x1 − y1 |, |y1 | < 1, z2 − y2 ≥ |y1 − z1 |}. Then A(x,z) is a bounded convex polytope, but the cohomology of kA(x,z) depends on (x, z) because we have two types of boundary conditions (open or closed). (ii) We define A′ , A± ⊂ q −1 (Z 2 ) ⊂ R6 by their fibers A′(x,z) = A(x,z) and A± (x,z) = ∂A(x,z) ∩ {y1 = ±1}. We have an exact sequence 0 → − kA → − kA′ → − kA+ ⊕ kA− → − 0. Now the fibers of A′ , A± are always compact convex polytopes. For any such polytope B we have Rq! (kB ) ≃ kq(B) and we deduce a resolution of K2 as the complex d

1 kq(A+ ) ⊕ kq(A− ) → − 0. We have q(A′ ) = W and 0→ − kq(A′ ) −→

q(A± ) = {(x, z) ∈ R4 ; z2 − x2 ≥ 2 ∓ (x1 + z1 ), |x1 | < 1, |z1 | < 1}. Hence ker(d1 ) ≃ kC1 , coker(d1 ) = kW2 and, up to a change of coordinates, we have the complex (VI.1.2) defining L. (iii) To prove the last assertion we can compute kC1 ◦ kW directly or use the fact that (kC1 )Z ≃ kC1 , because C1 ⊂ Z 2 , and Rγ (kC1 ) ≃ kC1 , because C1 is relatively compact and SS(kC1 ) ⊂ T ∗ V × (V × γ ◦a ). We deduce kC1 ◦ kW ≃ kC1 and this proves the lemma. We can see that − ◦ kW commutes with the translation Tc . More precisely Tc∗ (F ) ◦ kW ≃ Tc∗ (F ◦ kW ) ≃ F ◦ k(idV ×Tc )(W ) . The same holds for the map f and, since W2 = (idV × f )(W ), we can deduce kW2 ◦ kW from the triangle (VI.1.3). We find a similar triangle (VI.1.4)

kC2 → − k W2 ◦ k W → − kW3 [−1] → − kC2 [1],

∼ where C2 = (idV × f )(C1 ), W3 = (idV × f )(W2 ). Since kC1 ◦ kW −− → kC1 , applying − ◦ kW to (VI.1.3) gives the triangle (VI.1.5)

kC1 → − K3 → − kW2 ◦ kW [−1] → − kC1 [1].

The triangle (VI.1.4) implies that H 0 (kW2 ◦ kW ) ≃ kC2 and H 1 (kW2 ◦ kW ) ≃ kW3 . Then (VI.1.5) gives H 0 (K3 ) ≃ kC1 , H 1 (K3 ) ≃ kC2 and H 2 (K3 ) ≃ kW3 . Now we can compute kW3 ◦ kW in the same way and an induction gives the following result. Proposition VI.1.2. — For i ≥ 1 we Wi = (idV × f )i−1 (W ). Then, for any n ≥ 1,

define

Ci = (idV × f )i (C1 )

and

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(i) we have a distinguished triangle u

kC1 − → Kn+1 → − (idV × f )∗ (Kn )[−1] → − kC1 [1], (ii) Kn is concentrated in degrees 0, . . . , n−1 and H i Kn ≃ kCi+1 , for i = 0, . . . , n−2, H n−1 Kn ≃ kWn , (iii) setting ∆ti := {y1 = (−1)i x1 , y2 = x2 + 2i} × ∆V ′ and pi : R4 → − R2 , i (x1 , x2 , y1 , y2 ) 7→ (x1 − (−1) y1 , y2 − x2 − 2i), we have, in a small enough neighborhood of ∆ti and for i = 1, . . . , n − 1: Kn ≃ (p−1 i (L) ⊠ k∆V ′ )[1 − i], (iv) setting Yi = {y2 − x2 ≥ 2i}, we have, for i = 1, . . . , n − 1, (Kn )Yi ≃ (idV × f )i∗ (Kn−i )[−i]. Sn−1 Lemma VI.1.3. — We set Sn = supp(Kn ) = Wn ∪ i=1 Ci . Then we have an iso∼ morphism kSn −− → RHom (Kn , Kn ) mapping 1 to the identity morphism. Proof. — (i) The statement is local on V 2 . In a neighborhood of a point x0 which is away from the sets ∆ti defined in Proposition VI.1.2 our sheaf Kn is up to shift a constant sheaf on some subset Y of V 2 with Y = Ci or Y = Wn . Since the sets Ci and Wn are locally closed convex, we get RHom (Kn , Kn ) ≃ kSn near x0 . (ii) In a neighborhood of some ∆ti the lemma is reduced to the proof of RHom (L, L) ≃ kD , where D = supp(L). Let us set H = RHom (L, L). Since the morphism kD → − H is given, it only remains to check Hx ≃ k at any point x ∈ D. This is clear for x ̸= 0. We can compute H0 directly from the Definition (VI.1.2). Alternatively we can remark that SS(L) = ΛΨ ◦a T˙0∗ R (see the notation (II.1.5)), where Ψ is the Hamiltonian isotopy of T˙ ∗ R defined by Ψs (x; ξ) = (x − s ξ/|ξ|; ξ). It follows from Corollary II.1.5 that the restriction morphisms RHom(L, L) → − RHom(Ls , Ls ), where Ls = L| are all isomorphisms. We deduce that RΓ(R2 ; H) ≃ k. Since R×{s} H is a conic sheaf, we have H0 ≃ RΓ(R2 ; H), hence H0 ≃ k. We define an increasing sequence of open subsets Ui = {y2 − x2 < 2i} ⊂ R2n , i ∈ Z. By Lemma VI.1.3 the natural morphism uj : Kj+1 → − Kj , induced by kW → − k ∆V , ∼ gives an isomorphism Kj+1 | −− → Kj | if j > i. Hence we can define a sheaf Ui Ui K∞ ∈ D(kR2n ) such that K∞ | ≃ Kj | if j > i, for example as follows. We Ui

Ui

set Ui′ = Ui \ Ui−1 and Ui′′ = Ui \ Ui−2 . The natural restriction morphism and uj ′ ′ ′′ induce vi : (Ki+1 )Ui−1 → − (Ki+1 )Ui′′ and vi′ : (Ki+1 )Ui−1 → − ⊕(Ki )Ui−1 whose restric′ tions to Ui−1 are isomorphisms. Now we define K∞ by the distinguished triangle ! M M M vi−1 v +1 ′ ′′ (VI.1.6) (Ki )Ui−2 − → (Ki )Ui−1 → − K∞ −−→, v= . ′ vi−1 i i∈Z i∈Z

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Proposition VI.1.4. — We set S∞ = supp(K∞ ) = Then we have

S∞

i=1

91

Ci and Yi = {y2 − x2 ≥ 2i}.

RHom (K∞ , K∞ ) ≃ kS∞ and (K∞ )Yi ≃ RΓInt(Yi ) (K∞ ) ≃ (idV × f )i∗ (K∞ )[−i], for i ≥ 1. In particular RHom(K∞ , T4∗ (K∞ )) ≃ k[2].

(VI.1.7)

Proof. — The first assertion follows from Lemma VI.1.3. Proposition VI.1.2-(iv) gives (K∞ )Yi ≃ (idV × f )i∗ (K∞ )[−i]. The inductive description of Kn in (i) of the same ˙ proposition shows that SS(K∞ ) ⊂ {ξ2 + η2 ≥ 0}. Hence SS(K ∞ ) does not meet SS(kInt(Yi ) ) and Theorem I.2.13 gives (K∞ )Yi ≃ RΓInt(Yi ) (K∞ ). For the last assertion we remark that T4 = (idV × f )2 . Hence RHom (K∞ , T4∗ (K∞ )) ≃ RHom (K∞ , RΓInt(Y2 ) (K∞ )[2]) ≃ RΓInt(Y2 ) RHom (K∞ , K∞ [2]) ≃ RΓInt(Y2 ) (kS∞ )[2] ≃ (kS∞ ∩Y2 )[2] and the result follows. The morphism kW → − k∆V gives by iteration Ki → − k∆V and then K∞ → − k∆V . In particular for any F ∈ D(kV ) we have a natural morphism F ◦ K∞ → − F. We recall that we used the composition functor F 7→ F ◦ kγe rather than the cutoff functor Pγ since the composition is associative. The difference between these two functors is a switch between proper and usual direct image in the Definition (I.5.1) of the composition. Here we will also use the usual direct image. For later use we introduce a space of parameters which is a manifold N . Let q1 , q2 : N × V 2 → − N ×V and q3 : N × V 2 → − V 2 be the projections. For F ∈ D(kN ×V ) we define F ◦ K∞ , F ◦np K∞ ∈ D(kN ×V ) by (VI.1.8)

F ◦ K∞ = Rq2 ! (q1−1 F ⊗ q3−1 K∞ ),

(VI.1.9)

F ◦np K∞ = Rq2 ∗ (q1−1 F ⊗ q3−1 K∞ ).

Proposition VI.1.5. — Let F ∈ D(kN ×V ). We assume SS(F ) ⊂ T ∗ N × (V × γ ◦a ) and ∼ → F. supp(F ) ⊂ N × Z (recall Z = ]−1, 1[ × Rn−1 ). Then F ◦np K∞ −− 0 If, moreover, there exists x2 ∈ R such that supp(F ) ⊂ {x2 ≥ x02 }, then ∼ F ◦ K∞ −− → F. Proof. — The second assertion is a particular case of the first one, since the hypothesis gives that the map q2 in (VI.1.9) is proper on the support of q1−1 F ⊗ q3−1 K∞ . However we first prove the second assertion in (i) and (ii) below and we deduce the first one in (iii). (i) To prove that the morphism F ◦ K∞ → − F is an isomorphism, it is enough to see that the restrictions to {p} × V , for any p ∈ N , are isomorphisms. Using the base

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change formula and the bound of Theorem I.2.8 for SS(F | ), this means that we {p}×V can assume that N is a point. ∼ By Proposition III.1.7 we have Pγ (F ) −− → F , where Pγ is defined in (III.1.1) −1 by Pγ (F ) = Rq2 ∗ (kγe ⊗ q1 F ). Since supp(F ) ⊂ {x2 ≥ x02 } we can replace Rq2 ∗ ∼ by Rq2 ! in this expression and we find F ◦ kγe −− → F. ∼ We also have FZ −− → F . Indeed, the support condition implies F ≃ FZ and RΓ{φ≥0} F ≃ F where φ(x) = 1 + x1 . Now the microsupport condition implies (RΓ{φ≥0} F )x ≃ 0 if x ∈ {x1 = −1}. Hence F | ≃ 0. The same result holds {x1 =−1} ∼ for x1 = 1 and we get F | ≃ 0. Finally FZ −− → F as claimed. ∂Z ∼ ∼ We deduce F ◦ kW −− → F . Iterating i times we get F ◦ Ki −− → F for all i ∈ N. (ii) We recall that K∞ | ≃ Kj | if j > i, where Ui = {y2 − x2 < 2i} ⊂ R2n , i ∈ Z. Ui Ui For any sheaf K on V 2 supported in V 2 \ Ui we have supp(F ◦ K) ⊂ {x2 ≥ x02 + 2i}. +1 Using the triangle (K∞ )Ui → − K∞ → − (K∞ )V 2 \Ui −−→ and the same one with K∞ replaced by Ki , we deduce that, for any given x′ ∈ V , if j > i > x′2 − x02 , we have ∼ ∼ (F ◦K∞ )x′ −− → (F ◦(K∞ )Ui )x′ ≃ (F ◦(Kj )Ui )x′ ← −− (F ◦Kj )x′ . Hence (F ◦K∞ )x′ ≃ Fx′ which proves that the morphism F ◦ K∞ → − F is an isomorphism. (iii) Now we deduce the first assertion. The idea is to write F as the limit of FZk , k ∈ N, where Zk = {x2 ≥ −k}, and check that − ◦np q3−1 K∞ commutes with limits. Since FZk ◦np q3−1 K∞ ≃ FZk ◦ q3−1 K∞ , the result follows. Since we work in the derived category (and not the dg-derived category), we cannot Q u Q +1 use limits. Instead we use the distinguished triangle F → − k∈N FZk − → k∈N FZk −−→, where u = id − s and s is the product of the natural morphisms FZk → − FZk−1 . −1 It is then enough to check that − ◦np q3 K∞ commutes with products. The Definition (VI.1.9) is the composition of the three functors q1−1 , ⊗ and Rq2 ∗ . Since q1 is a submersion we have q1−1 ≃ q1! [−n] and this is a right adjoint. Hence q1−1 commutes with products; the same holds for Rq2 ∗ . The tensor product is not a right adjoint, but we can write q1−1 F ⊗ q3−1 K∞ ≃ RHom (D′ (q3−1 K∞ ), q1−1 F ) and q1−1 FZk ⊗ q3−1 K∞ ≃ RHom (D′ (q3−1 K∞ ), q1−1 FZk ). Indeed the microsupports of F , kZk and K∞ are all contained in {ξ2 > 0} (away from the zero section) and K∞ is constructible. Hence the claimed isomorphisms follow from several applications of Theorem I.2.13. Now we obtain F ◦np q3−1 K∞ ≃ Rq2 ∗ (RHom (D′ (q3−1 K∞ ), q1! F [−n])) (and the same with FZk instead of F ) and this is a composition of right adjoint functors. Hence it commutes with products and the result follows. For a real number c we let Tc : N ×V → − N ×V be the translation along the direction y2 of V , Tc (p, y1 , . . . , yn ) = (p, y1 , y2 + c, y3 , . . . , yn ) (p ∈ N ). For F ∈ D(kN ×V ) such that SS(F ) ⊂ {η2 ≥ 0} we let τc (F ) : F → − Tc∗ (F ) be the morphism (III.5.7). Corollary VI.1.6. — We have τc (K∞ ) = 0 for all c ≥ 4. In particular, for any manifold N and F ∈ D(kN ×V ) such that SS(F ) ⊂ T ∗ N × (V × γ ◦a ) and supp(F ) ⊂ N × Z, we have τc (F ) = 0 for all c ≥ 4.

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93

Proof. — By (VI.1.7) we have Hom(K∞ , Tc∗ K∞ ) ≃ 0 if c ≥ 4 and a fortiori ∼ τc (K∞ ) = 0. The second assertion then follows from the isomorphism F ◦np K∞ −− →F of Proposition VI.1.5 and the fact that τc (F ) coincides with idF ◦np τc (K∞ ).

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CHAPTER VI.2 NONSQUEEZING RESULTS

Here we use Corollary VI.1.6 to prove classical nonsqueezing results in the symplectic case. The morphism τc of (III.5.6), introduced by Tamarkin in [45], gives the following invariant, that we can call a displacement energy. We refer to [7] where a similar (more refined) invariant is used to obtain bounds on the displacement energy of some subsets of a cotangent bundle and to [54] for a survey of Tamarkin’s and Chiu’s results. We put coordinates (t; τ ) on T ∗ R. For a manifold M we thus consider {τ ≥ 0} as a subset of T ∗ (M × R) and we denote by Dτ ≥0 (kM ×R ) the category of sheaves F on M × R with SS(F ) ⊂ {τ ≥ 0}. Definition VI.2.1. — Let M be a manifold and F ∈ Dτ ≥0 (kM ×R ). We set e(F ) = sup{c ≥ 0; τc (F ) ̸= 0}. We check in Proposition VI.2.3 below that e(F ) is invariant by Hamiltonian isotopies of T ∗ M with compact support. We can reformulate Corollary VI.1.6 as follows: for any F ∈ D(kN ×V ) such that SS(F ) ⊂ (N × Z) × (V × γ ◦a ) we have e(F ) ≤ 4. Remark VI.2.2. — The morphism τc (F ) for F ∈ Dτ ≥0 (kM ×R ) is functorial in M : if N ⊂ M is a submanifold, then τc (F | ) = (τc (F ))| . This implies N ×R N ×R e(F | ) ≤ e(F ). N ×R

VI.2.1. Invariance of the displacement energy Let M be a manifold and let h : T ∗ M × I → − R be a function of class C ∞ . We ∗ assume that its Hamiltonian flow Φ : T M × I → − T ∗ M is defined and has compact support. As in Proposition II.3.1 we associate with h a homogeneous Hamiltonian isotopy Φ′ : T˙ ∗ (M × R) × I → − T˙ ∗ (M × R) lifting Φ and we let KΦ′ ∈ D(k(M ×R)2 ×I ) be the sheaf given by Theorem II.1.1. The isotopy Φ′ preserves the subset {τ ≥ 0} of T˙ ∗ (M × R) and commutes with the transpose derivative of Tc acting on T˙ ∗ (M × R) (also denoted Tc abusively, so Tc (x, t; ξ, τ ) = (x, t + c; ξ, τ )) for all c ∈ R. In other words Φ′ = T−c ◦ Φ′ ◦ Tc and the functor KΦ′ ◦ − coincides with T−c ∗ ◦ (KΦ′ ◦ −) ◦ Tc∗ . We obtain: for any F ∈ Dτ ≥0 (kM ×R ) we have KΦ′ ◦ F ∈ Dτ ≥0 (kM ×R×I ) and an isomorphism α : KΦ′ ◦ Tc∗ F ≃ Tc∗ (KΦ′ ◦ F ). We can then ask if the image of τc (F )

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by the composition functor KΦ′ ◦ − coincides with τc (KΦ′ ◦ F ) through α. To see this we recall the equivalence (II.1.6) of Corollary II.1.5. ∗ ∗ We use (II.1.6) with N = M × R, A = {τ ≥ 0} \ TM ×R (M × R) ⊂ T (M × R) and ′ ∗ ′ A ⊂ T (M × R × I) associated with A by Φ in the sense of (II.1.5). We let A′′ be the union of A′ and the zero section. Then A′′ is half of the hypersurface of T ∗ (M ×R×I) defined by the graph of the lift of h: A′′ = {((x, t; ξ, τ ), (s; −τ h(x, ξ/τ, s))); τ ≥ 0}. However we do not need a precise expression of A′ , we only need to know that i♯s (A′ ) = Φ′s (A) coincides with A, for all s ∈ I, where is is the inclusion of M × R × {s} in M × R × I. Corollary II.1.5 says that, for any s ∈ I, the inverse image functor i−1 − Dτ ≥0 (kM ×R ) s : DA′′ (kM ×R×I ) →

(VI.2.1)

is an equivalence of categories. Now we have A′′ ⊂ {τ ≥ 0} and any G ∈ DA′′ (kM ×R×I ) also comes with the morphism τc (G) : G → − Tc∗ G. By the functoriality of τc (Re−1 −1 mark VI.2.2) we have τc (i−1 s (G)) = is (τc (G)). Since is is an equivalence, we deduce −1 that, for any s ∈ I, τc (is (G)) vanishes if and only if τc (G) vanishes. We thus get the invariance of the displacement energy: Proposition VI.2.3. — Let Φ′ : T˙ ∗ (M × R) × I → − T˙ ∗ (M × R) be a homogeneous Hamiltonian isotopy lifting some Hamiltonian isotopy of T ∗ M in the sense of Proposition II.3.1. Let F ∈ Dτ ≥0 (kM ×R ). Then e(F ) = e(KΦ′ ◦ F ) = e(KΦ′ ,s ◦ F ), for any s ∈ I.

VI.2.2. Nonsqueezing for a flying saucer We put the natural Euclidean structure on Rn and T ∗ Rn ≃ R2n and we denote by B1 (E) and S1 (E) the closed unit ball and unit sphere of an Euclidean space E. We first define a subset Λ0 of S1 (T ∗ Rn ) which is the image of a Legendrian of J 1 (Rn ) whose front projection in Rn+1 is a flying saucer with conic points. We define Λ0 as the union of the graphs of the differential of two functions f1 , f2 : B1 (Rn ) → − R. We choose these functions to be rotation invariant with a differential belonging to the unit sphere of T ∗ Rn . In other words we write fi (x) = gi (∥x∥) for a function gi such that r2 + (gi′ (r))2 = 1. This determines fi up to a constant and we R ∥x∥ √ find f1 (x) = 0 1 − u2 du and f2 (x) = π/2 − f1 (x). We let W0 = {f1 (∥x∥) ≤ t < f2 (∥x∥)} be the region in Rn+1 bounded by the graphs of f1 , f2 and define ˙ W )) ⊂ T ∗ Rn . (VI.2.2) Λ0 = ρRn (SS(k 0

The functions fi are not differentiable at 0 and W0 has a conic point at (0, 0) and (0, π/2): ∗ ∗ SS(kW0 ) ∩ T(0,0) Rn+1 = SS(kW0 ) ∩ T(0,π) Rn+1 = {(ξ, τ ); τ ≥ ∥ξ∥}

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97

x for a non zero x. We deduce and we have dfi,x = gi′ (∥x∥) · ∥x∥ ( Λ0 = L0 ∪ B1 (T0∗ Rn ), (VI.2.3) L0 := {(x; ξ) ∈ S1 (T ∗ Rn ); ξ is a scalar multiple of x}.

We remark that L0 is the image of S1 (T0∗ Rn ) by the characteristic flow of S1 (T ∗ Rn ). More precisely we S1 (T ∗ Rn ) = {h = 1} where h(x; ξ) = ∥x∥2 + ∥ξ∥2 . The P have ∂ flow of Xh = 2 i ξi ∂xi − xi ∂ξ∂ i has orbits of period π. Identifying T ∗ Rn with Cn by (x; ξ) 7→ x + iξ the flow is given by Φh,s (x +P iξ) = exp(2si) · (x + iξ). For an orbit R γ of the flow the action is A = γ α where α = ξi dxi is the Liouville form. Here we find A = π for all orbits in S1 (T ∗ Rn ). We thus have (VI.2.4)

L0 = {exp(2si) · (0; ξ) ∈ T ∗ Rn ≃ Cn ; s ∈ [0, π], ∥ξ∥ = 1}.

If we smooth our functions fi near the origin (in which case the front looks like a flying saucer), we obtain an approximation of Λ0 by an immersed Lagrangian sphere with one double point. √ Proposition VI.2.4. — Let r < 1/ 2 be given and let Dr ⊂ R2 be the closed disk of radius r. There is no Hamiltonian isotopy Φ : R2n × I → − R2n such that 2n−2 Φ1 (Λ0 ) ⊂ Dr × R , where Λ0 is defined in (VI.2.2). Proof. — (i) Let us assume that such an isotopy Φ exists. We can find an isotopy Ψ of R2n = T ∗ Rn such that Ψ1 (Dr × R2n−2 ) ⊂ ]−a, a[2 × T ∗ Rn−1 , for some a with (2a)2 < π/2. Hence we can as well assume that Φ1 (Λ0 ) ⊂ ]−a, a[2 × T ∗ Rn−1 . We can also assume that Φ has compact support. We let Φ′ be a homogeneous isotopy of T ∗ Rn+1 lifting Φ as in Proposition II.3.1 and we let KΦ′ ∈ D(kR2n+2 ×I ) be the sheaf associated with Φ′ by Theorem II.1.1. ˙ 1 ) = Φ′ (SS(F ˙ 0 )) and (ii) We set F0 = kW0 and define F1 = KΦ′ ,1 ◦ F0 . Hence SS(F 1 ˙ 1 )) = Φ1 (Λ0 ). By Proposition VI.2.3 F1 has compact support. In particular ρRn (SS(F we have e(F0 ) = e(F1 ). ˙ 1 )) be the projection of SS(F ˙ 1 ) to the base. Since (iii) Let Γ1 = πRn+1 (SS(F 2 ∗ n−1 ˙ ρRn (SS(F1 )) ⊂ ]−a, a[ × T R , we have Γ1 ⊂ Za = ]−a, a[ × Rn . Hence F1 is locally constant outside Za . Since it has compact support, it has to vanish outside Za . Hence F1 satisfies the hypotheses of Corollary VI.1.6, up to a rescaling of Z and γ by a and we obtain e(F1 ) ≤ 4a2 < π/2. On the other hand RHom (F0 , Tc∗ (F0 )) ≃ kW0 ∩Tc (W0 ) for c ∈ [0, π/2[. Since the topology of W0 ∩ Tc (W0 ) is unchanged when c runs over [0, π/2[ we deduce that e(F0 ) = π/2. We thus have a contradiction.

VI.2.3. Nonsqueezing for L0 By (VI.2.3) the Lagrangian subset Λ0 of B1 (T ∗ Rn ) consists of two parts, B1 (T0∗ Rn ) and L0 which can be identified with the image of S1 (T0∗ Rn ) by the geodesic flow. The

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part B1 (T0∗ Rn ) corresponds to the microsupport of kW0 at the conic points (0, 0) and ˙ (0, π/2). We can define another sheaf G0 on Rn+1 such that ρRn (SS(G 0 )) = L0 . Let us recall the sheaf KΨ ∈ D(kR2n+1 ) of Example II.1.4 associated with the normalized geodesic flow of T˙ ∗ Rn . It fits in the distinguished triangle (II.1.4) which is an extension between constant sheaves on two opposite cones (one closed, the other open). If we restrict KΨ to a slice V0 = Rn ×{0}×R, we obtain a picture similar to the following. We define W1 = Tπ/2 (W0 ). Then there exists a diffeomorphism f from a neighborhood Ω of (0, π/2) in Rn+1 to a neighborhood of 0 in Rn+1 such that f (W0 ) = U ∩ V0 and f (W1 ) = Z ∩ V0 (with the notations U , Z of (II.1.4)). Then f −1 (KΨ | )[−n] extends V0 as a sheaf F which fits in a distinguished triangle (VI.2.5)

k W0 → − F → − kW1 [−n] → − kW0 [1].

˙ The microsupport of F is the image of SS(KΨ | ) by df and we find ρRn (SS(F |Ω )) ⊂ L0 , V0 as required. However F still has a too big microsupport at the conic points (0, 0) and (0, π). We can repeat the above gluing process at these points and iterate. We obtain a sheaf G0 ∈ Dlb (kRn+1 ), in the same way we defined K∞ (see (VI.1.6)), with the following property G0 | ≃ (Tk π2 )∗ (F )[−kn]| , for any k ∈ Z, where F is defined Ωk Ωk (VI.2.6) in (VI.2.5) and Ωk = Rn × ]k π2 , (k + 2) π2 [ ˙ and we have ρRn (SS(G 0 )) = L0 . Proposition VI.2.5. — Proposition VI.2.4 holds with Λ0 replaced by L0 . Proof. — It is enough to prove that e(G0 ) ≥ π/2; the rest of the proof of Proposition VI.2.4 works the same. By Remark VI.2.2 we have e(G0 ) ≥ e(G0 | ) for {x0 }×R L n any x0 ∈ R . The above description of G0 implies G0 | ≃ k∈Z k[k π2 ,(k+1) π2 [ [−kn]. {0}×R

Since e(k[a,b[ ) = b − a we obtain e(G0 ) ≥ π/2, as claimed. VI.2.4. Nonsqueezing for the ball We set B = Int(B1 (Rn )) and S = S1 (T ∗ Rn ). For y ∈ B, let Ly ⊂ S be the image of Ty∗ Rn ∩ S by the characteristic flow of S. Then Ly is a Lagrangian submanifold of T ∗ Rn . To prove the nonsqueezing for the ball we use the Lagrangian submanifolds Ly , y ∈ B, in family. We first prove that there exists a sheaf G on B × Rn+1 ˙ such that ρRn (SS(G . y )) = Ly , where Gy = G| {y}×Rn+1

Let us describe the family Ly . For a given y ∈ B we can see Ly as in (VI.2.4) as the image of a map fy : (R/πZ) × Sr (Ty∗ Rn ) → − T ∗ Rn , (s, ξ) 7→ exp(2si) · (y; ξ), p F where r = 1 − ∥y∥2 . We set L0 = y∈B Ly ⊂ Rn × T ∗ Rn . We can describe a Lagrangian submanifold L ⊂ (T ∗ Rn )2 above L0 as follows. In T ∗ R2n we consider

ASTÉRISQUE 440

VI.2.4. NONSQUEEZING FOR THE BALL

99

∗ 2n the hypersurface Z = T ∗ Rn × S and the Lagrangian submanifold T∆ R . Then ∗ 2n L1 = Z ∩ T∆ R is isotropic and its image L by the characteristic flow of Z is still isotropic, hence Lagrangian since it is of dimension 2n. We can see that L ∩ (T ∗ B × T ∗ Rn ) maps bijectively onto L0 through T ∗ B × T ∗ Rn → − B × T ∗ Rn . Like Ly , the Lagrangian L is the image of a map

(VI.2.7)

f : (R/πZ) × S → − (T ∗ Rn )2 , (s, (y; ξ)) 7→ ((y; −ξ), exp(2si) · (y; ξ)).

The important point is that f is injective whereas fy is not. Let L′ ⊂ J 1 (R2n ) = (T ∗ R2n ) × R be a Legendrian lift of L. Then L′ → − L is an infinite cyclic cover. A loop on L given by a characteristic flow line lifts to a path l on L′ with l(1) = Tπ (l(0)), where Tπ is as before the translation in the last variable by π. Hence Tπ (L′ ) = L′ . Since f is injective we have (VI.2.8)

Tc (L′ ) ∩ L′ = ∅

for 0 < c < π.

Moreover, we remark in (VI.2.7) that, if y ∈ ∂B, then ξ = 0 (because (y; ξ) ∈ S). Hence above ∂B × Rn we have (VI.2.9)

L′ ∩ ((∂B × Rn ) ×R2n J 1 (R2n )) ⊂ TB∗ B × T ∗ Rn × R.

For y ∈ B we let L′y ⊂ J 1 ({y}×Rn ) be the projection of L′ ∩({y}×Rn )×R2n J 1 (R2n ) to J 1 ({y} × Rn ). Then L′y is a Legendrian lift of Ly . For y ∈ B these manifolds L′y are all diffeomorphic to L′0 and there exists a C ∞ map u : B × L′0 → − J 1 (Rn ) such ′ ′ that u({y} × L0 ) = Ly . We can lift this map u to a contact isotopy (see for example Theorem 2.6.2 of [16]). Indeed this is possible for a family of compact Legendrian manifolds. However, since Tπ (L′y ) = L′y for all y ∈ B, we can find a family L′′y , y ∈ B, of compact Legendrian submanifolds of (T ∗ Rn ) × (R/πZ) such that L′y is a covering of L′′y . Then we find a contact isotopy of (T ∗ Rn ) × (R/πZ) and lift it to J 1 Rn . We thus have a contact isotopy Φ : B × J 1 (Rn ) → − J 1 (Rn ) such that Φy (L′0 ) = L′y , for any y ∈ B. A Legendrian submanifold of J 1 (M ) gives a conic Lagrangian submanifold of {τ > 0} ⊂ T ∗ (M × R). Hence L′ , L′y give conic Lagrangian submanifolds of T˙ ∗ R2n+1 or T˙ ∗ Rn+1 that we also denote L′ , L′y . The equivalence of categories (II.1.6) of Corollary II.1.5 applied with I = B, A0 = L, A′ = L′ gives a unique sheaf G ∈ D[L′ ] (kB×Rn+1 ) such that G| ≃ G0 , where G0 is defined in (VI.2.6). n+1 {0}×R

Remark VI.2.6. — A sheaf similar to G is constructed in another way in [11] and is shown to be a projector from sheaves on Rn+1 to sheaves with a microsupport ∗ n contained in ρ−1 Rn (B1 (T R )). Proposition VI.2.7. — Let r < 1 be given and let Dr ⊂ R2 be the closed disk of radius r. There is no Hamiltonian isotopy Φ : R2n × I → − R2n such that 2n 2n−2 Φ1 (B1 (R )) ⊂ Dr × R .

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Proof. — It is enough to prove that e(G) ≥ π; the rest of the proof of Proposition VI.2.4 works the same (recall that Corollary VI.1.6 works with a parameter space N which is B in our case). We set Ω = B × Rn+1 and let j : Ω → − R2n+1 be the inclusion. We let u ∈ Hom(Rj! G, Rj∗ G) be the natural morphism. It is enough to prove that uc := Rj∗ (τc (G)) ◦ u is non zero for 0 ≤ c < π. As in the proof of Proposition VI.2.5 we have uc ̸= 0 for small c (in fact for c < π/2). We have uc ∈ Hc := Hom(Rj! G, Tc∗ Rj∗ G). It is thus enough to see that the morphism Hε → − Hc is an isomorphism for 0 < ε ≤ c < π. We set LΩ = SS(kΩ ). We recall that LΩ is the union of the zero section over Ω ∗ ′ b ˙ and one half of T∂Ω R2n+1 . By Theorem I.2.15 we have SS(Rj LΩ and ! G) ⊂ L + ′ b a ˙ ˙ ˙ SS(Rj∗ G) ⊂ L + LΩ . Hence SS(Rj! G) ∩ SS(Tc∗ Rj∗ G) = ∅ for 0 < c < π: this follows from (VI.2.8) above Ω and from (VI.2.8) and (VI.2.9) above ∂Ω. The hypotheses of Corollary I.2.17 are then almost satisfied except for the compactness of supp(Rj! G). However G is π-periodic up to shift in the sense that Tπ ∗ (G) ≃ G[−2n] and we deduce that RHom (Rj! G, Tc∗ Rj∗ G) is π-periodic. Hence we can apply Lemma VI.2.8 below which a variant of Corollary I.2.17. Lemma VI.2.8. — Let M be a manifold and J an open interval of R. Let F , G ∈ D(kM ×R×J ). We assume (i) the projection supp(F ) ∩ supp(G) → − R × J is proper, (ii) F , G are non-characteristic for all maps iu : M × R × {u} → − M × R × J, u ∈ J, ∗ ∗ ˙ that is, SS(A) ∩ (TM (M × R) × T J) = ∅ for A = F , G, t ×R ˙ ˙ (iii) setting Λu = i♯u (SS(F )) and Λ′u = i♯u (SS(G)), we have Λu ∩ Λ′u = ∅ for all u ∈ J, (iv) H := RHom (F, G) is a-periodic for some a ∈ R in the sense that Ta−1 H ≃ H, where Ta is the translation Ta (x, t, u) = (x, t + a, u). −1 Then RHom(i−1 u F, iu G) is independent of u ∈ J.

Proof. — As in the proof of Corollary I.2.17 we set H = RHom (F, G) and the −1 −1 microsupport estimates imply RHom (i−1 u F, iu G) ≃ iu H for all u ∈ J. Hence −1 −1 −1 RHom(iu F, iu G) ≃ RΓ(M × R; iu H). Let e : R → − S1 = R/aZ be the projection. We also write e for idM ×e or idM ×e×idJ . Since H is periodic there exists H ′ ∈ D(kM ×S1 ×J ) such that H ≃ e−1 H ′ . Since e is a covering map, we have e∗ e−1 H ′ ≃ H ′ ⊗ LM ×S1 ×J , where LM ×S1 ×J is the local system LM ×S1 ×J = e∗ (kM ×R×J ). In the same way e∗ e−1 ju−1 H ′ ≃ ju−1 H ′ ⊗ LM ×S1 , where ju is the inclusion of M × S1 × {u}. Finally −1 −1 −1 ′ RHom(i−1 ju H ) u F, iu G) ≃ RΓ(M × R; e∗ e

≃ RΓ(M × S1 ; ju−1 (H ′ ⊗ LM ×S1 ×J )). Let q : M × S1 × J → − J be the projection. Then q is proper on supp(H ′ ) and the −1 ′ base change formula gives RHom(i−1 u F, iu G) ≃ (Rq∗ (H ⊗ LM ×S1 ×J ))u . Hence it ′ is enough to see that Rq∗ (H ⊗ LM ×S1 ×J ) is locally constant, which is proved by microsupport estimates as in the proof of Corollary I.2.17 (we remark that H ′ and H ′ ⊗ LM ×S1 ×J have the same microsupport since LM ×S1 ×J is locally constant).

ASTÉRISQUE 440

PART VII THE GROMOV-ELIASHBERG THEOREM

The Gromov-Eliashberg theorem (see [13, 19]) says that the group of symplectomorphisms of a symplectic manifold is C 0 -closed in the group of diffeomorphisms. This can be translated into a statement about the Lagrangian submanifolds which are graphs of symplectomorphisms. It can be deduced from the Gromov nonsqueezing theorem but we want to stress the relation with the involutivity theorem of Kashiwara-Schapira (stated here as Theorem I.3.6). Let us explain the idea of the proof. We assume for a while a stronger assumption than the Gromov-Eliashberg theorem: let M be a manifold and let ϕn be a sequence of homogeneous Hamiltonian isotopies of T˙ ∗ M which converges in C 0 norm to a diffeomorphism ϕ∞ of T˙ ∗ M . Let Kn ∈ D(kM 2 ) be the sheaf associated with ϕn by The0 2 ˙ orem II.1.1. Hence SS(K n ) is Γϕn , the graph of ϕn , and H (M ; Kn ) ≃ k. We define L Q +1 a kind of limit K∞ by the distinguished triangle n∈N Kn → − n∈N Kn → − K∞ −−→. Q L 0 2 ˙ Then we can check that SS(K ∞ ) ⊂ Γϕ∞ and H (M ; K∞ ) ≃ k∈N k/ k∈N k. Hence K∞ is not the zero sheaf. If we could prove moreover that K∞ is not locally constant, we would deduce from the involutivity theorem that Γϕ∞ is coisotropic, then that ϕ∞ is a symplectic map. The Gromov-Eliashberg theorem is a local non homogeneous version of the previous (unproved) statement and the ϕn are only symplectic diffeomorphisms. It is not difficult to modify the ϕn away from a given point to turn them into Hamiltonian isotopies and make them homogeneous by adding a variable. Our sheaves Kn live now on M 2 × R. The problem is that the convergence of Γϕn to Γϕ∞ is now true only in a neighborhood of a point and we have no control on Γϕn away from this point; more precisely we have a subset Γ′n of Γϕn such that Γ′n converges to a subset of Γϕ∞ . We use a cut-off lemma of Part III to split Kn in a small ball B as Kn = Kn′ ⊕ Kn′′ , with SS(Kn′ ) ⊂ Γ′n . The main difficulty is to prove that the above “limit” of Kn′ is not locally constant. For this we restrict Kn to a line D = {x0 } × R and decompose L Kn as a sum of constant sheaves on intervals using Corollary IV.4.3, say Kn | ≃ a∈An kIan [dna ]. D To ensure that Kn′ does not vanish when n → − ∞, we prove that there are intervals Ian bigger than B ∩ D as follows. Let π be the projection from T ∗ (M 2 × R) to the base.

102

If an interval Ian is contained in B, the splitting Kn = Kn′ ⊕ Kn′′ prevents it from having one end in π(Γ′n ) and the other in π(Γϕn \ Γ′n ). In other word, the intervals with exactly one end in π(Γ′n ) are big. Hence it is enough to see that the projection of Γ′n to M 2 is of degree one to find a big interval.

ASTÉRISQUE 440

CHAPTER VII.1 THE INVOLUTIVITY THEOREM

The main tool in our proof of the Gromov-Eliashberg theorem is the involutivity theorem of [28]. We recall its statement (see Theorem 6.5.4 of [28] restated here as Theorem I.3.6). For a manifold N , a subset S of N and p ∈ N , we use the notations Cp (S), Cp (S, S) = C(S, S) ∩ Tp N of (I.1.2) and(I.1.3) for the tangent cones of S at p. Let M be a manifold, k any coefficient ring and F ∈ D(kM ). The involutivity theorem says that the microsupport S = SS(F ) of F is a coisotropic subset of T ∗ M in the sense that (Cp (S, S))⊥ωp ⊂ Cp (S), for all p ∈ S. We quote the following lemmas. Lemma VII.1.1. — Let X be a symplectic manifold and let S ⊂ S ′ be locally closed subsets of X. Let p ∈ S. We assume that S is coisotropic at p. Then S ′ is also coisotropic at p. Proof. — This is obvious since we have the inclusions (Cp (S ′ , S ′ ))⊥ωp ⊂ (Cp (S, S))⊥ωp ⊂ Cp (S) ⊂ Cp (S ′ ). We recall the map ρM : T ∗ M × T˙ ∗ R → − T ∗ M , (x, t; ξ, τ ) 7→ (x; ξ/τ ), defined in (II.3.1). Lemma VII.1.2. — Let M be a manifold and S ⊂ T ∗ M a locally closed subset. Let −1 p ∈ S and q ∈ ρ−1 M (p). Then S is coisotropic at p if and only if ρM (S) is coisotropic at q. Proof. — We use coordinates (x, t; ξ, τ ) on T ∗ (M × R) and corresponding coordinates (X, T ; Ξ, Σ) on Tq T ∗ (M × R). We set S ′ = ρ−1 M (S) and we write q = (x0 , t0 ; ξ0 , τ0 ). We have dρM,q (X, T ; Ξ, Σ) = (X; τ10 Ξ − τξ02 Σ). Since S ′ is conic, 0 we may assume τ0 = 1. Using the symplectic transformations (x; ξ) 7→ (x; ξ − ξ0 ) on T ∗ M and (x, t; ξ, τ ) 7→ (x, t + ⟨ξ0 , x⟩; ξ − τ ξ0 , τ ) on T ∗ (M × R), which commute with ρM , we may also assume ξ0 = 0. Then we have dρq (X, T ; Ξ, Σ) = (X; Ξ) and we deduce Cq (S ′ ) = Cp (S) × T(t0 ;1) T ∗ R and Cq (S ′ , S ′ ) = Cp (S, S) × T(t0 ;1) T ∗ R. Now the result follows easily.

SOCIÉTÉ MATHÉMATIQUE DE FRANCE 2023

CHAPTER VII.2 APPROXIMATION OF SYMPLECTIC MAPS

Let (E, ω) be a symplectic vector space which we identify with R2n . We recall a stanE dard application of the Alexander trick which says that a symplectic map φ : BR → − E defined on some ball of E coincides with a Hamiltonian isotopy of E on some smaller ball BrE . We endow E with the Euclidean norm of R2n . For an open subset U ⊂ E and a map ψ : U → − E we set (VII.2.1)

∥ψ∥U = sup{∥ψ(x)∥; x ∈ U },

(VII.2.2)

∥ψ∥1U = sup{∥ψ(x)∥, ∥dψx (v)∥; x ∈ U, ∥v∥ = 1}, if ψ is C 1 .

E → − E be a symLemma VII.2.1. — Let R > r and ε be positive numbers. Let φ : BR 1 ′ E plectic map of class C . Then there exists R > r and a symplectic map ψ : BR − E ′ → ∞ which is of class C such that ∥φ − ψ∥BrE ≤ ε. E → − E of Proof. — We set r1 = (R+r)/2 and we choose a (non symplectic) map φ′ : BR ∞ ′ 1 ′ ′∗ ′ class C such that ∥φ−φ ∥B E ≤ ε. We set ω = φ (ω). We have ω −ω = (φ−φ′ )∗ ω. r1

Hence, if we consider ω and ω ′ as maps from E to ∧2 E and we endow ∧2 E with the Euclidean structure induced by E, we have ∥ω − ω ′ ∥BrE ≤ Cε, where the constant C 1 only depends on n. We set r2 = (r1 +r)/2. By Moser’s lemma we can find a flow Φ : BrE1 ×[0, 1] → − E such that Φt (BrE2 ) ⊂ BrE1 for all t ∈ [0, 1] and ω | E = Φ∗1 (ω ′ )| E . The flow Φ is the flow Br

2

Br 2 E Br1 , where

of a vector field Xt which satisfies ιXt (ωt ) = −α over ωt = tω ′ − (1 − t)ω ′ and dα = ω − ω. We can assume that α satisfies the bound ∥α∥BrE ≤ C ′ ∥ω ′ − ω∥BrE 1 1 for some C ′ > 0 only depending on r1 . Hence Xt satisfies ∥Xt ∥BrE ≤ C ′′ ε, for some 1 constant C ′′ > 0 and all t ∈ [0, 1]. We may assume from the beginning that C ′′ ε < r1 − r2 . Hence Φ1 (BrE2 ) ⊂ BrE1 and we have ∥Φ1 − id∥BrE ≤ C ′′ ε. The map ψ = φ′ ◦ Φ1 : BrE2 → − E is a symplectic 2 ′′ map such that ∥φ − ψ∥BrE ≤ (1 + C )ε, which gives the lemma (up to replacing ε by ε/(1 + C ′′ )).

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E Proposition VII.2.2. — Let R > r > 0 be given. Let φ : BR → − E be a symplectic map of class C ∞ . Then there exists a Hamiltonian isotopy Φ : E × R → − E of class C ∞ and with compact support such that Φ1 | E = φ| E . Br

Br

Proof. — (i) Up to composing with a translation or a symplectic linear map, we may assume that φ(0) = 0 and dφ0 = idE . We first show that there exists a Hamiltonian isotopy Φ : E × R → − E with compact support such that Φ−1 ◦ φ = id near 0. We choose a symplectic isomorphism E 1 E 2 ≃ T ∗ ∆ where ∆ is the diagonal. Through this isomorphism the graph of φ is a Lagrangian subset, say Λ, of T ∗ ∆. Since φ(0) = 0 and dφ0 = idE , the set Λ is tangent to the zero section at 0 and there exists a 1-form α on ∆ such that Λ coincides with the graph of α near 0. Since Λ is Lagrangian, α is closed and we can find a function f : ∆ → − R with compact support such that Λ coincides with the graph of df in some neighborhood U of 0. Up to restricting U we can assume that f is small enough in C 2 -norm so that the graph Λt of t df , viewed as a subset of E 2 , is the graph of a diffeomorphism Φt : E → − E for all t ∈ [−1, 2]. Then Φ is a Hamiltonian isotopy with compact support and Φ−1 1 ◦ φ = idE near 0. (ii) By (i) we can assume that φ| E = idE for some small ball BεE . We define Bε U ⊂ E × ]0, +∞[ and ψ : U → − E by U = {(x, t); ∥x∥ < R/t},

ψ(x, t) = t−1 φ(tx).

Then ψ(·, t) is a symplectic map for all t > 0. We let V = {(ψ(x, t), t); (x, t) ∈ U } be the image of U by ψ × idR . Then V is contractible and we can find h : V → − R such that ψ is the Hamiltonian flow of h. We define U0 ⊂ U by U0 = {(x, t); t > 0, ∥x∥ < ε/t}. Since φ| E = idE , we have Bε

ψ(x, t) = x for all (x, t) ∈ U0 . Hence U0 ⊂ V . Moreover h is constant on U0 . We can assume h| = 0 and extend h by 0 to a C ∞ function defined on V ′ = ]−∞, 0] ∪ V . U0 We set Z = {(ψ(x, t), t); t ∈ ]0, 1] and ∥x∥ ≤ r}. For t ≤ ε/r and ∥x∥ ≤ r we have ψ(x, t) = x. Hence Z = (BrE × ]0, ε/r]) ∪ (ψ × idR )(BrE × [ε/r, 1]) and it follows that Z is compact. We choose a C ∞ function g : E × R → − R with compact support such that g = h on Z. Then the Hamiltonian isotopy Φ defined by g has compact support contained in C and satisfies Φ1 = φ on BrE . This proves the proposition.

ASTÉRISQUE 440

CHAPTER VII.3 DEGREE OF A CONTINUOUS MAP

We recall the definition of the degree of a continuous map. Let M, N be two oriented manifolds of the same dimension, say d. We assume that N is connected. ∼ → Z. − Z and an isomorphism Hcd (N ; ZN ) −− We have a morphism Hcd (M ; ZM ) → d Let f : M → − N be a proper continuous map. Applying Hc (N ; ·) to the morphism ZN → − Rf∗ f −1 ZN ≃ Rf! ZM we find ∼ Z← −− H d (N ; ZN ) → − H d (M ; ZM ) → − Z. c

c

The degree of f , denoted deg f , is the image of 1 by this morphism. Lemma VII.3.1. — Let M, N be two oriented manifolds of dimension d. We assume that N is connected. (i) Let f : M → − N be a proper continuous map and let V ⊂ N be a connected open subset. Then deg f = deg(f | −1 : f −1 (V ) → − V ). f

(V )

(ii) Let I ⊂ R be an interval. Let U ⊂ M × I, V ⊂ N × I be open subsets and let f: U → − V be a continuous map which commutes with the projections U → − I and V → − I. We set Ut = U ∩ (M × {t}), Vt = V ∩ (N × {t}) and ft = f | : Ut → − Vt , for Ut all t ∈ I. We assume that f is proper and that V and all Vt , t ∈ I, are non empty and connected. Then deg f = deg ft , for all t ∈ I. Proof. — (i) and (ii) follow respectively from the commutative diagrams Z

Z Z

Z

∼

Hcd (N ; ZV )

Hcd (M ; Zf −1 (V ) )

Z

∼

Hcd (N ; ZN )

Hcd (M ; ZM )

Z,

∼

Hcd (Vt ; ZVt )

Hcd (Ut ; ZUt )

Z

∼

Hcd+1 (V ; ZV )

Hcd+1 (U ; ZU )

Z.

Proposition VII.3.2. — Let BR be the open ball of radius R in Rd . Let U, V ⊂ Rd be open subsets and let f : U → − BR , g : V → − BR be proper continuous maps. We assume

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that there exists r < R such that f −1 (Br ) ⊂ U ∩ V , and that d(f (x), g(x)) < r/2, for all x ∈ U ∩ V . Then deg f = deg g. Proof. — (i) We define h : (U ∩ V ) × [0, 1] → − Rd+1 by h(x, t) = (tf (x) + (1 − t)g(x), t). −1 Let us prove that h (Br/2 × [0, 1]) is compact. Since f −1 (Br ) is compact and contained in U ∩ V , it enough to prove that h−1 (Br/2 × [0, 1]) ⊂ f −1 (Br ) × [0, 1]. Let (x, t) ∈ (U ∩ V ) × [0, 1] be such that ∥h(x, t)∥ ≤ r/2. Since h(x, t) belongs to the line segment [f (x), g(x)] which is of length < r/2, we deduce f (x) ∈ Br , as required. (ii) We define W = h−1 (Br/2 × [0, 1]), Wt = W ∩ (Rd × {t}) for t ∈ [0, 1] and h′t = h| : Wt → − Br/2 . By (i) h| : W → − Br/2 × [0, 1] is proper. Hence Wt W ′ ′ Lemma VII.3.1 (ii) implies that deg h0 = deg h1 . We conclude with Lemma VII.3.1 (i) which implies deg h′0 = deg g and deg h′1 = deg f .

ASTÉRISQUE 440

CHAPTER VII.4 THE GROMOV-ELIASHBERG THEOREM

Let (E, ω) be a symplectic vector space which we identify with R2n . We endow E E with the Euclidean norm of R2n . For R > 0 we let BR be the open ball of radius R E E }. − E we set ∥ψ∥BRE = sup{∥ψ(x)∥; x ∈ BR and center 0. For a map ψ : BR → For a map f : E → − E we denote by if : E → − E × E the embedding x 7→ (x, if (x)) and by Γf = if (E) the graph of f . Lemma VII.4.1. — Let V = Rn and E = T ∗ V . Let f : E → − E be a map of class C 1 and 0 < R be given. Then there exist a Hamiltonian isotopy Θ : T ∗ V 2 × R → − T ∗V 2 with compact support, ε > 0 and three balls centered at 0: BV ⊂ V 2 and B0∗ ⊂ B1∗ ⊂ (V ∗ )2 such that for any other map g : E → − E with ∥f − g|| E < ε BR

we have (a) (b) (c) (d)

Θ1 ◦ if (0) = (0; 0), Θ1 (Γg ) ∩ (BV × (B1∗ \ B0∗ )) = ∅, E Γ′g := Θ1 (Γg ) ∩ (BV × B0∗ ) is contained in Θ1 (ig (BR )), ′ ′ − BV of degree 1. the restriction of πV 2 to Γg gives a proper map Γg →

Proof. — (i) We set p = (0, f (0)) ∈ E 2 , q = (0; 0) ∈ T ∗ V 2 and F = Tp Γf . We can find a symplectic map ψ : Tp E 2 → − Tq E 2 such that d(πV 2 )q : Tq E 2 → − T0 V 2 induces an 2 ∼ isomorphism ψ(F ) −−→ T0 V . We choose a Hamiltonian isotopy Θ such that Θ1 (p) = q and dΘ1 = ψ. (ii) We can find balls BV , B0∗ , B1∗ such that (b-d) hold for g = f . Indeed we first E choose a neighborhood W ⊂ BR of 0 in E such that πV 2 ◦ Θ1 ◦ if induces a dif′ feomorphism from W to W = πV 2 (Θ1 (if (W ))). Then πV 2 is a diffeomorphism from Γ′ = Θ1 (if (W )) to W ′ and it is easy to find BV , B0∗ such that (c) and (d) hold. Up to shrinking BV , B0∗ , we can also find B1∗ such that (b-d) hold. (iii) For g close enough to f the property (d) holds by Proposition VII.3.2 (apply the proposition with f ′ , g ′ , U ′ , V ′ where f ′ := πV 2 ◦ Θ1 ◦ if , g ′ := πV 2 ◦ Θ1 ◦ ig , U ′ = f ′−1 (BV ), V ′ = g ′−1 (BV )). ∗ E To check (c), we ask the additional condition Θ−1 1 (BV × B1 ) ⊂ BR × E, which is ∗ satisfied up to shrinking BV , B1 .

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For (b) we choose balls BV′ , B1′ slightly smaller than BV , B1∗ and B0′ slightly bigger than B0∗ . Let us assume that there exists x ∈ E such that Θ1 (x, g(x)) ∈ BV′ ×(B1′ \B0′ ). E Then (x, g(x)) ∈ BR × E and ∥f (x) − g(x)∥ < ε. For ε small enough this implies Θ1 (x, f (x)) ∈ BV × (B1∗ \ B0∗ ) which is empty. Hence, up to replacing BV , B0∗ , B1∗ by BV′ , B0′ , B1′ , we also have (b) for g. Now we can give a proof of the Gromov-Eliashberg rigidity theorem (see [13, 19]). E E Theorem VII.4.2. — Let R > 0. Let φk : BR → − E, k ∈ N, and φ∞ : BR → − E be 1 C maps. We assume

(i) φk is a symplectic map, that is, φ∗k (ω) = ω, for all k ∈ N, (ii) ∥φk − φ∞ ∥BRE → − 0 when k → − ∞, E (iii) dφ∞,x : Tx E → − Tφ∞ (x) E is an isomorphism, for all x ∈ BR . Then φ∞ | E is a symplectic map. B R

E . Proof. — (i) We will prove dφ∞ | is a symplectic linear map for any given x ∈ BR x Up to composition with a translation we can as well assume x = 0. By Lemma VII.2.1 and Proposition VII.2.2 we can also assume, up to shrinking R, that φk = Φk1 | E is the BR restriction of (the time 1 of) a globally defined Hamiltonian isotopy Φk : E × R → − E with compact support, for each k ∈ N. Let us choose an isomorphism E ≃ T ∗ V where V = Rn . We let Γk ⊂ T ∗ V 2 be the graph of Φk1 , twisted by (x, x′ ; ξ, ξ ′ ) 7→ (x, x′ ; ξ, −ξ ′ ). We let Γ∞ be the graph of φ∞ with the same twist. It is enough to prove that Γ∞ is coisotropic at (0; φ∞ (0)). We assume n ≥ 2 (the case n = 1 is about volume preserving maps and is easy). Then Φk can be defined by a compactly supported Hamiltonian function.

˙ k ) is a conic (ii) By Corollary II.3.2 there exists Fk ∈ Db (kV 2 ×R ) such that SS(F ∗ 2 ˙ Lagrangian submanifold of T (V × R) which is the graph of a homogeneous lift of Φk1 . Applying the cut-off functor Pγ of (III.1.1), with γ = {(0, t) ∈ V 2 × R; t ≤ 0}, ˙ k ) ⊂ {τ > 0}. We recall the map ρ : T ∗ V 2 × T˙ ∗ R → we can as well assume SS(F − T ∗V 2, ˙ k )) = Γk . (x, t; ξ, τ ) = (x; ξ/τ ), defined in (II.3.1). Then ρ(SS(F (iii) We apply Lemma VII.4.1 with the function f : E → − E given by f (x; ξ) = φ∞ (x; ξ)a . It yields a Hamiltonian isotopy Θ with compact support and three balls centered at 0: BV ⊂ V 2 and B0∗ ⊂ B1∗ ⊂ (V ∗ )2 such that the conditions (a-d) of the lemma hold for g(x; ξ) = φk (x; ξ)a if k is big enough. e By Proposition II.3.1 we can lift Θ to a homogeneous Hamiltonian isotopy Θ ∗ 2 ˙ of T (V × R) such that the diagram (II.3.2) commutes. Then Theorem II.1.1 gives ˙ e 1 . We set Gk = K ◦ Fk and K ∈ Db (k(V 2 ×R)2 ) such that SS(K) is the graph of Θ b ˙ e 1 (SS(F ˙ k )). We still Λk = SS(Gk ) for each k ∈ N. Then Gk ∈ D (kV 2 ×R ) and Λk = Θ have Λk ⊂ {τ > 0}, ρ(Λk ) = Θ1 (Γk ).

ASTÉRISQUE 440

CHAPTER VII.4. THE GROMOV-ELIASHBERG THEOREM

111

(iv) We can find a point x0 ∈ BV ⊂ V 2 , as closed to 0 as we want, such that, for any k, the Lagrangian submanifold Λk ⊂ T ∗ (V 2 × R) is in generic position with respect to the line D = {x0 } × R in the following sense: there exists a neighborhood Wk of D and a hypersurface Sk of Wk meeting D transversely such that Λk ∩ T˙ ∗ Wk = T˙S∗k Wk ∩ {τ > 0}. (This implies that x0 is not on the diagonal.) Since Φk and Θ have compact supports and x0 is not on the diagonal, Sk has finitely many connected components. Then Gk | is a constructible sheaf and we can decompose it as a finite sum of D L k constant sheaves on intervals Gk | ≃ k [dα ] by Corollary IV.4.3. Since α∈Ak kIα D SS(Gk ) ⊂ {τ ≥ 0}, the Iαk are of the form [a, b[ or ]−∞, b[ with b ∈ R ∪ {+∞}. The finite ends of the intervals Iαk are in bijection with Ek = Θ1 (Γk ) ∩ Tx∗0 (V 2 ) = (Λk ∩ (Tx∗0 (V 2 ) × T ∗ R))/R>0 .

(v) We set Γ0k = Θ1 (Γk ) ∩ (BV × B0∗ ) and Ek0 = Ek ∩ Γ0k . By Lemma VII.4.1 the map Γ0k → − BV is of degree 1 and it follows that Ek0 is of odd cardinality. Hence there exists one interval Iαk with one end, say ak , in Ek0 and the other, say bk , in (Ek \ Ek0 ) ∪ {±∞}. We translate Gk vertically so that ak = 0 and we shift its degree so that dkα = 0 (we still have ρ(Λk ) = Θ1 (Γk )). Now Gk | has one direct summand which is kIαk with Iαk = [0, bk [ or ]bk , 0[, bk finite D or infinite. One of these possibilities occurs infinitely many times and, up to taking a subsequence, we assume Iαk = [0, bk [ with bk ∈ R, for all k, the other cases being similar. ∗ ∗ ˙ (vi) By Lemma VII.4.1 again ρ(SS(G k )) ∩ (BV × (B1 \ B0 )) = ∅. Hence, by Proposition III.3.2, there exists an open ball W with center (x0 , 0) and radius r such that, for any k, we have a distinguished triangle on W

G′k ⊕ G′′k → − Gk |

+1

W

→ − Lk −−→,

′ ′ ˙ where Lk is a constant sheaf, SS(G k ) = Λk with

Λ′k = Λk ∩ ρ−1 (BV × B0∗ ) ∩ T ∗ W ′′ ∗ ′ ′′ ∗ ˙ ˙ and SS(G k ) = (Λk ∩ T W ) \ Λk . In particular SS(Gk ) does not meet T(x0 ,0) W and G′′k is constant near (x0 , 0). In the same way, if (x0 , bk ) belongs to W , then G′k is constant near (x0 , bk ). On the other hand, if (x0 , bk ) belongs to W , Lemma VII.4.3 below implies that the direct summand k[0,bk [ | of Gk | is also a direct summand of H with D∩W D∩W H = G′k | or H = G′′k | . Then H would be non constant at (x0 , 0) and (x0 , bk ), D∩W D∩W which gives a contradiction. It follows that (x0 , bk ) ̸∈ W and G′k | has a direct summand which is k[0,r[ D∩W (recall r is the radius of W ).

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(vii) We define G ∈ D(kW ) by the distinguished triangle M Y +1 (VII.4.1) G′k → − G′k → − G −−→ . k∈N

k∈N

L Q +1 For any N ∈ N we also have the triangle k≥N G′k → − k≥N G′k → − G −−→. We have S Q L ′ ′ ′ ˙ ˙ SS( k≥N Λk and the same bound holds for SS( k≥N Gk ). Hence the k≥N Gk ) ⊂ ˙ same bound also holds for SS(G) and, when N → − ∞, we obtain −1 ˙ SS(G) ⊂ ρ (Θ1 (Γ∞ ) ∩ (W × B0∗ )). We let i : D ∩ W → − W be the inclusion. We remark that i is non-characteristic Q for ρ−1 (W × B0∗ ). Hence i is non-characteristic for F = G′k or F = k∈N G′k and L Q Theorem I.2.8 gives i−1 F ≃ i! F [1]. Since i−1 commutes with and i! with , we deduce the following distinguished triangle by applying i−1 to (VII.4.1) M Y +1 (G′k | ) → − (G′k | ) → − G| −−→ . D

k∈N

D

D

k∈N

We set l = k∈N k/ k∈N k. Since k[0,r[ is a direct summand of G′k | , the sheaf D∩W l[0,r[ is a direct summand of G| . In particular G is not constant around (x0 , 0) D∩W ˙ and SS(G) ∩ T∗ (V 2 × R) ̸= ∅. Q

L

(x0 ,0)

∗ ˙ We choose p ∈ SS(G) ∩ T(x (V 2 × R). By the involutivity Theorem and 0 ,0) Lemma VII.1.2 we obtain that Θ1 (Γ∞ ) is coisotropic at ρ(p). Since ρ(p) ∈ Tx∗0 V 2 and Θ1 (Γ∞ ) ∩ Tx∗0 V 2 consists of a single point, we have ρ(p) = (x0 ; ξ(x0 )). The point x0 can be chosen arbitrarily close to 0. Hence Θ1 (Γ∞ ) is coisotropic at (0; ξ(0)) and it follows that Γ∞ is coisotropic at (0; φ∞ (0)), as required.

Lemma VII.4.3. — Let k be a field. Let I be an interval in R and let a, b ∈ I with a < b. ˙ Let F1 , F2 , F, L ∈ Db (kI ). We assume that SS(L) = ∅, that we have a distinguished triangle u v +1 F1 ⊕ F2 − →F − → L −−→ and that k[a,b[ is a direct summand of F . Then k[a,b[ is a direct summand of F1 or F2 . Proof. — By hypotheses there exist i : k[a,b[ → − F and p : F → − k[a,b[ such that p ◦ i = idk[a,b[ . Since L has constant cohomology sheaves, we have Hom(k[a,b[ , L) ≃ 0. Hence v ◦ i = 0 and there exists i′ = ( ii12 ) : k[a,b[ → − F1 ⊕ F2 such that i = u ◦ i′ . Let p1 : F1 → − k[a,b[ , p2 : F2 → − k[a,b[ be the components of p ◦ u. Then idk[a,b[ = p1 ◦ i1 + p2 ◦ i2 and we deduce p1 ◦ i1 ̸= 0 or p2 ◦ i2 ̸= 0. Since Hom(k[a,b[ , k[a,b[ ) = k, we can multiply our morphisms by a scalar to have p1 ◦ i1 or p2 ◦ i2 equals id, proving the lemma.

ASTÉRISQUE 440

PART VIII THE THREE CUSPS CONJECTURE

In [6] Arnol’d states a theorem of Möbius “a closed smooth curve sufficiently close to the projective line (in the projective plane) has at least three points of inflection” and conjectures that “the three points of flattening of an immersed curve are preserved so long as under the deformation there does not arise a tangency of similarly oriented branches”. This is a statement about oriented curves in RP2 . Under the projective duality it is turned into a statement about Legendrian curves in the projectivized cotangent bundle of RP2 : if {Λs }s∈[0,1] is a generic path in the space of Legendrian knots in P T ∗ (RP2 ) = (T ∗ RP2 \ RP2 )/R× such that Λ0 is a fiber of the projection π : P T ∗ (RP2 ) → − RP2 , then the front π(Λs ) has at least three cusps. This statement is given by Chekanov and Pushkar in [40], where the authors prove a local version, replacing RP2 by the plane R2 , and also another similar conjecture “the four cusps conjecture”. Here we prove the conjecture when the base is the sphere S2 (which implies the case of RP2 since we can lift a deformation in P T ∗ (RP2 ) to a deformation in P T ∗ S2 = T˙ ∗ S2 /R× ). Let us give the precise statement. Let {Λs }s∈[0,1] be a path of Legendrians in P T ∗ S2 starting at Λ0 = T˙x∗0 S2 /R× for some point x0 ∈ S2 . We let Σs be the projection of Λs to S2 . Theorem VIII.0.1. — Let s ∈ [0, 1] be such that Σs is a curve with only cusps and double points as singularities. Then Σs has at least three cusps. Here is a sketch of the proof. A first remark is that, for a connected curve Σ in S2 with only cusps and double points as singularities and smooth part Σreg , the closure of T˙Σ∗reg S2 in T˙ ∗ S2 is connected if and only if the number of cusps is odd. Hence we only have to show that Σs does not have only one cusp. For any path of Legendrians {Λs }s∈[0,1] there exists a contact isotopy {Φs }s∈[0,1] of the ambient contact manifold such that Λs = Φs (Λ0 ) (see for example Theorem 2.6.2 of [16]). We will apply Theorem II.1.1 to this contact isotopy. To fit with the framework of this − P T ∗ S2 to a homogeneous Hamiltonian theorem we lift the isotopy Φ : P T ∗ S2 ×[0, 1] → ∗ 2 ∗ 2 ˙ ˙ isotopy Φ : T S × [0, 1] → − T S which satisfies Φ0 = id and Φs (x; λ ξ) = λ · Φs (x; ξ)

114

for all λ ∈ R× (not only for λ ∈ R>0 ) and all (x; ξ) ∈ T˙ ∗ S2 . By Theorem II.1.1, for each s ∈ [0, 1] we obtain an auto-equivalence, say RΦ,s = KΦ,s ◦ −, of the category ˙ ˙ D(kS2 ), with the property that SS(R Φ,s (F )) = Φs (SS(F )) for all F ∈ D(kS2 ). We ∗ 2 ˙ s) = ˙ ˙ set F0 = k{x0 } and Fs = RΦ,s (F0 ). Hence SS(F0 ) = Tx0 S and it follows that SS(F ∗ 2 Φs (T˙x0 S ). The fact that Φ is homogeneous for the action of R× , and not only R>0 , implies that RΦ,s commutes with the duality functor DS2 . The sheaf F0 is self-dual and simple. Hence so is Fs . Now we prove the following result (see Theorem VIII.6.1 and the beginning of the proof of Theorem VIII.7.3): if F ∈ D(kS2 ) satisfies (a) (b) (c) (d)

F is self-dual, F is simple, RΓ(S2 ; F ) ≃ k, ˙ Λ = SS(F )/R>0 is a smooth curve whose projection to S2 is a generic curve Σ with only one cusp,

then Hom(F, F [1]) ̸= 0. Since Hom(F0 , F0 [1]) = 0 there cannot exist an autoequivalence R of D(kS2 ) such that R(F0 ) = F . We describe the hypotheses of Theorem VIII.6.1 and see at the same time how (a-d) imply them. We choose a Morse function q : S2 → − R with only two critical points and sufficiently generic with respect to Σ. We decompose Rq∗ (F ) using Corollary IV.4.3. The hypothesis (c) implies that all intervals, but one, appearing in the corollary are half-closed. Then (a) implies that Lthe non half-closed interval is reduced to a point. We thus have Rq∗ (F ) ≃ k{t0 } ⊕ a∈A knIaa [da ] where the Ia are half-closed intervals, da ∈ Z and t0 is some point in R. We can assume t0 = 0 and, choosing q sufficiently generic with respect to Σ, we can assume that 0 ̸∈ Ia , for all a. Then the decomposition of Rq∗ (F ), restricted to a neighborhood of 0, gives the Hypothesis (VIII.2.8) of Theorem VIII.6.1 (up to rescalling), that is, Rq∗ (F )| ≃ k{0} ⊕ B]−1,1[ for ]−1,1[ some B ∈ Db (k). We write Ct = q −1 (t). The decomposition of Rq∗ (F ) and Proposition I.2.4 imply that Σ is tangent to C0 . For q generic, we can choose a diffeomorphism q −1 (]−1, 1[) ≃ S1 × ]−1, 1[ such that Σ ∩ q −1 (]−1, 1[) is the union of one branch, say Γ0 , tangent to C0 and contained in q −1 ([0, 1[), and other branches of the form {θ} × ]−1, 1[ (see (VIII.2.5), (VIII.2.6) and Fig. VIII.2.1). We are now in the settings of Theorem VIII.6.1, which says that, if Σ only has one cusp (hypothesis (d) above), then Hom(F, F [1]) ̸= 0. The proof of Theorem VIII.6.1 distinguishes two cases and uses two criteria to ensure the non-vanishing of Hom(F, F [1]). We decompose F | using Corollary IV.4.3 C1/2 L ′ ′ as F | ≃ L ⊕ a∈A′ e∗ (kIa )[da ], where L is locally constant, the Ia′ are intervals C1/2

of R, d′a ∈ Z and e : R → − C1/2 ≃ R/2πZ is the quotient map. The branch Γ0 meets C1/2 in two points, say θ1 , θ2 , and some intervals Ia′ have one end in e−1 ({θ1 , θ2 }) (a priori there are four such intervals but some could coincide). Now the two cases depend on these intervals.

ASTÉRISQUE 440

115

If one of the intervals with one end in e−1 ({θ1 , θ2 }) is closed or open, then the non vanishing of Hom(F, F [1]) is given by Propositions VIII.2.3 and VIII.2.5. This result is in fact local around C0 and we do not use the fact that Σ only has one cusp. In the other case all intervals with one end in e−1 ({θ1 , θ2 }) are half-closed. Here we apply a more global criterion to ensure Hom(F, F [1]) ̸= 0. We define notions of F -linked and F -conjugate points of Λ in Section VIII.3. Our criterion is roughly that, if there exist a pair (p0 , p1 ) of F -linked points and a pair (q, q ′ ) of F -conjugate points such that (p0 , p1 ) and (q, q ′ ) are intertwined on the circle Λ/R>0 , then Hom(F, F [1]) ̸= 0 (see Proposition VIII.3.5). Examples of conjugate points are the points of Λ corresponding to the ends of one interval Ia in the decomposition of F | C1/2 recalled above. In Proposition VIII.5.1 we see that in our situation we can situate some F -linked points. The hypothesis that Σ has only one cusp is used as follows. The sheaf F has a shift (or Maslov potential—see Example I.4.5) at each point of Λ, which is a half integer and changes by 1 when we cross a cusp. If Σ has only one cusp, then Λ/R>0 is decomposed in two arcs, say Λ+ and Λ− , according to the value of the shift. In the above decomposition of F | , the points above the ends of an C1/2

interval Ia′ which is half-closed are not in the same component Λ± . This gives several pairs of conjugate points (q+ , q− ) with q± ∈ Λ± . Using the linked points of Proposition VIII.5.1 it is then possible to find two intertwined pairs of linked/conjugate points and apply Proposition VIII.3.5. This part is organized as follows. In Section VIII.1 we describe the sheaf associated with the standard curve with three cusps and give an example of a curve with one cusp whose conormal bundle is the microsupport of a sheaf. In Section VIII.2 we give the first criterion (local around C0 ) for the non vanishing of Hom(F, F [1]). In Section VIII.3 we introduce the notions of linked and conjugate points and prove that the existence of two intertwined pairs of linked/conjugate points also implies the non vanishing of Hom(F, F [1]). Examples of conjugate points are given in the next section. In Section VIII.5 we prove the existence of three linked points (under the same hypotheses as in Section VIII.2). In the next two sections we apply these criteria to prove the three cusps conjecture. In the last section we give a sketch of proof of the four cusp conjecture with the same method. In this part we assume that k is an infinite field (we use it in Proposition VIII.5.1 and we use Gabriel’s theorem).

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CHAPTER VIII.1 EXAMPLES

A first idea to prove Theorem VIII.0.1 could be the more ambitious statement: if Λ ⊂ T˙ ∗ S2 is a smooth conic Lagrangian submanifold and Σ = πS2 (Λ) is a curve with only one cusp and otherwise only double points as singularities, then there is no ˙ F ∈ D(kS2 ) such that SS(F ) = Λ. Indeed Theorem II.1.1 would imply that there is no homogeneous isotopy Φ of T˙ ∗ S2 such that Λ = Φ1 (T˙x∗0 S2 ). But this statement is false; we give a counterexample in this section. The examples given here are in fact in R2 . We put coordinates (x, y) on R2 and (x, y; ξ, η) on T ∗ R2 . Sheaf associated with a cusp. — Let Σcusp = {(x, y); x2 = y 3 } be the ordinary cusp in R2 . Let Λcusp be the closure of T˙Σ∗cusp \{0} (R2 \{0}) in T˙ ∗ R2 . Then Λcusp is a smooth conic Lagrangian submanifold of T˙ ∗ R2 consisting of two connected components, say Λ1 = {(t3 , t2 ; −2u, 3tu); t ∈ R, u ∈ R>0 } and Λ2 = Λa1 . We −y 3/2 < x ≤ y 3/2 , y > 0} and W2 = {(x, y); −y 3/2 ≤ x < ˙ W) Fig. VIII.1.1). By Example 5.3.4 of [28] we know that SS(k i (outside T0∗ R2 this follows from Example I.2.3-(iii) and a direct the microsupport over 0). W1

F = kW1 [d1 ] ⊕ kW2 [d2 ]

W2

L

L z

z

′

z

set W1 = {(x, y); y 3/2 , y > 0} (see = Λi , for i = 1, 2 computation gives

z′

F | ≃ k[z,z′ [ [d1 ] ⊕ k]z,z′ ] [d2 ] L

Figure VIII.1.1.

˙ Lemma VIII.1.1. — Let F ∈ D(kR2 ) be such that SS(F ) = Λcusp and F is simple along Λcusp . Then there exist E ∈ D(k) and d1 , d2 ∈ Z such that F ≃ kW1 [d1 ] ⊕ kW2 [d2 ] ⊕ ER2 .

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CHAPTER VIII.1. EXAMPLES

With the notations of Lemma VIII.1.1, if supp(F ) ⊂ W1 and L is a horizontal line cutting Σcusp in two points z, z ′ , then F | ≃ k[z,z′ [ [d1 ] ⊕ k]z,z′ ] [d2 ]. L

Proof. — This follows from the description of sheaves on the plane in [43] (see for example (3.3) in the proof of Thm. 3.12 or Prop. 5.8) but we give a sketch of proof for the reader convenience. ˙ (i) We first consider the case where SS(F ) = Λ1 and supp(F ) is contained in W1 . The set U = W2 ∪ {x < 0} is open with a boundary of class C 1 . We set Z = R2 \ U . The microsupport condition gives (RΓZ (F ))p ≃ 0 for all p ∈ ∂U , and then for all p ∈ Z because supp(F ) ⊂ U . Hence RΓZ (F ) ≃ 0 and we obtain, by excision, F ≃ RΓU (F ) ≃ Rj∗ (F | ), where j is the inclusion of U in R2 . Using Example I.2.10 U and the fact that F is simple, we see that F | ≃ kU,W1 [d1 ] for some d1 ∈ Z. We U deduce F ≃ kR2 ,W1 [d1 ]. ˙ ) = Λ1 but we assume nothing on supp(F ). We set (ii) Now we assume SS(F q = (0, −1) and let Br be the open disk with center q and radius r. By Corol∼ lary I.2.16 we have isomorphisms RΓ(Br ; F ) −− → RΓ(Bs ; F ) for all r ≥ s > 0. 2 ′ ∼ We deduce RΓ(R ; F ) −−→ Fq . We set E = RΓ(R2 ; F ). We have a natural morphism u : ER′ 2 → − F and uq is an isomorphism. Hence the cone of u, say F ′ , satisfies ′ ˙ SS(F ) = Λ1 and Fq′ ≃ 0. By (i) we have F ′ ≃ kW1 [d1 ] for some d1 ∈ Z. We can check that Hom(F ′ , ER′ 2 [1]) ≃ 0 and it follows that F ≃ F ′ ⊕ ER′ 2 . ˙ (iii) Now we assume SS(F ) = Λcusp . By Proposition III.3.2 there exists a neighbor+1

hood V of 0 and a distinguished triangle in V , F1 ⊕ F2 → − F| → − ER′′2 −−→, where V ˙ i ) = Λi and E ′′2 is a constant sheaf. We can find an isotopy from R2 to a small SS(F R

neighborhood of 0 which preserves Λcusp . Hence Proposition I.2.9 implies that the distinguished triangle can be extended to R2 . By (ii) we know F1 and F2 and, using Hom(kR2 , Fi [1]) ≃ 0, we deduce the result. Sheaf associated with a deltoid. — We deform T˙0∗ R2 by the Hamiltonian flow, say Φ, of the function h(x, y; ξ, η) = η 3 /(ξ 2 + η 2 ) (which is one of the simplest example of function homogeneous of degree 1). We set Λdelt = Φ1 (T˙0∗ R2 ) and Σdelt = πR2 (Λdelt ). Then Σdelt is a curve which bounds a star shaped domain, say D, and which has three cusps, all pointing to the outward direction (Σdelt is a kind of deltoid):

Let KΦ be the sheaf associated with Φ by Theorem II.1.1. The composition with KΦ,1 gives an equivalence between simple sheaves with microsupport T˙0∗ R2 and simple sheaves with microsupport Λdelt . We set F = KΦ,1 ◦ k{0} . Then ˙ ˙ {0} )) = Λdelt and F is simple along Λdelt . Since k{0} has comSS(F ) = Φ1 (SS(k ˙ pact support, the same holds for F . Since SS(F ) = Λdelt , F is locally constant outside Σdelt . Hence F must be zero outside D. By Lemma VIII.1.1 we have

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supp(F ) = D and, if L is a line cutting Σdelt transversely in two points z, z ′ , then F | ≃ k[z,z′ [ [d1 ] ⊕ k]z,z′ ] [d2 ] for some integers d1 , d2 . L Lemma VIII.1.2. — Let F = KΦ,1 ◦ k{0} ∈ D(kR2 ) be the simple sheaf along Λdelt corresponding to k{0} . Then F is concentrated in degree −1 and its restriction to a line cutting transversely Σdelt at z, z ′ is (k[z,z′ [ ⊕ k]z,z′ ] )[1]. We summarize these results in the following picture (where the deltoid is slightly deformed) L2

L3 (F [−1])|

Li

≃k

⊕k

B L1 Figure VIII.1.2.

Proof. — We have already seen that the restriction of F to a transverse line is of the form k[z,z′ [ [d1 ] ⊕ k]z,z′ ] [d2 ] for some integers d1 , d2 . Let us first prove that d1 = d2 . Let E1 , E2 , E3 be the edges of the domain D. We set Ui = R2 \ (Ei−1 ∪ Ei+1 ) (with E4 = E1 ). Then F | is of the form Fid1 ,d2 := Ui

d2 ,d1 kUi ∩D [d1 ] ⊕ kUi ∩D [d2 ] or Fid2 ,d1 . Moreover, if F | ≃ Fid1 ,d2 , then F | ≃ Fi+1 . Ui Ui+1

Turning once around D we obtain F1d1 ,d2 ≃ F1d2 ,d1 , which gives d1 = d2 . Now we prove that d1 = 1. Let q : R2 → − R be the projection q(x, y) = y. By Corollary II.1.5 we know that RΓ(R2 ; F ) ≃ RΓ(R2 ; k{0} ) ≃ k. We compute RΓ(R2 ; F ) using RΓ(R2 ; F ) ≃ RΓ(R; Rq∗ (F )) and we will deduce d1 . The line q −1 (y) is transverse to Σdelt except for one value y = y0 . For y ̸= y0 , F | −1 is a sheaf on R which is the sum of two or four sheaves of the type kI [d1 ], where q (y) I is a half closed interval. Hence (Rq∗ (F ))y ≃ RΓ(q −1 (y); F | −1 ) ≃ 0 for y ̸= y0 . q

(y)

It follows that RΓ(R; Rq∗ (F )) ≃ (Rq∗ (F ))y0 . Let z, z ′ be the transverse intersections of Σdelt and q −1 (y0 ) and let z ′′ be their tangent intersection. Then F | −1 is of the q (y0 ) type (k[z,−[ ⊕ k]z,−[ )[d1 ] near z,

(k]−,z′ [ ⊕ k]−,z′ ] )[d1 ] near z ′ ,

(k]−,z′′ [ ⊕ k]z′′ ,−[ ⊕ kR )[d1 ] near z ′′ . By Corollary IV.4.3 we have F | −1 ≃ (kI1 ⊕ kI2 ⊕ kI3 )[d1 ], where the interq (y0 ) vals I1 , I2 , I3 have two closed ends and four open ends. This leaves two possibilities: (1) two of these intervals are half-closed and one is open, or (2) two are open and one is closed. In the first case we find (Rq∗ (F ))y0 ≃ k[d1 − 1] and in the second

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case (Rq∗ (F ))y0 ≃ k2 [d1 − 1] ⊕ k[d1 ]. Since the result is k, this excludes the second case and we obtain d1 = 1. Sheaf associated with a curve with one cusp. — We define a new curve Σ from Σdelt as follows. We cut Σdelt by the rectangle B of Fig. VIII.1.2; the bottom edge ∂ − B of B cuts Σdelt in one pair of points near the bottom cusp and the top edge ∂ + B of B cuts Σdelt in two pairs of points near the top cusps. We put two small copies of B, say B2 and B3 , above another copy B1 , so that the corresponding copies of Σdelt ∩ B glue together, as in Fig. VIII.1.3. We attach a cusp to the pair of points of Σdelt ∩ ∂ − B1 and we attach four arcs of circles, say C1 , . . . , C4 , to the four pairs of points of Σdelt ∩ (∂ + B2 ∪ ∂ + B3 ) so that the resulting curve, say Σ, is connected. The closure of the conormal of the regular part of Σ is a smooth conic Lagrangian submanifold of T˙ ∗ R2 , say Λ. C1 ∂ + B2

D

C2

D′

C3 ∂ + B3

C4

B2

B3

B1 ∂ − B1 Figure VIII.1.3.

Let F be the sheaf described in Lemma VIII.1.2. Let Fi be a copy of F | in the B rectangle Bi . By the description of F , the sheaves F1 , F2 , F3 glue into a sheaf, G, on B1 ∪ B2 ∪ B3 . We can then glue G with a sheaf associated with the cusp to obtain a sheaf G′ . We let D ≃ C1 × ]0, 1[ be the domain bounded by C1 , C2 and we set D1 = D ∪ C1 , D2 = D ∪ C2 . We define similarly the strips D′ , D3′ , D4′ associated with the two other arcs C3 , C4 . Now we glue G′ with the sheaf (kD1 ⊕ kD2 ⊕ kD3′ ⊕ kD4′ )[1] ′ ˙ and we obtain a sheaf F ′ such that SS(F ) = Λ.

ASTÉRISQUE 440

CHAPTER VIII.2 SIMPLE SHEAF AT A GENERIC TANGENT POINT

In this section we consider a sheaf F on a surface M whose microsupport ˙ SS(F ) = Λ ⊂ T˙ ∗ M is a smooth Lagrangian submanifold. We give a criterion for the non vanishing of Hom(F, F [1]) (see Propositions VIII.2.3 and VIII.2.5). The geometric situation and the hypotheses on F are described in Section VIII.2.2. Our criterion is in fact local around an embedded circle of M . We begin with a general statement to go from a local non vanishing to a global one (Lemma VIII.2.1).

VIII.2.1. Local cohomology We make general remarks about the (local) extensions of sheaves. Let M be a manifold and Z a closed subset of M . Let F, G ∈ D(kM ) be given. We will often use the isomorphisms (see Proposition I.1.1-(a-c)) (VIII.2.1)

Hom(F, G[i]) ≃ H i RHom(F, G) ≃ H i (M ; RHom (F, G)),

where RHom denotes the derived Hom functor, with values in D(k), and Hom denotes the Hom functor in D(kM ), with values in Mod(k). We also have the following isomorphisms, deduced from (I.1.5) and the adjunction (⊗, RHom ) (VIII.2.2)

RΓZ RHom (F, G) ≃ RHom (F, RΓZ (G)) ≃ RHom (FZ , G).

Since RΓM (F ) ≃ F ≃ FM , the morphism kM → − kZ induces the natural morphisms (VIII.2.3) and then (VIII.2.4) by composition (VIII.2.3) (VIII.2.4)

iZ (F ) : RΓZ (F ) → − F,

jZ (F ) : F → − FZ ,

i

H RHom(FZ , RΓZ (G)) → − H i RHom(F, G).

The next result gives conditions about some morphisms of sheaves supported on Z to ensure the non triviality of H i RHom(F, G). Lemma VIII.2.1. — Let i ∈ Z and u : FZ → − RΓZ (G)[i] be given. We assume that jZ (G) ◦ iZ (G) ◦ (u[−i]) ̸= 0. Then the image of u in H i RHom(F, G) by (VIII.2.4) is non zero.

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Proof. — Composing with jZ (F ), iZ (G) and jZ (G) gives the commutative diagram iZ (G)◦ − ◦jZ (F )

RHom(FZ , RΓZ (G))

RHom(F, G)

jZ (G)◦iZ (G)◦ −

RHom(FZ , GZ )

jZ (G)◦ − ∼ − ◦jZ (F )

RHom(F, GZ ),

where the top arrow is (VIII.2.4) and the left arrow is composing with jZ (G) ◦ iZ (G). Since GZ is supported in Z, we have RΓZ (GZ ) ≃ GZ and the isomorphisms (VIII.2.2) show that the bottom arrow is an isomorphism. Taking H i (−) gives the result.

VIII.2.2. Generic tangent point—notations and hypotheses We consider a surface M , a smooth closed conic Lagrangian submanifold Λ ⊂ T˙ ∗ M ˙ and F ∈ Db (kM ) such that SS(F ) = Λ. We introduce some hypotheses on Λ ∩ T ∗ U and F |U , where U is an open subset of M diffeomorphic to a cylinder. Since these hypotheses are not completely obvious, we give a quick justification why we introduce them (see also the introduction and Lemma VIII.7.2). Let us assume for a while (as we will do when we apply the results of this section) that Σ = π˙ M (Λ) is an immersed curve, with only cusps and transverse double points as singularities. We choose a Morse function q : M → − R generic enough so that the fibers q −1 (t) are tangent to Σ only for finitely many values t1 , . . . , tN and that q −1 (ti ) only contains one tangent point and no cusp nor double points. By LCorollary IV.4.3 there exists a decomposition of Rq∗ (F ) as a finite sum Rq∗ (F ) ≃ a∈A knIaa [da ], where the Ia ’s are intervals. By Proposition I.2.4 the ends of these intervals can only be the points ti and the critical values of q. In our application the sheaf F ∈ Db (kM ) will satisfy DM (F ) ≃ F (for this, it is necessary that Λa = Λ, by Theorem I.2.13) and RΓ(M ; F ) ≃ k. By Lemma VIII.7.2 this implies that all intervals Ia are half-closed except one which is reduced to a point. We can check that this point cannot be a critical value, hence it is ti0 for some i0 . Thus there is one special tangent fiber, with respect to F , and in this section we try to understand F around this fiber. Up to adding −ti0 to q we assume ti0 = 0. Then q −1 (0) is a union of circles, one of which is tangent to Σ. Restricting to a neighborhood U of this circle, we obtain the situation described in (VIII.2.5) and (VIII.2.6) below, with Γ = Σ ∩ U ,L and F |U satisfies (VIII.2.7) and (VIII.2.8) (we restrict the isomorphism Rq∗ (F ) ≃ a∈A knIaa [da ] to a neighborhood J of 0 which does not contain the ends of the Ia ’s, for Ia ̸= {0}, and rescale to have J = ]−1, 1[). In Fig. VIII.2.1 we give a typical curve π˙ M (Λ) in M and more precisely the situation in the open subset U . We set J = ]−1, 1[ and we choose a diffeomorphism U ≃ S1 × J. We let q: U → − J

ASTÉRISQUE 440

VIII.2.2. GENERIC TANGENT POINT—NOTATIONS AND HYPOTHESES

123

be the projection. We take the coordinates θ ∈ ]−π, π] on S1 , t on J and (θ; ξ) on T ∗ S1 , (t; τ ) on T ∗ J. We set   C = q −1 (t), t ∈ J, U0 = ]−1, 1[ × J,   t 2 Γ0 = {(θ, t) ∈ U0 ; t = θ }, Ω = {(θ, t) ∈ U0 ; t > θ2 }, (VIII.2.5)   Λ = T ∗ U , p± = (0, 0; 0, ±1) ∈ Λ . 0

Γ0

0

0

We assume that Λ satisfies (VIII.2.6) ( Λa = Λ (that is, Λ is stable by the antipodal map (x; ξ) 7→ (x; −ξ)), Λ ∩ T ∗ U = T˙ ∗ U , where Γ = ({θ1 , . . . , θN } × J) ∪ Γ0 and θ1 , . . . , θN ∈ S1 \ ]−1, 1[. Γ

M J U

1 0 −1

q

1

≃S ×J

U

Ω Γ0

C0

θi

−1

1

U0

θj

≃ ]−1, 1[×J

Figure VIII.2.1.

In fact, if Λ ⊂ T˙ ∗ M is any smooth closed conic Lagrangian submanifold such that Λa = Λ and πM (Λ) is a smooth curve in a neighborhood of C0 which has one tangent point with C0 with a tangency of order 1, then we can find an isotopy {φt }t∈[0,1] of M such that dφ1 (Λ) satisfies (VIII.2.6), up to maybe changing q to −q. We will often make the following hypotheses on the restriction of F to U (VIII.2.7)

DU (F |U ) ≃ F |U ,

(VIII.2.8)

Rq∗ (F |U ) ≃ k{0} ⊕ BJ ,

for some B ∈ Db (k),

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where BJ denotes the constant sheaf on J with stalk B (that is, BJ = a−1 J (B) where aJ : J → − {pt} is the map to a point). VIII.2.3. Local study around C0 We prove that H 1 RHom(F, F ) is non zero for a sheaf F satisfying (VIII.2.7), (VIII.2.8) and some additional hypothesis (see Proposition VIII.2.5). For this we use Lemma VIII.2.1 with G = F and Z = C0 . To construct the morphism u of the lemma we use the decomposition of FC0 and RΓC0 (F ) (which are sheaves on the circle) given by Corollary IV.4.3 and, more specifically, the fact that a non zero locally constant sheaf with unipotent monodromy appears in the decomposition of FC0 . This fact is proved in Proposition VIII.2.3. We begin with a lemma which describes the restriction of F to the open set U0 defined in (VIII.2.5). For this lemma and for Proposition VIII.2.3 we consider a sheaf satisfying (VIII.2.8) but maybe not (VIII.2.7) (the argument of the proof constructs an intermediate sheaf which inherits (VIII.2.8) but a priori not (VIII.2.7)). In this section we use the notations (VIII.2.5) and (VIII.2.6), in particular U , U0 , Γ, Ω, Λ, Λ0 . Lemma VIII.2.2. — Let Λ ⊂ fold satisfying (VIII.2.6). Let L i as F ≃ i∈Z (H F )[−i] and F satisfies (VIII.2.8), then

T˙ ∗ M be a smooth closed conic Lagrangian submaniF ∈ Db[Λ∩T ∗ U ] (kU ) be given. Then F is decomposed i ˙ SS(H F ) ⊂ Λ ∩ T ∗ U , for each i ∈ Z. Moreover, if

• (H −1 F )[1] satisfies (VIII.2.8) (for some other B ∈ D(k)), • Rq∗ (H i F ) is a constant sheaf on J, for all i ̸= −1, • H −1 F | ≃ kpU0 ⊕ kqU0 \Ω ⊕ krU \Ω with p, q, r ∈ N, q, r ≥ 1. U0 0 Proof. — (i) Since Λ ∩ T ∗ U ⊂ TΓ∗ U where Γ is a smooth curve, F is weakly constructible for the stratification U = Γ ∪ (U \ Γ) by Proposition I.2.20 and Remark I.2.22. It follows that the same holds for all H i F . By Corollary I.2.21, for two such weakly constructible sheaves G, G′ , the complex H = RHom (G, G′ ) is also weakly constructible for the same stratification. If G, G′ are concentrated in degree 0, then HU \Γ is concentrated in degree 0 and HΓ in degrees 0 and 1. Moreover HΓ has stalks 0 outside Γ and HΓ |Γ is locally constant; similarly HU \Γ has stalks 0 outside U \ Γ and HU \Γ | is locally constant. Since Γ consists only of segments, we deduce U \Γ

H 2 (U ; HΓ ) ≃ 0. We can check that the same vanishing holds for H 2 (U ; HU \Γ ) and L we deduce by excision that Ext2 (G, G′ ) ≃ 0. Lemma IV.4.1 gives F ≃ i (H i F )[−i] i ˙ and this implies the bound for SS(H (F )). (ii) Now we assume that F satisfies (VIII.2.8). L i i By (i) we have Rq∗ F ≃ Rq ∗ (H F )[−i]. Each Rq∗ (H F ) is weakly coni∈Z structible for the stratification J = {0} ⊔ (J \ {0}) and the same argument as in (i) shows that Rq∗ (H i F ) is decomposed by the cohomological degree. Hence there exists

ASTÉRISQUE 440

VIII.2.3. LOCAL STUDY AROUND C0

125

i0 such that Rq∗ (H i0 F )[−i0 ] has k{0} as a direct summand and Rq∗ (H i F ) are constant sheaves on J for i ̸= i0 . Let us check that i0 = −1. We see H i0 F as an object of Db (kU ) concentrated in degree 0. Since the fibers of q have dimension 1, the direct image Rq∗ (H i0 F ) is concentrated in degrees 0 and 1 and so is its summand k{0} [i0 ]. Hence i0 can only be 0 or −1. If i0 = 0, then taking H 0 of Rq∗ (H 0 F ) we find that k{0} is a direct summand of H 0 Rq∗ (H 0 F ) (in the category Mod(kJ ) of non derived sheaves). We recall that H 0 Rq∗ (H 0 F ) ≃ q∗ (H 0 F ) (the non derived direct image). It follows that there exists s ∈ Γ(J; q∗ (H 0 F )) with supp(s) = {0}. Since Γ(J; q∗ (H 0 F )) ≃ Γ(U ; H 0 F ) this section s gives a section s′ of H 0 F with supp(s′ ) ⊂ q −1 (0). But we have seen that H 0 F is constructible for the stratification U = Γ ∪ (U \ Γ). The support of s′ should be a union of strata and there is no stratum contained in q −1 (0). This excludes the case i0 = 0 and we have i0 = −1. (iii) We recall that U0 is an open subset of U which contains the curve Γ0 and no other component of Γ (see (VIII.2.5)). We can find a submersion f : U0 → − R whose fibers are −1 ˙ intervals and such that Ω = f −1 (]0, +∞[). Since SS(H (F )) ⊂ Λ, Proposition I.2.9 implies that H −1 (F )| ≃ f −1 G for some sheaf G on R. Then G must be concentrated U0 ˙ in degree 0 and we have SS(G) ⊂ T0∗ R. By Corollary IV.4.3 G is decomposed as a sum of the sheaves kR , k]0,+∞[ , k[0,+∞[ , k{0} , k]−∞,0[ and k]−∞,0] with some multiplicities. Hence H −1 (F )| is decomposed as a sum of the sheaves kU0 , kΩ , kΩ , kΓ0 , kU0 \Ω and U0 kU0 \Ω . The sheaves kΩ , kΩ , kΓ0 have a support which is closed in U (not only in U0 ). It follows that, if one of them, say F ′ , is a summand of H −1 (F )| , then it is also a U0 summand of H −1 (F ) and Rq∗ (F ′ ) is a summand of Rq∗ (H −1 (F )). Since H −1 (F )[1] satisfies (VIII.2.8), the summands of Rq∗ (H −1 (F )) can only have {0} or J as possible support. On the other hand Rq∗ kΩ , Rq∗ kΩ , Rq∗ kΓ0 all have support [0, 1[. Hence kΩ , kΩ , kΓ0 cannot appear in the decomposition of H −1 (F )| . U0

The only possible summands of H −1 (F )| are then kU0 , kU0 \Ω and kU0 \Ω . If the U0 −1 ˙ last two do not appear both, then SS(H (F )) does not meet Λ0 or only contains one −1 ˙ of the two components of Λ0 . By Proposition I.2.4 we obtain that SS(Rq (F )) ∗ (H ˙ ˙ does not meet T0 R or only contains one half of T0 R, which contradicts the fact that k{0} [−1] is a summand of Rq∗ (H −1 (F )). This concludes the proof. ˙ Let F ∈ Db (kU ) be given such that SS(F ) = Λ ∩ T˙ ∗ U and F is simple. We assume that F is constructible (for the stratification U = Γ⊔(U \Γ)—see Section I.2.3). Then F| is constructible and we can decompose F | according to Corollary IV.4.3: C1/2 C1/2 there exist L ∈ Db (kC1/2 ) and {(Ia , da )}a∈A , where L has locally constant cohomology sheaves of finite rank and A is a finite family of bounded intervals and integers, such that M (VIII.2.9) F| ≃L⊕ e∗ (kIa )[da ], C1/2

a∈A

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where e : R → − C1/2 ≃ R/2πZ is the quotient map. Since F is simple, so is F | C1/2 by Corollary 7.5.13 of [28] (see also Lemma VIII.4.2 below). In particular, the intervals Ia appear with multiplicity 1 and, for any two distinct intervals Ia , Ib , we have ˙ ∗ (kI )) ∩ SS(e ˙ ∗ (kI )) = ∅. This implies: SS(e a b if x is an end of Ia , y an end of Ib and there exists π > ε > 0 such (VIII.2.10) that e(Ia ∩ ]x − ε, x + ε[) = e(Ib ∩ ]y − ε, y + ε[), then Ia = Ib . In the next proposition we consider a sheaf F ∈ Ds,f [Λ] (kU ) satisfying (VIII.2.8). By Lemma VIII.2.2 we know that F is decomposed according to the cohomological degree and that H −1 F | has kU0 \Ω and kU0 \Ω as direct summands. We remark U0 √ √ that C1/2 ∩ (U0 \ Ω) is the union of the two intervals ]−1, −1/ 2] and [1/ 2, 1[ of √ , there length l = 1 − 1/ 2. By (VIII.2.10), in the decomposition (VIII.2.9) of F | C1/2

are two intervals (uniquely defined but maybe equal) in the family {Ia }a∈A , say Ib , with right end x and Ic , with left end y, such that √ e(Ib ∩ ]x − l, +∞[) = ]−1, −1/ 2], (VIII.2.11) √ e(Ic ∩ ]−∞, y + l[) = [1/ 2, 1[. We recall that a locally constant sheaf G on the circle is described up to isomorphism by its stalk Gθ at a given point and its monodromy m : Gθ → − Gθ . We then have H 0 (C0 ; G) ≃ {v ∈ Gθ ; m(v) = v}. Hence a locally constant sheaf has a section if and only if its monodromy has a unipotent factor. Proposition VIII.2.3. — Let Λ ⊂ T˙ ∗ M be a smooth closed conic Lagrangian submanifold satisfying (VIII.2.6). We use the notations (VIII.2.5). Let F ∈ Db[Λ] (kU ) be a simple sheaf satisfying (VIII.2.8). We assume that F is constructible (see Section I.2.3). We consider the intervals Ib , Ic appearing in the decomposition (VIII.2.9) of F | C1/2

which are defined in (VIII.2.11). We assume that Ib or Ic is closed. Then there exists a decomposition F | ≃ F ′ ⊕ L[1] such that L ∈ Mod(kC0 ) is a locally constant sheaf C0 with unipotent monodromy and L ̸= 0. Proof. — (i) We assume that Ib is closed, the other case being similar. We argue by contradiction and assume that there exists F satisfying the hypotheses of the proposition but not the conclusion. We choose F (among those F contradicting the proposition) so that the integer ⌊l(Ib )/2π⌋ is minimal, where l(Ib ) is the length of Ib . We write Ib = [α, β]. We set C = C1/2 for short. By Lemma VIII.2.2 we know that F is decomposed according to the cohomological degree, that (H −1 F )[1] satisfies (VIII.2.8) and that H −1 F | has kU0 \Ω and kU0 \Ω as direct summands. Since F is simple, only U0

this degree −1 term involves intervals with one end in e−1 (Γ0 ∩C). We can then assume from the beginning that F is concentrated in degree −1 and we write F = F0 [1], where F0 ∈ Mod(kU ) is seen as usual as an object of Db (kU ) that is concentrated in degree 0. The hypotheses say that we can write F0 | ≃ e∗ (k[α,β] ) ⊕ F1 . We define C

ASTÉRISQUE 440

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VIII.2.3. LOCAL STUDY AROUND C0

s = (1, 0) ∈ H 0 (C; F0 | ) according to this decomposition, where 1 is the natural C section of e∗ (k[α,β] ). By the base change formula we have H 0 (C; F0 | ) ≃ (q∗ (F0 ))1/2 (the stalk of the C non derived direct image q∗ (F0 ) at the point 1/2 ∈ J). The Hypothesis (VIII.2.8) implies that q∗ (F0 ) ≃ H 0 (Rq∗ F0 ) is a constant sheaf on J, hence the restriction morphism H 0 (J; q∗ (F0 )) → − (q∗ (F0 ))1/2 is an isomorphism. Since H 0 (J; q∗ (F0 )) = 0 ′ H (U ; F0 ), there exists s ∈ H 0 (U ; F0 ) such that s′ | = s. We interpret the sections s C and s′ as morphisms u : kC → − F0 | , u′ : kU → − F0 . C

(ii) If ⌊(β − α)/2π⌋ = 0, then I ′ = e([α, β]) is an arc of C (smaller than C). Then S = supp(s′ ) is a closed subset of U which satisfies S ∩ C = I ′ . By Lemma VIII.2.2 and the fact that F is simple we have F0 | ≃ kpU0 ⊕ kU0 \Ω ⊕ kU0 \Ω . Hence S ∩ U0 can U0 only be ∅, U0 or U0 \ Ω. The first two cases are excluded because one end of the arc I ′ is in ∂Ω and I ′ ̸= C. Hence S ∩ U0 = U0 \Ω. It follows that I ′ = C \(C ∩Ω). Outside U0 the sheaf F0 is constant on the vertical segments {θ} × J, by Theorem I.2.8 and the hypotheses (VIII.2.6) on Λ, and S \ (S ∩ U0 ) must be a union of such segments. Hence we have S = U \ Ω. The morphism u′ factorizes through kU → − kS and gives v : kS → − F0 . By construc− F0 | , is the morphism induced by the tion the restriction of v to U0 , v | : kU0 \Ω → U0

U0

above decomposition of F0 | . Let us define G ∈ Db (kU ) by the distinguished triangle U0

v

+1

kS − → F0 → − G −−→ .

(VIII.2.12)

Then G| ≃ kpU0 ⊕ kU0 \Ω . The image of the triangle (VIII.2.12) by Rq∗ gives U0 w

+1

Rq∗ (kS ) − → k{0} [−1] ⊕ BJ [−1] → − Rq∗ (G) −−→,

(VIII.2.13)

0 with B ∈ Db (k) as in (VIII.2.8) and w = Rq∗ (v). Let us write w = ( w w1 ). The 1 morphism w0 has target k{0} [−1] and thus is determined by H (w| ). By the base

{0}

change formula H 1 (w| ) = H 1 (C0 ; v | ). We have kS | ≃ kC0 and we have asC0 {0} C0 sumed that F0 | does not have a direct summand which is locally constant with C0

unipotent monodromy. This implies that H 1 (C0 ; v | ) vanishes. Indeed, decomposing C0 L P F0 | ≃ L0 ⊕ a∈A′ e∗ (knJaa ) as in (IV.3.3), we write v | = v0 + a∈A′ va . Our C0 C0 hypothesis says that L0 has no global section, hence v0 = 0. If va ̸= 0, then the corresponding interval Ja is closed and H 1 (C0 ; e∗ (knJaa )) ≃ 0, hence H 1 (C0 ; va ) = 0. Finally H 1 (C0 ; v | ) = 0, as claimed. C0 We thus have w0 = 0 and we deduce from (VIII.2.13) that k{0} [−1] is a direct summand of Rq∗ (G). On the other hand, the triangular inequality for the microsupport together with (VIII.2.12) and G| ≃ kpU0 ⊕ kU0 \Ω give the bound U0

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− ˙ SS(G) ⊂ (Λ ∩ T ∗ U ) \ Λ− 0 , where Λ0 is the connected component of Λ0 which con˙ tains p− . This implies SS(Rq ∗ (G)) ⊂ {(0; τ ); τ > 0} and contradicts the fact that k{0} [−1] is a direct summand of Rq∗ (G).

(iii) Now we assume ⌊(β − α)/2π⌋ > 0. We define G ∈ Db (kU ) by the distinguished u′

+1

triangle kU −→ F0 → − G −−→. Applying Rq∗ gives the distinguished triangle (VIII.2.14)

+1

Rq∗ (kU ) → − k{0} [−1] ⊕ BJ [−1] → − Rq∗ (G) −−→ .

The same argument as in (ii) works and we find that k{0} [−1] is a direct summand of Rq∗ (G). Since Rq∗ (kU ) ≃ kJ ⊕ kJ [−1] the other summands of Rq∗ (G) are ˙ U ) = ∅, we also have constant sheaves on J. Hence G[1] satisfies (VIII.2.8). Since SS(k ˙ ˙ 0 ) and G is simple. By (iv) below we have G| ≃ e∗ (k[α,β−2π] ) ⊕ F1 . SS(G) = SS(F C Hence G[1] satisfies the same hypotheses as F with (β − α)/2π replaced by (β − α)/2π − 1. Hence G| has a direct summand, say Lu , which is a locally C0 constant sheaf with unipotent monodromy (recall that F was chosen to contradict L the result with β − α minimal). As in (ii) we write F0 | ≃ L0 ⊕ a∈A′ e∗ (knJaa ) and C0 P ′′ ′ ⊂ A′ is the set of closed intervals. Setting we write u′ | = ′′ ua , where A a∈A C0 L L F0′ = L0 ⊕ a∈A′ \A′′ e∗ (knJaa ), F0′′ = a∈A′′ e∗ (knJaa ) we have F0 | ≃ F0′ ⊕ F0′′ and C0 cannot = (0, u′′ ). Hence G| ≃ F0′ ⊕ coker(u′′ ). The summand Lu of G| u′ | C0 C0 C0 appear in F0′ (by the assumption on F0 | ) hence it appears in coker(u′′ ). Hence Lu is C0 a quotient of F0′′ . On the other hand, if Ja is closed, then Hom(e∗ (knJaa ), Lu ) = 0 and we have a contradiction. This proves the proposition. (iv) It remains to explain the decomposition of G| . We have β − α ≥ 2π and we C set I = [α, β], J = [α, β − 2π], J ′ = [α + 2π, β]. We have the canonical isomorphisms Hom(kC , e∗ (kI )) ≃ Hom(kR , kI ) ≃ Γ(R; kI ) ≃ k where the first one is given by the adjunction (e−1 , e∗ ). Let i : kC → − e∗ (kI ) be the morphism corresponding to 1 ∈ k. We have a natural isomorphism φ : e∗ (kJ ) ≃ e∗ (kJ ′ ) and two restriction morphisms r : kI → − kJ , r′ : kI → − kJ ′ . For θ ∈ C the vector space (e∗ (kI ))θ has a basis {p1 , . . . , pn } identified with I ∩ e−1 (θ) (n depends on θ). We order the basis by the order induced by R, that is, I ∩e−1 (θ) = {p1 < · · · < pn }. In the same way, a basis of (e∗ (kJ ))θ is {p1 , . . . , pn−1 }. The morphisms e∗ (r) and φ−1 ◦ e∗ (r′ ) induce in the stalks the morphisms (x1 , . . . , xn ) 7→ (x1 , . . . , xn−1 ) and (x1 , . . . , xn ) 7→ (x2 , . . . , xn ). We also have (kC )θ ≃ k and the morphism i induces the diagonal morphism in the stalks, x 7→ (x, . . . , x). It follows that the sequence i

e∗ (r)−φ−1 ◦e∗ (r ′ )

0→ − kC → − e∗ (kI ) −−−−−−−−−−−→ e∗ (kJ ) → − 0 is exact in the stalks, hence exact. Example VIII.2.4. — Here is an example of a sheaf satisfying the hypotheses of Proposition VIII.2.3. We consider the closed subset S = U \Ω of U introduced in part (ii) of the proof and define its interior S ′ = U \ Ω. Then Hom(kS ′ , kS [1]) ≃ H 1 (S ′ ; kS ) ≃ k u +1 and we define F ∈ Db (kU ) by the distinguished triangle kS ′ − → kS [1] → − F −−→, where

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u is the image of 1 by this isomorphism. The image of the triangle by Rq∗ gives the distinguished triangle v

+1

kJ [−1] ⊕ k]−1,0[ − → kJ [1] ⊕ k]−1,0] → − Rq∗ F −−→, where the morphism v is non zero because its stalk at a point t ∈ ]−1, 0[ is the image of the morphism kS1 → − kS1 [1] by the functor of global sections and this is non zero. Writing v as a 2 × 2 matrix of morphisms, the only possibly non zero entry is v22 : k]−1,0[ → − k]−1,0] and v22 must be a multiple of the canonical morphism k]−1,0[ → − k]−1,0] whose cone is k{0} . Since the other entries of v are zero, we find Rq∗ F ≃ k{0} ⊕ kJ ⊕ kJ [1], which is the Hypothesis (VIII.2.8). Proposition VIII.2.5. — Let Λ ⊂ T˙ ∗ M be a Lagrangian submanifold satisfying (VIII.2.6) and let F ∈ Db[Λ] (kU ) be a simple sheaf. We assume that F is constructible (see Section I.2.3) and satisfies (VIII.2.7) and (VIII.2.8). We also assume that H −1 F | has a direct summand which is a (non zero) locally constant C0 sheaf with unipotent monodromy. Then H 1 RHom(F, F ) is non zero. Proof. — (i) We will use Lemma VIII.2.1 and look for u : FC0 [−1] → − RΓC0 (F ) such that i ◦ u ̸= 0, where i = jC0 (F ) ◦ iC0 (F ) is the composition of the natural morphisms iC0 (F ) : RΓC0 (F ) → − F , jC0 (F ) : F → − FC0 . In fact, to ensure that i ◦ u ̸= 0 we first find w : kC0 → − RΓC0 (F ) satisfying i ◦ w ̸= 0 and prove the existence of a factorization w = u ◦ v as in the diagram w

kC0 v

RΓC0 (F )

i

FC0

u

FC0 [−1]. It is then clear that i◦u ̸= 0. We first define w such that i◦w ̸= 0. Then in parts (ii-iv) of the proof we actually show that any w : kC0 → − RΓC0 (F ) can be written w = u ◦ v as above. The decomposition (VIII.2.8) gives a morphism k{0} → − Rq∗ (F ), hence w′ : k{0} → − RΓ{0} Rq∗ (F ). Using RΓ{0} Rq∗ (F ) ≃ Rq∗ RΓC0 (F ) and the adjunction (q −1 , Rq∗ ), we see that w′ induces a morphism w : kC0 → − RΓC0 (F ). Then −1 ′ i ◦ w : q (k{0} ) ≃ kC0 → − FC0 corresponds to Rq∗ (i) ◦ w : k{0} → − Rq∗ (FC0 ) via the adjunction. Through the isomorphisms RΓ{0} Rq∗ (F ) ≃ Rq∗ RΓC0 (F ) and (Rq∗ (F )){0} ≃ Rq∗ (FC0 ) the direct image Rq∗ (i) : RΓ{0} Rq∗ (F ) → − (Rq∗ (F )){0} coincides with the natural morphism j{0} (Rq∗ (F ))◦i{0} (Rq∗ (F )). It follows that Rq∗ (i)◦w′ induces the identity morphism on the summand k{0} of Rq∗ F . Hence Rq∗ (i) ◦ w′ is non zero and its image by the adjunction, i ◦ w, is also non zero. (ii) The existence of v, u only depend on the restriction of F to a neighborhood of C0 . So from now on we restrict over U , but still write F instead of F |U . We recall that DF = DU F = RHom (F, ωU ) and, since U is oriented of dimension 2, DU F ≃ (D′U F )[2], where D′U F = RHom (F, kU ).

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L Using Lemma VIII.2.2 we have F ≃ i∈Z H i F [−i] and H −1 F satisfies (VIII.2.8). L i i We deduce DF ≃ i∈Z D(H F [−i]) where D(H F [−i]) is concentrated in de−1 gree 2 − i. Hence H F satisfies (VIII.2.7). Since RHom(H −1 F [−1], H −1 F [−1]) is a direct summand of RHom(F, F ), we may assume from the beginning that F is concentrated in degree −1. (iii) We recall that a locally constant sheaf G on the circle is described up to isomorphism by its stalk Gθ at a given point and its monodromy m : Gθ → − Gθ . The stalk of D′ G is Hom(Gθ , k) and its monodromy is t m−1 . Two locally constant sheaves with isomorphic stalks and conjugate monodromies are isomorphic. In particular, if G has unipotent monodromy, then D′ G ≃ G. Using Corollary IV.4.3 we decompose F | as C0 M e∗ (kIa )[1], (VIII.2.15) F | ≃ Luni [1] ⊕ Lnu [1] ⊕ C0 a∈A′

where Luni , Lnu are locally constant sheaves, Luni has unipotent monodromy, no summand of Lnu has unipotent monodromy, and A′ is a finite family of bounded intervals or R. Since DU F ≃ F we have RΓC0 (F )| ≃ RΓC0 (DU F )| ≃ DC0 (F | ). C0 C0 C0 Since D′ Luni ≃ Luni , we obtain M e∗ (kIa∗ ), (VIII.2.16) RΓC0 (F )| ≃ Luni ⊕ D′ Lnu ⊕ C0 a∈A′

where

Ia∗

′

is the interval such that D kIa ≃ kIa∗ .

(iv) We see w as a section of RΓC0 (F ) using Hom(kC0 , RΓC0 F ) ≃ H 0 (C0 ; RΓC0 (F )). For a locally constant sheaf G as in (iii) with monodromy m : Gθ → − Gθ , we have 0 H (C0 ; G) ≃ {v ∈ Gθ ; m(v) = v}. Hence a locally constant sheaf has a section if and only if its monodromy has a unipotent factor. Using the decomposition (VIII.2.16) we P can then write w = (wuni , 0, a wa ) where wuni is a section of Luni and wa a section of e∗ (kIa∗ ). 1 We choose a non zero section vuni of Luni (recall that Luni ̸= 0 by assumption) and define vuni by vuni = wuni

if wuni ̸= 0,

1 vuni = vuni

if wuni = 0.

Using the decomposition (VIII.2.15) we set v = (vuni , 0, 0) : kC0 → − FC0 [−1] (viewing vuni as a morphism kC0 → − Luni ). Now, for each a ∈ A′ , we look for ua : Luni → − e∗ (kIa∗ ) such that wa = ua ◦ vuni . The morphism vuni is injective (at each stalk, it is a non zero morphism (kC0 )θ = k → − (Luni )θ , hence injective). Let L′ be its cokernel. For each a ∈ A′ , the exact − 0 yields the following part of a long exact sequence 0 → − kC0 → − Luni → − L′ → sequence φ

Hom(Luni , e∗ (kIa∗ )) − → Hom(kC0 , e∗ (kIa∗ )) → − Ext1 (L′ , e∗ (kIa∗ )), where φ is the morphism f 7→ f ◦ vuni . By the adjunction (e−1 , e∗ ) we have Ext1 (L′ , e∗ (kIa∗ )) ≃ Hom(L′ , e∗ (kIa∗ )[1]) ≃ Hom(e−1 (L′ ), kIa∗ [1]).

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Now L′ is locally constant, hence e−1 (L′ ) is constant, say e−1 (L′ ) ≃ kN R , and we have Hom(e−1 (L′ ), kIa∗ [1]) ≃ (H 1 (R; kIa∗ ))N . We remark that a sheaf kIa∗ has a non zero section if and only if Ia∗ is closed, in which case we have H 1 (R; kIa∗ ) ≃ 0 and the morphism φ is surjective. Hence for any a ∈ A′ there exists ua : Luni → − e∗ (kIa∗ ) such that wa = ua ◦ vuni . Using the decompositions (VIII.2.15), (VIII.2.16) we define u : FC0 [−1] → − RΓC0 (F ) by    0 0 0 idLuni 0 0 u = P0 0 0 if wuni = 0. u = P0 0 0 if wuni ̸= 0, ua 0 0

ua 0 0

The equality u ◦ v = w follows; indeed it reads respectively in the cases wuni ̸= 0 and wuni = 0:     wuni   0 0 0   v1   0  idLuni 0 0 wuni uni 0 0 = P w , = P0 . P0 0 0 00 P0 0 ua 0 0

0

a

a

ua 0 0

0

a

wa

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CHAPTER VIII.3 MICROLOCAL LINKED POINTS

Let M be a manifold (in Proposition VIII.3.5 M will be a surface) and let Λ be a smooth conic Lagrangian submanifold of T˙ ∗ M . We consider F ∈ Db (kM ) such ˙ that SS(F ) = Λ and F is simple along Λ. In this section we give a criterion which implies that H 1 RHom(F, F ) is non zero (see Proposition VIII.3.5). For U, V two open subsets of M such that M = U ∪ V and for G ∈ Db (kM ) we denote by (VIII.3.1)

H 1 ({U, V }; G) = H 0 (U ∩ V ; G)/(H 0 (U ; G) × H 0 (V ; G))

the first Čech group of G associated with the covering {U, V } of M (we do not assume that U and V are connected). By the Mayer-Vietoris long exact sequence we have an injective map (VIII.3.2)

H 1 ({U, V }; G) ,→ H 1 (M ; G).

In particular it is enough for our purpose to find a covering {U, V } with H 1 ({U, V }; RHom (F, F )) ̸= 0. To better understand this latter group, we use the natural morphism from RHom (F, F ) to Rπ˙ M ∗ µhom(F, F ). Since F is simple we have a canonical isomorphism ∼ kΛ −− → µhom(F, F )| ˙ ∗ T M sending 1 to idF . We deduce morphisms RHom (F, F ) → − Rπ˙ M ∗ (kΛ ) and H 1 ({U, V }; RHom (F, F )) → − H 1 ({U ′ , V ′ }; kΛ ), where U ′ = T ∗ U ∩ Λ, V ′ = T ∗ V ∩ Λ. However we lose too much information in this way and we consider a map to another Čech group (see (VIII.3.9)). The construction of this map relies on the notion of linked points in Λ introduced in Definition VIII.3.1 below. We first introduce a general notation. For G, G′ ∈ Db (kM ) we recall the canonical isomorphism (I.3.2) (VIII.3.3)

RHom (G, G′ ) ≃ RπM ∗ µhom(G, G′ ).

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For an open subset W of M and p ∈ T ∗ W , we deduce the morphisms (VIII.3.4)

Hom(G| , G′ | ) → − H 0 (T ∗ W ; µhom(G, G′ )), W W

u 7→ uµ+ ,

(VIII.3.5)

Hom(G| , G′ | ) → − H 0 µhom(G, G′ )p , W W

u 7→ uµ+ p .

˙ For G = G′ = F with F simple and for p ∈ SS(F ) we obtain 0 ˙∗ Hom(F | , F | ) → − H (T W ; kΛ ), (VIII.3.6) W

(VIII.3.7)

W

Hom(F | , F | ) → − k, W W

u 7→ uµ , u 7→ uµp

and we have uµ+ = uµp · (idF )µ+ p p , where · is the scalar multiplication (indeed the 0 identification H µhom(F, F )p ≃ k sends (idF )µ+ to 1). p Definition VIII.3.1. — Let W ⊂ M be an open subset and let p, q ∈ Λ ∩ T ∗ W be given points. We say that p and q are F -linked over W if uµp = uµq for all u ∈ Hom(F | , F | ). W W ∼ Because of the isomorphism kΛ −− → µhom(F, F )| ˙ ∗ the scalar uµp only depends T M ∗ on the component of Λ ∩ T W containing p and we could also speak of F -linked connected components of Λ ∩ T ∗ W . In particular, if Λ is connected and W = M all points of Λ are F -linked over M . The notion of F -linked points will be used when Λ ∩ T ∗ W is a priori non connected. Remark VIII.3.2. — (a) Let p = (x; ξ) ∈ Λ. Let φ : M → − R be a function of class C ∞ such that Λ and Λφ intersect transversely at p. We set Z = {φ ≥ φ(x)}. For G, G′ ∈ Db[Λ] (kM ), we have by (I.4.6) H 0 µhom(G, G′ )p ≃ Hom((RΓZ (G))x , (RΓZ (G′ ))x ). If G = G′ = F with F simple at p, then (RΓZ (F ))x is k in some degree. For u ∈ Hom(F, F ) the morphism (RΓZ (u))x is the multiplication by the scalar uµp . (b) We keep the notations in (a). We assume that uµp ̸= 0. Defining H ∈ Db[Λ] (kM ) u

+1

by the distinguished triangle F − →F → − H −−→, we thus have (RΓZ (H))x ≃ 0. On ˙ the other hand SS(H) ⊂ Λ by the triangular inequality for the microsupport and the ˙ vanishing of (RΓZ (H))x implies that SS(H) does not meet the connected component of Λ containing p. (c) We keep the notations in (a). We assume that u factorizes through H ∈ Db (kM ) and p ̸∈ SS(H). Then (RΓZ (u))x factorizes through (RΓZ (H))x ≃ 0 and we obtain uµp = 0. µ By Theorem I.3.8 the functors u 7→ uµ+ p or u 7→ up are well-defined in the quotient b category D (kM ; p). We can also express this result as follows. ′ ˙ ˙ Lemma VIII.3.3. — Let G, G′ ∈ Db (kM ) be such that SS(G), SS(G ) ⊂ Λ and let g +1 p ∈ Λ be given. We assume that there exists a distinguished triangle G − → G′ → − H −−→,

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135

with p ̸∈ SS(H). Then the composition with g induces isomorphisms ∼ ∼ µhom(G, G)p −− → µhom(G, G′ )p ← −− µhom(G′ , G′ )p a

b

and we have = = In particular, if G and G′ are simple ′ and u : G → − G and v : G → − G satisfy v ◦ g = g ◦ u then uµp = vpµ . a((idG )µ+ p )

gpµ+ ′

b((idG′ )µ+ p ).

Proof. — We apply the functor µhom(G, ·) to the given distinguished triangle and we take the stalks at p. By the bound (I.3.6) we have µhom(G, H)p ≃ 0 and we deduce the isomorphism a. The isomorphism b is obtained in the same way. Then the last µ µ+ assertion follows from the relations uµ+ and vpµ+ = vpµ · (idG′ )µ+ p = up · (idG )p p . We recall that U, V are two open subsets of M such that M = U ∪ V . Let γ : [0, 1] → − Λ be a path such that (VIII.3.8) ( p0 = γ(0) and p1 = γ(1) belong to (T ∗ U \ T ∗ V ) ∩ Λ and are F -linked over U ; moreover im(γ) is not entirely contained in T ∗ U ∩ Λ. We define a circle C by identifying 0 and 1 in [0, 1] and let r : [0, 1] → − C be the quotient map. The natural orientation of [0, 1] induces an orientation on C. We set U ′ = r(γ −1 (T ∗ U ∩ Λ)), V ′ = r(γ −1 (T ∗ V ∩ Λ)). We have U ′ ∪ V ′ = C and, by (VIII.3.8), U ′ and V ′ are neither empty nor equal to C. Hence they are unions of non trivial arcs of C and we have a canonical isomorphism H 1 ({U ′ , V ′ }; kC ) ≃ H 1 (C; kC ) ≃ k. For u ∈ H 0 (U ; RHom (F, F )) the inverse image of uµ by γ gives a welldefined section of H 0 (U ′ ; kC ) because p0 and p1 are F -linked over U . An element of H 0 (V ; RHom (F, F )) also induces a section of H 0 (V ′ ; kC ) because p0 , p1 ̸∈ T ∗ V ∩Λ. We deduce a well-defined map (VIII.3.9)

mγ : H 1 ({U, V }; RHom (F, F )) → − H 1 ({U ′ , V ′ }; kC ) ≃ k.

Now we describe a situation where the map mγ is surjective. We first introduce a notion of conjugate pair of points. Definition VIII.3.4. — Let M be a manifold, k a field and F ∈ Db (kM ). Let ˙ q0 = (x0 ; ξ0 ) and q1 = (x1 ; ξ1 ) be given points of SS(F ), generating two distinct half-lines R>0 · q0 ̸= R>0 · q1 . Let I be either an open interval of R or the circle S1 and let i : I → − M be an embedding. We say that q0 and q1 are F -conjugate with respect to i if (i) x0 , x1 ∈ im(i); we write t0 = i−1 (x0 ), t1 = i−1 (x1 ), (ii) for k = 0, 1, there exist a neighborhood Uk ⊂ M of xk and a hypersurface Nk ⊂ Uk such that SS(F ) ∩ T˙ ∗ Uk ⊂ T˙N∗ k Uk and i is transverse to Nk at tk , (iii) F is simple along SS(F ) at qk for k = 0, 1,

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(iv) i−1 F has a direct summand F ′ such that (tk ; id (ξk )) ∈ SS(F ′ ) for k = 0, 1 and F ′ is isomorphic to ( kJ [d] for some interval J of I if I is an interval, e∗ (kJ )[d] for some interval J of R if I = S1 , where d ∈ Z and e : R → − S1 is the covering map. We will see in Proposition VIII.4.4 that the notion of conjugate and linked points are related. Proposition VIII.3.5. — Let M be a surface and let Λ be a smooth conic Lagrangian submanifold of T˙ ∗ M such that Λ/R>0 is a circle. Let F ∈ Db[Λ] (kM ) such that F is simple along Λ. We assume that there exists an embedding of the circle i : S1 → − M and p0 , p1 , q0 , q1 ∈ Λ such that (a) (b) (c) (d) (e)

im(i) meets πM (Λ) at smooth points and transversely, M \ im(i) has two connected components U , V , q0 and q1 are F -conjugate with respect to i, p0 and p1 are F -linked over U , the pairs {[p0 ], [p1 ]} and {[q0 ], [q1 ]} in Λ/R>0 are intertwined, i.e., each of the two arcs from [p0 ] to [p1 ] contains exactly one point of {[q0 ], [q1 ]} (where [p] denotes the image of p ∈ Λ in Λ/R>0 ).

Let γ be a path from p0 to p1 in Λ lifting an arc in Λ/R>0 . Then we can increase U , V so that the map mγ of (VIII.3.9) is defined and surjective. In particular H 1 RHom(F, F ) ̸= 0. The hypotheses are illustrated in Fig. VIII.6.1 and VIII.6.2 for the proof of Theorem VIII.6.1: the embedding i is the inclusion of the circle C1/2 and the points p+ l , ql are F -conjugate with respect to i; the points p+ , p0 are F -linked over the lower open set bounded by C1/2 ; the dotted path in Fig. VIII.6.1 is im(πM ◦ γ). Proof. — (i) By the hypothesis (a) we can assume that im(i) has a neighborhood N of the type N = S1 × K, where K = ]−1, 1[ and i corresponds to the embedding ∗ K for some subset Λ0 ⊂ T˙ ∗ S1 . Since p0 , of S1 × {0}, such that Λ ∩ T ∗ N = Λ0 × TK ∗ 1 p1 ∈ T (M \S ) we can also assume that p0 , p1 ̸∈ T ∗ N . By Theorem I.2.8 we can write F | ≃ p−1 F0 where p : N → − S1 is the projection and F0 ∈ D(kS1 ). We must have N F0 = i−1 F . We write q0 = (a, 0; α, 0), q1 = (b, 0; β, 0) with a, b ∈ S1 and α, β ̸= 0. By the definition of conjugate points, F0 has a direct summand isomorphic to e∗ (kJ )[d], for some interval J of R whose endpoints are mapped to a and b by e : R → − S1 and ˙ ∗ (kJ )) = R>0 · (a; α) ⊔ R>0 · (b; β). Hence we can write such that SS(e (VIII.3.10)

F|

N

≃ F ′ ⊕ F ′′ ,

′ ′ ˙ ˙ ∗ (kJ )) × T ∗ K. In other words SS(F ˙ where F ′ = p−1 (e∗ (kJ )[d]) and SS(F ) = SS(e ) K ∗ consists of the two components of Λ ∩ T N containing q0 and q1 . We remark that ∗ the components of Λ ∩ T ∗ N are of the form Λi = R>0 · (ai ; αi ) × TK K, i = 1, . . . , n,

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137

with (ai ; αi ) ∈ T ∗ S1 . Since γ : [0, 1] → − Λ lifts an arc of Λ/R>0 whose two ends, p0 and ∗ p1 , do not belong to T N , it follows that Ii = γ −1 (Λi ) is either empty or an interval of [0, 1]. Moreover Ii does not contain 0, 1. The hypothesis (e) says that im(γ) meets ′ ˙ exactly one of the two components of SS(F ). Hence γ −1 (SS(F ′ )) consists of a single −1 interval. We choose the indices so that γ (SS(F ′ )) = I1 , im(γ) meets Λ1 , . . . , Λn′ and does not meet Λn′ +1 , . . . , Λn , for some n′ ≤ n. (ii) We set U + = U ∪ N and V + = V ∪ N . Hence U + ∩ V + = N . Let x ∈ k be given. Using the decomposition (VIII.3.10) we define a morphism α(x) : F | → − F | by N N ! x · idF ′ (VIII.3.11) α(x) = . 0F ′′ As described after (VIII.3.8) we set C = [0, 1]/(0 ∼ 1) and let r : [0, 1] → − C be the quotient map. We set U ′ = r(γ −1 (T ∗ U + ∩ Λ)) and V ′ = r(γ −1 (T ∗ V + ∩ Λ)). Then Fl Fm U ′ and V ′ are unions of possibly several arcs of C, say U ′ = i=1 Ui′ , V ′ = i=1 Vi′ . ′ F n We have U ′ ∩ V ′ = r(γ −1 (Λ ∩ T ∗ N )) = i=1 r(Ii ) and in fact m = l = n′ /2. By 1 ′ ′ definition H ({U , V }; kC ) is the cokernel of the morphism d:

l M i=1

H

0

(Ui′ ; kC )

⊕

m M i=1

′

H

0

(Vi′ ; kC )

→ −

n M

H 0 (r(Ii ); kC ),

i=1

0

where each H (−; kC ) is isomorphic to k and d is induced by the obvious restriction maps (we remark that H 0 (Ui′ ; kC ) → − H 0 (r(Ij ); kC ) is the identity map if r(Ij ) ⊂ Ui′ , the zero map else). We know that H 1 ({U ′ , V ′ }; kC ) ≃ H 1 (C, kC ) ≃ k and the quotient ∼ map induces H 0 (r(Ii ); kC ) −− → k, for each i = 1, . . . , n′ . By the definition of α(x), the section α(x)µ of µhom(F, F ) ≃ kΛ is the scalar x ′ ˙ over the two components of Λ ∩ T ∗ N given by SS(F ) and 0 over all other compo∗ nents of Λ ∩ T N . Taking the pull-back by γ and the image by r, and remembering that γ −1 (SS(F ′ )) = I1 is a single interval, we obtain that mγ ([α(x)]) is represented Ln′ ′ 0 by (xi )ni=1 ∈ i=1 H (r(Ii ); kC ) with x1 = x and xi = 0 for i ̸= 1. It follows that mγ ([α(x)]) is non zero if x ̸= 0, hence mγ is surjective. In particular H 1 ({U, V }; RHom (F, F )) ̸= 0 and, by (VIII.3.2), we have H 1 RHom(F, F ) ≃ H 1 (M ; RHom (F, F )) ̸= 0.

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CHAPTER VIII.4 EXAMPLES OF MICROLOCAL LINKED POINTS

We begin with the following easy example over R. In higher dimension we give Propositions VIII.4.4 and VIII.4.5 below which reduce to this case by inverse or direct image. Proposition VIII.4.1. — Let t0 ≤ t1 ∈ R and p0 = (t0 ; τ0 ), p1 = (t1 ; τ1 ) ∈ T˙ ∗ R be given. Let F ∈ Db (kR ) be a constructible sheaf with p0 , p1 ∈ SS(F ) such that F is simple at p0 , p1 . We assume that there exists a decomposition F ≃ G ⊕ kI [d] where G ∈ Db (kR ), d ∈ Z and I is an interval with ends t0 , t1 such that p0 , p1 ∈ SS(kI ). Then p0 and p1 are F -linked over any open interval containing I. Proof. — Let i : kI [d] → − F and q : F → − kI [d] be the morphism associated with the decomposition of F . Then i and q give a morphism Hom(F, F ) → − Hom(kI , kI ) ≃ k. Since F is simple at pk , for k = 0, 1, and pk ∈ SS(kI ), we have pk ̸∈ SS(G). Let u: F → − F be given and let I u : kI [d] → − kI [d] be the morphism induced by u. Then µ µ upk = (I u)pk by Lemma VIII.3.3. Since Hom(kI , kI ) ≃ k we have (I u)µp0 = (I u)µp1 for any u defined in a neighborhood of I. The result follows. Let f : M → − N be a morphism of manifolds. We recall the notations fd : M ×N T ∗ N → − T ∗ M and fπ : M ×N T ∗ N → − T ∗ N for the natural maps induced on the cotangent bundles. We study easy cases of inverse or direct images of simple sheaves. We let ˙ F ∈ Db (kM ) be such that Λ = SS(F ) is a smooth Lagrangian submanifold and F is simple along Λ. More general, but local, statements are given in [28] (see Corollaries 7.5.12 and 7.5.13). Lemma VIII.4.2. — Let i : L → − M be an embedding and let x0 be a point of L. We assume that there exist a neighborhood U ⊂ M of i(x0 ) and a submanifold N ⊂ U such that Λ ∩ T˙ ∗ U ⊂ T˙N∗ U and N is transverse to L at x0 . Then, up to shrinking U −1 ˙ around x0 , the set Λ′ = SS((i F )| ) is contained in T˙N∗ ∩L L and (i−1 F )| is U ∩L U ∩L ′ simple along Λ .

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Moreover, if u : F → − F is defined in a neighborhood of x0 and v : i−1 F → − i−1 F is the induced morphism, then for any p = (i(x0 ); ξ0 ) ∈ Λ and q = (x0 ; id (ξ0 )) we have vqµ = uµp . We note that the inclusion Λ∩T ∗ U ⊂ T˙N∗ U implies that Λ is a union of components of T˙N∗ U (hence this is an equality if N is connected of codimension ≥ 2). Proof. — Up to shrinking U we can find a submersion f : U → − L ∩ U such that N = f −1 (L ∩ N ) and f ◦ i = idL∩U . By Proposition I.2.9 we can write F | ≃ f −1 G U for some G ∈ Db (kL∩U ). We must have G = (i−1 F )| . U ∩L Then µhom(F, F ) ≃ fd ∗ fπ−1 (µhom(G, G)) over U and the lemma follows. Lemma VIII.4.3. — Let q : M → − R be a function of class C 2 and let t0 be a regular value of q. We assume that there exist an open interval J around t0 , a connected hypersurface L of U := q −1 (J) and a connected component Λ0 of T˙L∗ U such that (a) (b) (c) (d)

q| : U → − J is proper on supp F , U q | is Morse with a single critical point x0 and q(x0 ) = t0 , L Λ0 ⊂ Λ ∩ T ∗ U and Tx∗0 M ∩ Λ = Tx∗0 M ∩ Λ0 , ((Λ ∩ T ∗ U ) \ Λ0 ) ∩ qd (M ×R T˙ ∗ R) = ∅.

Let p = (x0 ; ξ0 ) be the point of Λ0 (ξ0 is unique up to a positive scalar) above x0 . We have p ∈ im(qd ) and we set p′ = (t0 ; τ0 ) = qπ (qd−1 (p)). Then p′ ∈ SS(Rq∗ F ) and Rq∗ (F ) is simple at p′ . Moreover, for any morphism u : F | → − F | , denoting U U by v = Rq∗ (u) : Rq∗ F | → − Rq∗ F | the induced morphism, we have vpµ′ = uµp . J

J

−1

Proof. — (i) We set N = q (t0 ). Then N is a smooth hypersurface of M and, up to shrinking J and restricting to a neighborhood of supp(F ) ∩ N , we can assume that U = N × J. The hypersurfaces N and L of U are tangent at the point x0 . We choose a distance function dN on N and define f : U → − R, (x, t) 7→ dN (x0 , x). For r > 0 we set Vr = f −1 ([0, r[) and Zr = U \ Vr . We have a distinguished triangle +1 FVr → − F → − FZr −−→. By (c) we can find a ball B around x0 in U such that T ∗ B ∩Λ = ∗ ˙ Z ) ∩ T ∗ B ⊂ Λ(r) where T B ∩ Λ0 . By Theorem I.2.13 we obtain SS(F r ∗,out ∗,out −1 Λ(r) = ((Λ0 ∩ πU (Zr )) ∪ T∂V U ∪ (Λ0 + T∂V U )) ∩ T ∗ B, r r ∗,out and T∂V U is the outer conormal bundle of ∂Vr in U . We claim that Λ(r) ∩ TN∗ U = ∅ r for a dense set of r. Indeed it is enough to see that (TL∗ U + Tf∗−1 (r) U ) ∩ TN∗ U ∩ T ∗ B = ∅. Since x0 is the only critical point of q | , L ∩ N is smooth outside x0 and L ∗ (TL∗ U + Tf∗−1 (r) U ) ∩ TN∗ U is empty if and only if TL∩N U ∩ Tf∗−1 (r) U is empty; this means that r is a regular value of f | and happens for a dense set of r. L∩N

(ii) We choose r and ε0 > 0 such that Λ(r) ∩ TN∗ U = ∅ and Vr ∩ q −1 ([−ε0 , ε0 ]) ⊂ B. Then, for 0 < ε < ε0 small enough, we have Λ(r) ∩ qd ((N × [−ε, ε]) ×R T˙ ∗ R) = ∅. It ˙ Z ) ∩ qd ((N × [−ε, ε]) ×R T˙ ∗ R) = ∅. Hence Rq∗ (FZ )| follows from (d) that SS(F is r r [−ε,ε] a constant sheaf and we can assume from the beginning that F = FVr .

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141

(iii) We set W = Vr ∩ (N × [−ε, ε]). Taking r and ε smaller if necessary we are in the situation of Example I.2.10 and there exist E, E ′ ∈ Db (k) and a distinguished u +1 ′ triangle EW − → EW ′ → − F | −−→ in Db (kW ), where W ′ is the closure of one compoW nent of W \ L. Since F is simple, we must have E = k[d] for some integer d. Since ′ (Rq∗ (EW ))| is constant, we deduce (RΓY (Rq∗ (FVr )))t0 ≃ (RΓY (Rq∗ (kW ′ [d])))t0 , [−ε,ε]

where Y = ]−∞, 0] or Y = [0, +∞[ according to the sign of τ0 . Now the problem is reduced to the computation of Rq∗ (kW ′ ) in a neighborhood of t0 , which is a classical computation in Morse theory (in our case this is done in the proof of Proposition 7.4.2 of [28]). Proposition VIII.4.1 and Lemmas VIII.4.2 and VIII.4.3 imply the following results. Proposition VIII.4.4. — Let M be a manifold, k a field and F ∈ Db (kM ). Let ˙ p0 , p1 ∈ SS(F ) be given. We assume that there exists an embedding i : I → − M of the circle or an open interval of R such that p0 and p1 are F -conjugate with respect to i. Then p0 and p1 are F -linked over any open subset containing i(I). Proposition VIII.4.5. — Let q : M → − R be a map of class C 2 and let t0 ≤ t1 be regular values of q. For k = 0, 1 we assume that there exist an open interval Jk around tk , a connected hypersurface Lk of Uk :=q −1 (Jk ) and a connected component Λk of T˙L∗k Uk such that the hypotheses (a)-(d) of Lemma VIII.4.3 are satisfied. We assume moreover that Rq∗ F has a decomposition Rq∗ F ≃ G ⊕ kJ [d] where G ∈ Db (kR ), d ∈ Z and J is an interval with ends t0 , t1 . Let pk = (xk ; ξk ) ∈ Λ be such that ˙ J ). Then p0 and p1 are F -linked over any open q(xk ) = tk and qπ (qd−1 (pk )) ∈ SS(k −1 subset containing q ([t0 , t1 ]). Now we check that in a generic situation a given point has a unique conjugate. Proposition VIII.4.6. — Let M be a manifold, let Λ ⊂ T˙ ∗ M be a smooth conic Lagrangian submanifold and let i : S1 → − M be an embedding of the circle. We assume that, in some neighborhood U of i(S1 ), there exists a hypersurface N of U which is ˙ transverse to i and such that Λ ⊂ TN∗ U . Let F ∈ Db (kM ) be such that SS(F ) = Λ, −1 1 F is simple along Λ and i F is constructible. Then, for any q0 ∈ Λ∩(i(S )×M T ∗ M ) there exists a unique q1 ∈ Λ ∩ (i(S1 ) ×M T ∗ M ) (unique up to the action of R>0 ) such that q0 and q1 are F -conjugate with respect to i. ′ Proof. — We set Λ′ = id i−1 π (Λ). The hypothesis on Λ implies that Λ is in 1 ∗ bijection with Λ ∩ (i(S ) ×M T M ). Up to shrinking U we can find a diffeo∼ morphism f : U −− → S1 × Rd , d = dim M − 1, and a finite subset Z of S1 such that f (N ) = Z × Rd and f ◦ i is the inclusion of S1 × {0}. By Proposition I.2.9 we can write f∗ (F | ) ≃ p−1 i−1 (F ) where p : S1 × Rd → − S1 is the projection. It follows U that SS(i−1 (F )) = Λ′ . Moreover i−1 (F ) is simple along Λ′ by Lemma VIII.4.2. L Using Corollary IV.4.3, we write i−1 (F ) ≃ L⊕ a∈A e∗ (knIaa )[da ], where L is locally constant and A is a finite family of bounded intervals of R. Since the Ia ’s are bounded we can identify the set of ends of these intervals with B = A × {left, right}. Taking the

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microsupport gives a map µ from B to the set of half-lines of T˙ ∗ S1 . More precisely, for an interval Ia with Ia = [x, y] we have µ(Ia , left) = {(e(x); ξ); εξ > 0} with ε = 1 if Ia is closed near x, ε = −1 if Ia is open near x, and µ(Ia , right) = {(e(y); ξ); εξ > 0} with ε = −1 if Ia is closed near y, ε = 1 if Ia is open near y. Since i−1 (F ) is simple, the integers na are all equal to 1 and the map µ is injective, inducing a bijection between B and Λ′ /R>0 . Now, two points q0 , q1 ∈ Λ ∩ (i(S1 ) ×M T ∗ M ) are F -conjugate with respect to i if and only if their images in Λ′ correspond, via µ, to the right and left ends of the same interval. The result follows.

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CHAPTER VIII.5 GENERIC TANGENT POINT—GLOBAL STUDY

In this section we find some F -linked points in order to be able to apply Proposition VIII.3.5 in the proof of Theorem VIII.6.1. We first describe the geometric situation. We assume now that M = S2 is the sphere. We keep the notations and assumptions of §VIII.2.2: U ⊂ M is an open subset diffeomorphic to S1 × J, with J = ]−1, 1[, and Λ ⊂ T˙ ∗ M is a smooth closed conic Lagrangian submanifold satisfying (VIII.2.6). We will also use the notations Γ, Ω, Λ0 and p± ∈ Λ0 in (VIII.2.5) and (VIII.2.6). Since M is the sphere, the boundary S1 ×{−1} of U bounds a disk D. For t ∈ [−1, 1] we set (VIII.5.1)

Mt = D ∪ (S1 × [−1, t[).

Proposition VIII.5.1. — Let F ∈ Db[Λ] (kM ) be a simple sheaf. We assume that F | is U constructible (see Section I.2.3) and satisfies (VIII.2.8). Then there exists p ∈ Λ ∩ T ∗ M0 such that p, p− and p+ are F -linked over Mt , for any t ∈ ]0, 1]. Proof. — (i) By (VIII.2.5) and (VIII.2.6) the group Hom(F | , F | ) is independent Mt Mt of t ∈ ]0, 1] and we can assume t = 1. By Proposition VIII.4.5 (applied with t0 = t1 = 0 and J = {0}) we know that p− and p+ are F -linked over M1 . Hence it is enough to find p ∈ Λ ∩ T ∗ M0 which is F -linked with p+ over M1 . We argue by contradiction and assume that there exists no such p. Since Λ/R>0 is compact and Λ ∩ T ∗ U has the form given in (VIII.2.6), we see that Λ ∩ T ∗ M1 is a finite union of connected components, say N G Λ ∩ T ∗ M1 = Λi , i=0

where Λ0 is already defined in (VIII.2.5). The components of Λ ∩ T ∗ M0 are then Λi ∩ T ∗ M0 , i = 1, . . . , N . Before we go on we discuss the idea of the proof since it is rather long. The statement implies in particular that N ≥ 1. We could indeed see now that N ≥ 1 as follows: ˙ if N = 0, then F | is a simple sheaf on the disk M1 with SS(F |M1 ) = Λ0 . Using M1 Example I.2.10 we could describe F and see that it cannot satisfy the decomposition

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Hypothesis (VIII.2.8). The idea is to try to reduce to this situation and obtain a contradiction. We define u : F | → − F| satisfying (VIII.5.2) below. The cone of u, M1 M1 ˙ say C(u), is such that SS(C(u)) = Λ0 but a priori C(u) does not satisfy (VIII.2.8). We modify u in (ii) and (iii) below to obtain another morphism, say u1 , (u1 = u + v in (VIII.5.6)) such that u1 is an isomorphism on M0 , and not only a “microlocal isomorphism” along Λ ∩ T ∗ M0 as is the case for u. (For this we modify u by a morphism v which factorizes through a constant sheaf.) Now the cone of u1 still does not satisfy (VIII.2.8) but it is zero on M0 and we easily obtain a contradiction in (iv), using (VIII.2.8) for F (in fact the case N = 0 was the motivation for introducing u, u1 and their cones, but the proof is finally not a reduction to N = 0). Now we resume the course of the proof and we build u. We have p+ ∈ Λ0 and we choose pi ∈ Λi ∩ T ∗ M0 for each i = 1, . . . , N . By Definition VIII.3.1 there exists ui : F | → − F| such that (ui )µpi ̸= (ui )µp+ , for each i = 1, . . . , N . Adding a multiple M1 M1 of idF and rescaling we can even assume (ui )µpi = 1 and (ui )µp+ = 0. Since k is infinite, we can find a = (ai )i=1,...,N ∈ kN outside the union of hyperS P planes i=1,...,N Pi , where Pi = {a; j=1,...,N aj (uj )µpi = 0} (we remark that Pi ̸= kN P satisfies because the coefficient of ai is 1). Then u = j=1,...,N aj uj : F | → − F| M1

(VIII.5.2)

uµp+

uµpi

= 0 and

M1

̸= 0 for all i = 1, . . . , N .

(ii) Since F | is constructible the algebra Hom(F | , F | ) is finite dimensional and U U U there exists a non zero polynomial P ∈ k[X] such that P (u| ) = 0. Let us write U P (X) = Q(X) · X k , where Q(0) ̸= 0. Then R(X) = Q(X) + X is prime with P (X) and it follows that R(u| ) is an isomorphism (we can find A, B ∈ k[X] with U AR + BP = 1 and we obtain A(u)R(u) = idF ). It follows that R(u) is an isomorphism (on M1 ). Indeed, let F ′ be the cone of R(u). By the triangular inequality ′ ˙ for the microsupport we have SS(F ) ⊂ Λ ∩ T ∗ M1 . Since R(u| ) is an isomorphism U ′ ˙ we also have SS(F ) ∩ T ∗ U = ∅. But a microsupport cannot be a proper subset of a smooth connected Lagrangian submanifold by Corollary I.3.9. Since all components ′ ˙ of Λ ∩ T ∗ M1 meet T ∗ U we deduce SS(F ) = ∅. Hence F ′ is locally constant on M1 . Since it vanishes on U , it vanishes everywhere on M1 . This means that R(u) is an isomorphism on M1 . such that (iii) We claim that there exists v : F | → − F| M1 M1 (VIII.5.3)

v|

M0

= Q(u)|

M0

and vpµ = 0 for any p ∈ Λ ∩ T ∗ M1 .

To construct v we first define G ∈ Db (kM1 ) and u′ by the distinguished triangle u′

G −→ F |

uk

M1

−→ F |

M1

→ − G[1].

˙ By the triangular inequality for the microsupport we have SS(G) ⊂ Λ ∩ T ∗ M1 . ˙ By (VIII.5.2) and Remark VIII.3.2-(b) SS(G) avoids the components Λi for ˙ ˙ i = 1, . . . , N . Hence SS(G) ⊂ Λ0 . In particular SS(G| ) is empty and G| is locally M0

ASTÉRISQUE 440

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145

constant, hence constant since M0 is a disk. Let us write G| = KM0 = a−1 M0 (K) for M0 b some K ∈ D (k) where aM0 : M0 → − {pt} is the projection, Since uk ◦ Q(u) = 0 there exists, w′ : F → − G such that Q(u) = u′ ◦ w′ . Now both morphisms u′ | : KM0 → − F| and w′ | : F | → − KM0 extend to M1 M0 M0 M0 M0 ′′ ′′ as u : KM1 → − F| and w : F | → − KM1 . To see this we use KM0 = a−1 M0 (K), M1

M1

! KM0 ≃ a!M0 (K)[−2] and the adjunctions (a−1 M0 , RΓ(M0 ; −) and (RΓc (M0 ; −), aM0 ), which yield

(VIII.5.4)

Hom(KM0 , F |

(VIII.5.5)

Hom(F |

M0

M0

) ≃ Hom(K, RΓ(M0 ; F )),

, KM0 ) ≃ Hom(RΓc (M0 ; F ), K[−2]).

We have similar isomorphisms with M0 replaced by M1 . We extend the projection q : U → − J as q : M1 → − R by setting q(D) = −1. Using the Hypothesis (VIII.2.8) and RΓ(M1 ; F ) ≃ RΓ(R; Rq∗ F ), we have RΓ(M1 ; F ) ≃ RΓ(M0 ; F ) ⊕ k. Hence the image of u′ by (VIII.5.4) can be extended to M1 and thus u′ also can be extended to M1 . In the same way RΓc (M1 ; F ) ≃ RΓc (M0 ; F )⊕k and w′ can be extended to M1 . We choose such extensions u′′ , w′′ and we set v = u′′ ◦w′′ . Then v | = (u′ ◦w′ )| = M0 M0 ˙ Q(u)| . Since v factorizes through KM and SS(K M ) is empty, Remark VIII.3.2-(c) 1

M0

1

gives vpµ = 0 for any p ∈ Λ ∩ T ∗ M1 . (iv) We define G′ ∈ Db (kM1 ) by the distinguished triangle (VIII.5.6)

G′ → − F|

u+v

M1

−−−→ F |

M1

′ ˙ Then SS(G )∩T ∗ M1 ⊂ Λ∩T ∗ M1 . Since (u+v)| ′

→ − G′ [1].

= (u+Q(u))| = (R(u))| is an M0 M0 ′ ˙ ≃ 0. In particular SS(G ) ∩ T ∗ M0 = ∅ and we obM0

isomorphism by (ii), we have G | M0 ′ ˙ tain that SS(G )∩T ∗ M1 ⊂ Λ0 . Then G′ is locally constant on M1 \Ω, hence zero since it

is zero on M0 . Using Proposition I.2.9 as in the part (iii) of the proof of Lemma VIII.2.2 we deduce that G′ is a direct sum of sheaves of the type kΩ , kΩ and kΓ0 with some shifts in degree. It follows easily that either G′ ≃ 0 or supp(Rq∗ (G′ )) = [0, 1[. On the other hand, applying Rq∗ to the triangle (VIII.5.6), we obtain the w +1 distinguished triangle Rq∗ (G′ )| → − (k{0} ⊕ BJ ) − → (k{0} ⊕ BJ ) −−→. Whatever J the morphism w this implies supp(Rq∗ (G′ )) ∩ J = J, {0} or ∅. This prevents supp(Rq∗ (G′ )) = [0, 1[ and proves that G′ ≃ 0. Hence u + v is an isomorphism. But we also have (u + v)µp = uµp + vpµ = 0 for any p ∈ Λ0 . This gives a contradiction and concludes the proof.

SOCIÉTÉ MATHÉMATIQUE DE FRANCE 2023

CHAPTER VIII.6 FRONT WITH ONE CUSP

In this section we keep the hypotheses of Section VIII.5 and add genericity hypotheses on Σ = π˙ M (Λ). So M = S2 is the sphere, Λ ⊂ T˙ ∗ M is a smooth closed conic Lagrangian submanifold and there exists a cylinder U = S1 × J in M such that Λ satisfies (VIII.2.6). We assume moreover that Σ has exactly one simple cusp c and that Σ ∩ (M \ {c}) is an immersed curve, with transverse double points. Since Λ is stable by the antipodal map, Λ ∩ Tc∗ M consists of two opposite half lines and Λ \ (Λ ∩ Tc∗ M ) has two connected components. The antipodal map has no fixed point and it follows that it exchanges the two connected components of Λ \ (Λ ∩ Tc∗ M ). We consider F ∈ Ds,f [Λ] (kM ) satisfying (VIII.2.7) and (VIII.2.8). Since F is simple along Λ, it has a shift s(p) ∈ 21 Z at any p ∈ Λ (see (I.4.4)). As recalled in Example I.4.5, the shift is locally constant outside the cusps and changes by 1 when p crosses a cusp. Hence s(p) takes two distinct values over Λ \ (Λ ∩ Tc∗ M ), one for each connected component. We recall that we have distinguished two points p± of Λ (see notations (VIII.2.5)) and we have p+a = p− . Hence F has different shifts at the points p+ and p− . We denote by Λ± the connected component of Λ \ (Λ ∩ Tc∗ M ) containing p± . By Example I.4.5 the sheaf k]−∞,0[ on R has shift − 21 at (0; 1) and the sheaf k]−∞,0] has shift 12 at (0; −1). In particular the constant sheaf on a closed or open interval has the same shifts at both ends. The constant sheaf on a half-closed interval has different shifts at both ends. Theorem VIII.6.1. — Let M be the sphere and let Λ ⊂ T˙ ∗ M be a closed conic Lagrangian submanifold such that Λ satisfies (VIII.2.6). We assume that Σ = π˙ M (Λ) is a curve with only cusps and ordinary double points as singularities. We assume that Σ has exactly one cusp. Let F ∈ Db[Λ] (kM ) be a simple sheaf. We assume that F | is constructible and satisfies (VIII.2.7) and (VIII.2.8). Then U H 1 RHom(F, F ) ̸= 0. Proof. — The notations introduced in (i) are illustrated in Fig. VIII.6.1, where the curve is Σ, the dotted path is im(πM ◦ γ) and ε = +.

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c

p+ l

C 21 ql C0

p+ r p+ p−

p0

Figure VIII.6.1.

(i) We let i be the inclusion of S1 = C1/2 in M . We use the notation Mt of (VIII.5.1). By Proposition VIII.5.1 there exists p0 = (z0 ; ξ0 ) ∈ Λ ∩ T ∗ M0 such that p+ , p− and p0 are F -linked over M1/2 . The point p0 is in Λ+ or Λ− . We define ε = + or ε = − so that p0 ∈ Λε . √ √ We denote by zl = (−1/ 2, 1/2), zr = (1/ 2, 1/2) the intersections of Γ0 and C1/2 . ± We also denote by p± l , pr the points of Λ0 (well-defined up to a positive multiple) above zl , zr in the same connected component as p± . Let γ : [0, 1] → − Λ be an embedding such that γ(0) = pε , γ(1) = p0 , γ([0, 1]) ⊂ Λε and πM ◦γ is an immersion (hence πM ◦γ describes the portion of the curve Σ between (0, 0) and z0 which does not contain the cusp). Then the image of γ meets either the half-line R+ · pεl or the half-line R+ · pεr . We assume it meets R+ · pεl , the other case being similar. (In Fig. VIII.6.2 the circle represents Λ/R>0 and the positions of the points correspond to Fig. VIII.6.1; the points c′ , c′′ are the preimages of the cusp c; we have ε = +, the image of the path γ is the arc (p+ p0 ), which contains p+ l .) By Proposition VIII.4.6 there exists a unique ql ∈ Λ (up to a positive scalar) which is F -conjugate to pεl with respect to the embedding i of C1/2 in the sense of Definition VIII.3.4. This means that there exists an interval Ia in the decomposi+ ε + ˙ ∗ (kI )) = id i−1 tion (VIII.2.9) of F | such that SS(e π (R · pl ⊔ R · ql ). Let x be a C1/2

the right end of Ia . Then e(x) = zl . If Ia is closed near x, then Ia is the same as Ib introduced in (VIII.2.11). (ii) If the interval Ia is closed, then the non vanishing of H 1 RHom(F, F ) follows from Propositions VIII.2.3 and VIII.2.5.

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149

c′ ql p− l

p+ r

p−

p+ p+ l

p− r p0

c′′

Figure VIII.6.2.

(iii) We assume that Ia is open. Then D′C1/2 (e∗ (kIa )) ≃ e∗ (kIa ). Since DM (F ) ≃ F , the interval Ia , maybe translated by a multiple of 2π, also appears in the decomposition of F | . Hence we can still apply Propositions VIII.2.3 and VIII.2.5. C1/2

(iv) If the interval Ia is half-closed, then F has different shifts at the points pεl and ql . Hence ql belongs to Λ−ε and the path γ does not meet R+ · ql . This means that the pairs {[pεl ], [ql ]} and {[pε ], [p0 ]} are intertwined in Λ/R>0 (see Fig. VIII.6.2). The result follows from Proposition VIII.3.5 (applied with the embedding i of C1/2 in M and the path γ).

SOCIÉTÉ MATHÉMATIQUE DE FRANCE 2023

CHAPTER VIII.7 PROOF OF THE THREE CUSPS CONJECTURE

We first make a general remark about the sheaf KΦ associated with a homogeneous Hamiltonian isotopy. Lemma VIII.7.1 is a general statement and M is any manifold (we come back to M = S2 for the three cusps conjecture), I is an open interval containing 0 and Φ : T˙ ∗ M × I → − T˙ ∗ M is a Hamiltonian isotopy such that Φ0 = id. We assume that Φs (x; λ ξ) = λ · Φs (x; ξ) for all λ ∈ R× (not only λ ∈ R>0 ) and all (x; ξ) ∈ T˙ ∗ M , where we set as usual Φs (·) = Φ(·, s). Theorem II.1.1 associates with Φ a sheaf KΦ ∈ Dlb (kM 2 ×I ) and the next lemma says that it is self-dual in the sense KΦ ≃ RHom (KΦ , ωM ⊠ kM ×I ). Let us first recall some fact about the dualizing sheaf. For a manifold M we set ωM = a!M (k), where aM : M → − {pt} is the projection. We have ωM ≃ orM [dM ], where orM is the orientation sheaf and dM the dimension of M . In our relative case ωM ⊠ kM ×I ≃ p! (kM ×I ), where p : M 2 × I → − M × I is (x, y, s) 7→ (y, s). Lemma VIII.7.1. — Let KΦ ∈ Dlb (kM 2 ×I ) be the sheaf associated with Φ by Theorem II.1.1. Then KΦ ≃ RHom (KΦ , ωM ⊠ kM ×I ). Moreover, for any F ∈ D(kM ) and any s ∈ I, we have DM (KΦ,s ◦ F ) ≃ KΦ,s ◦ DM (F ). Proof. — (i) We use the uniqueness part of Theorem II.1.1. We set K ′ = RHom (KΦ , ωM ⊠ kM ×I ). Since ωM is locally constant, Theorem I.2.13 ′ a a ˙ ˙ gives SS(K ) = (SS(K Φ )) = ΛΦ , where ΛΦ is the graph of Φ, as defined in (II.1.2). ′ a ˙ ) = ΛΦ . The hypothesis on Φ implies ΛΦ = ΛΦ , hence SS(K 2 2 In particular the inclusion i : M × {0} → − M × I is non-characteristic for SS(K ′ ) and Theorem I.2.8 gives i! (K ′ ) ≃ i−1 (K ′ ) ⊗ ωi , where ωi = i! (kM 2 ×I ) ≃ kM 2 [−1]. By adjunction we have i! (K ′ ) ≃ RHom (i−1 KΦ , i! (ωM ⊠ kM ×I )) ≃ RHom (k∆M , ωM ⊠ kM [−1]). Now RHom (k∆M , −) ≃ Rδ∗ δ ! (−), where δ is the inclusion of ∆M . We also have ωM ⊠ kM ≃ p!2 (kM ), where p2 : M 2 → − M is the second projection. Since p2 ◦ δ = idM we obtain RHom (k∆M , ωM ⊠ kM ) ≃ k∆M . Hence i−1 (K ′ ) ≃ k∆M . Summing up we

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CHAPTER VIII.7. PROOF OF THE THREE CUSPS CONJECTURE

′ ˙ have SS(K ) = ΛΦ and K ′ | 2 ≃ k∆M . The uniqueness property of KΦ gives M ×{0} ′ K ≃ KΦ .

(ii) The proof of the second statement is the same as (i). We define two sheaves F ′ , F ′′ on M × I by F ′ = KΦ ◦ DM (F ) and F ′′ = RHom (KΦ ◦ F, ωM ⊠ kI ). Then F ′ | ≃ F ′′ | and F ′ , F ′′ have the same microsupport, which is M ×{0}

M ×{0}

ΛΦ ◦a SS(F ) in the notation (II.1.5). Then Corollary II.1.5 gives F ′ ≃ F ′′ . In particular F ′| ≃ F ′′ | , which is the required isomorphism. M ×{s}

M ×{s}

Lemma VIII.7.2. — Let G ∈ Db (kR ) be a constructible sheaf. We assume that G has compact support, that there exists an isomorphism G ≃ DR (G) and that RΓ(R; G) ≃ k. L Then there exist t0 ∈ R and a decomposition G ≃ k{t0 } ⊕ a∈A knIaa [da ] where the Ia are half-closed intervals. Proof.L— By Corollary IV.4.3 there exists of G as a finite sum L a decomposition na na G ≃ a∈A kIa [da ]. Then RΓ(R; G) ≃ a∈A RΓ(R; kIa ) [da ]. Since RΓ(R; G) ≃ k all the intervals Ia but one, say Ib , have cohomology zero, which means that they are half-closed. For α < β and I = ]α, β[, J = [α, β], we set I ∗ = J, J ∗ = I. We have DR (kI [d]) ≃ kI ∗ [1 − d]. If the remaining interval Ib is open or is closed with non empty interior, then Ib∗ also appears in the family Ia , a ∈ A, because DR (G) ≃ G. Then L kIb∗ also contributes to RΓ(R; G) and it would imply that k∈Z H k (R; G) has dimension at least 2. Hence Ib is reduced to one point, say t0 . Using DR (k{t0 } [d]) ≃ k{t0 } [−d] we deduce the lemma. Now we can prove the three cusps conjecture. We let P T ∗ S2 = T˙ ∗ S2 /R× be the projectivized cotangent bundle of S2 . Let I be an open interval containing 0 and let Φ : P T ∗ S2 × I → − P T ∗ S2 be a contact isotopy of P T ∗ S2 . It lifts to a homogeneous Hamiltonian isotopy Φ : T˙ ∗ S2 × I → − T˙ ∗ S2 which satisfies Φ0 = id and Φs (x; λ ξ) = × λ · Φs (x; ξ) for all λ ∈ R (not only for λ ∈ R>0 ) and all (x; ξ) ∈ T˙ ∗ S2 . Let x0 ∈ S2 be a given point. We set Λ0 = T˙x∗0 S2 , Λs = Φs (Λ0 ) and Σs = πS2 (Λs ). Theorem VIII.7.3. — Let s ∈ I be such that Σs is a curve with only cusps and double points as singularities. Then Σs has at least three cusps. Proof. — (i) We remark that the number of cusps of Σs is odd because Λs is connected. Hence we have to prove that Σs does not have one cusp. (ii) Let K = KΦ ∈ Dlb (k(S2 )2 ×I ) be the sheaf associated with Φ by Theorem II.1.1. For s ∈ I we set Ks = K | 2 2 ∈ Db (k(S2 )2 ). We also set F0 = k{x0 } and (S ) ×{s} ˙ s ) = Λs . We have DS2 (F0 ) ≃ F0 and by Lemma VIII.7.1 Fs = Ks ◦ F0 . Then SS(F we deduce that DS2 (Fs ) ≃ Fs . By Corollary II.1.5 we also have RΓ(S2 ; Fs ) ≃ RΓ(S2 ; F0 ) ≃ k for all s ∈ I. (iii) Now we consider a given s ∈ I so that Σs satisfies the hypotheses of the theorem. We choose a Morse function q : S2 → − R with only one minimum x− and one maximum x+ such that x− , x+ ̸∈ Σs . We can choose q such that any fiber q −1 (t), t ∈ R,

ASTÉRISQUE 440

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153

contains at most one of the following special points: tangent point between q −1 (t) and Σs , double point or cusp (in particular a cusp is not tangent to q −1 (t)). We set G = Rq∗ (Fs ). We have isomorphisms DR (G) ≃ Rq∗ (DS2 Fs ) ≃ G and RΓ(R; G) ≃ RΓ(S2 ; Fs )L≃ k. By Lemma VIII.7.2 there exist t0 ∈ R and a decomposition G ≃ k{t0 } ⊕ a∈A knIaa [da ] where the Ia are half-closed intervals. We set t± = q(x± ). Since Fs is constant near x− , we have G ≃ L ⊗ (k[t− ,+∞[ ⊕ k]t− ,+∞[ [−1]) near t− , where L = (Fs )x− . The same holds near t+ . Hence t0 ̸= t− and t0 ̸= t+ . Since k{t0 } is a direct summand of G, we have Tt∗0 R ⊂ SS(G). By Proposition I.2.4 this implies Λs ∩ Tq∗−1 (t0 ) S2 ̸= ∅. By the assumption on q it follows that q −1 (t0 ) is tangent to Σs at a single point and that q −1 (t0 ) contains no cusp or double point. By Lemma VIII.4.3 the intervals Ia cannot have an end at t0 . Up to a change of coordinates we can assume that t0 = 0 and that Λ ∩ T ∗ U , with U = q −1 (]−1, 1[), satisfies (VIII.2.6) and Fs satisfies (VIII.2.7) and (VIII.2.8). In particular Fs | is weakly constructible (see Proposition I.2.20 and Remark I.2.22). U Moreover F = K ◦ F0 ∈ Dlb (kS2 ×I ) has finite dimensional stalks at any point of ˙ (S2 × I) \ π˙ S2 ×I (SS(F )) by Lemma I.4.7. Hence Fs | is constructible. By TheoU rem VIII.6.1 we deduce Hom(Fs , Fs [1]) ̸≃ 0. By Corollary II.1.5 we have Hom(Fs , Fs [1]) ≃ Hom(F0 , F0 [1]) ≃ 0 and this gives a contradiction.

SOCIÉTÉ MATHÉMATIQUE DE FRANCE 2023

CHAPTER VIII.8 THE FOUR CUSPS CONJECTURE

In this short section we sketch a proof of Arnol’d’s four cusps conjecture along the same lines as the proof of Theorem VIII.7.3. It is proved by Chekanov and Pushkar in [40] and says the following. Let M = R2 and f : M → − R, x 7→ ∥x∥. Let S1 = f −1 (1) ± 1 be the unit circle and let Λ = {(x; λdf ); x ∈ S , ±λ > 0} be the outer and inner conormal bundles of S1 . Let Φ : T˙ ∗ M × I → − T˙ ∗ M be a homogeneous Hamiltonian − isotopy with 0, 1 ∈ I. We set Λs = Φs (Λ ) for s ∈ I and we assume that Λ1 = Λ+ . We make the same genericity hypotheses as in [40]: • for s ∈ I outside a finite set {s1 , . . . , sN }, Σs = π˙ M (Λs ) is a curve with only cusps and ordinary double points as singularities; • for s ∈ {s1 , . . . , sN }, Σs may have a birth/death of swallowtail, a triple point, a self-tangency or a double point where one of the points is a cusp. Then there exists s ∈ I such that Σs has at least four cusps. Here is a sketch of proof. (i) We let F0 = k{f ≤1} be the constant sheaf on the closed unit ball. Hence ˙ 0 ) = Λ0 . We let Λ′ ⊂ T˙ ∗ (M × I) be the image of Λ0 by the whole isotopy SS(F (see (II.1.5)). It is a conic Lagrangian submanifold which is non-characteristic for ∗ all inclusions is : M × {s} → − M × I (that is, Λ′ ∩ (TM M × Ts∗ I) = ∅) and sat′ ♯ isfies is (Λ ) = Λs . By Corollary II.1.5 there exists a unique F ∈ D(kM ×I ) such −1 ˙ that SS(F ) = Λ′ and i−1 0 F ≃ F0 . We set Fs = is F . We know that F is simple ˙ s ) = Λs and RΓ(M ; Fs ) is independent of s. along Λ′ , Fs has compact support, SS(F ˙ 1 ) = Λ+ , hence F1 is locally constant on M \ S1 . (ii) We compute F1 . We have SS(F Since F1 has compact support, it must be supported on the unit ball. The microsupport condition then implies that it is of the form F1 ≃ E{f 0 → − M is generically one to one). The fact that M is R2 and not S2 makes the situation easier. In particular an argument as in the step (ii) in the proof of Proposition VIII.2.3 gives + ∗ −1 −1 the following: let p+ q (±ε), for ε > 0 small 0 , p1 be the two points in Λs0 ∩TΓ+ M ∩π enough (we write ±ε because we do not know which side of q −1 (0) the branch Γ+ is + −1 situated). Let q0+ , q1+ be the conjugate points of p+ (±ε). Then 0 , p1 with respect to q + + Fs0 has different shifts at pi and qi . (See Fig. VIII.8.2 where we give two examples of possible Σs0 and some notations—we do not claim that there actually exist sheaves corresponding to these pictures.) − We have similar pairs {p− i , qi }. Using these four pairs of conjugate points we can now prove directly an analog of Theorem VIII.6.1 (there is no need for a third linked point as in Proposition VIII.5.1) as follows. Let us assume that Σs0 has only two cusps, say c1 , c2 . Then Λs0 \ (Λs0 ∩ (Tc∗1 M ∪ Tc∗2 M )) consists of two connected components, corresponding to two different shifts, say Λ0 , Λ1 . We assume that p+ ∈ Λ0 . By Proposition VIII.4.5 (with t0 = t1 = 0 and J = {0}) we know that p− and p+ are F -linked over any open subset containing q −1 (0). We distinguish three cases, (a), (b-i), (b-ii) (see Fig. VIII.8.3—the first picture in Fig. VIII.8.2 is compatible with (b-i) and the second one with (b-ii)): + (a) If p− ∈ Λ0 , the arc of Λ0 from p+ to p− contains p+ 0 or p1 . We choose the notations + + so that it contains p0 (see Fig. VIII.8.3 (a)). Then q0 ∈ Λ1 and we see that the + pairs {[p+ ], [p− ]} and {[p+ 0 ], [q0 ]} in Λs0 /R>0 are intertwined. By Proposition VIII.3.5 1 we deduce H RHom(Fs0 , Fs0 ) ̸= 0, contradicting H 1 RHom(F0 , F0 ) = 0. − (b) If p− ∈ Λ1 , we choose the notations so that p+ 0 , p0 and c1 belong to the same − + − arc of Λs0 /R>0 joining p and p . We remark that the open arc ]p− 0 , p1 [ is contained + + − −1 in q (]−ε, ε[). Since q(q0 ) = ±ε, q0 belongs to the arc ]c1 , p0 ] or to the arc ]c2 , p− 1 ]. + − It could happen that q0+ = p− , but this implies that Γ and Γ are on the same side 0

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c2

p+ 0

q

−1

(ε)

q

−1

(0)

q0+

p+ 1 p+

p−

q −1 (−ε)

p− 1

p− 0

c1 c1 c2

q

−1

q

−1

q

−1

q0−

p+ 1

(ε)

p− 0

(0)

p+ 0

q0+

p+

p− 1 p−

(−ε)

Figure VIII.8.2. Two examples of Σs0 – the branches Γ± are the dotted parts

p+ + 1

c1

p

q0+

p+ 0 p− 0 p−

p− 1

c1

c1 q0− p− 0 p− p− 1

p+ 0 p+ p+ 1

c2

c2

(a)

(b-i)

q0+

Figure VIII.8.3.

ASTÉRISQUE 440

q0+ p− 0 p− p− 1

p+ 0 p+ p+ 1 c2 (b-ii)

CHAPTER VIII.8. THE FOUR CUSPS CONJECTURE

159

+ − − of q −1 (0) and that πM (p+ 1 ) < πM (p0 ) < πM (p0 ) < πM (p1 ) for the order on the line + − + − q −1 (ε); hence we cannot have both q0 = p0 and q1 = p1 . Up to switching 0 and 1, + − we can assume q0+ ̸= p− 0 . In other words p0 and p0 are not conjugate, and we also + − + + have q0− ̸= p+ 0 . Similarly as q0 , q0 belongs to the arc ]c1 , p0 [ or to the arc ]c2 , p1 ]. + − + − (b-i) If q0 belongs to ]c2 , p1 ] (see Fig. VIII.8.3 (b-i)), then the pairs {[p ], [p ]} and + − + + − {[p+ 0 ], [q0 ]} are intertwined. If q0 belongs to ]c2 , p1 ], then the pairs {[p ], [p ]} and − − {[p0 ], [q0 ]} are intertwined. In both cases we conclude as in case (a). − (b-ii) If none of these two cases occurs, then q0+ belongs to ]c1 , p− 0 [, q0 be+ + + − − longs to ]c1 , p0 [ and the pairs {[p0 ], [q0 ]} and {[p0 ], [q0 ]} are intertwined (see Fig. VIII.8.3 (b-ii)). Since F -conjugate points are F -linked (see Proposition VIII.4.4), we can again apply Proposition VIII.3.5, after moving the pairs so that they are not −1 ′ on the same line (choose p− (ε ) for ε′ slightly different from ε). 0 on a line q

SOCIÉTÉ MATHÉMATIQUE DE FRANCE 2023

PART IX TRIANGULATED ORBIT CATEGORIES FOR SHEAVES

In the study of Lagrangian exact submanifolds of cotangent bundles in Part XIII we will use triangulated orbit categories. For a triangulated category T , the triangulated envelope of the orbit category of T is a triangulated category T ′ with a functor ι : T → − T ′ such that ι(F ) ≃ ι(F )[1] for any F ∈ T and L HomT ′ (ι(F ), ι(G)) ≃ i∈Z HomT (F, G[i]). Such categories are constructed by Keller in [31]. We specify his construction in the case of categories of sheaves and check that we can define a microsupport in this framework.

CHAPTER IX.1 DEFINITION OF TRIANGULATED ORBIT CATEGORIES

We will use a very special case of the triangulated hull of an orbit category as described by Keller in [31]. More precisely Definition IX.1.1 below is inspired by §7 of [31] that we apply to the simple case where we quotient Db (kM ) by the autoequivalence F 7→ F [1] (in [31] much more general equivalences are considered). However we apply this construction for sheaves instead of modules over an algebra. We use the name triangulated orbit category for the category D/[1] (kM ) introduced in Definition IX.1.1 by analogy with the categories introduced by Keller. However we only show that we have a functor ιM : Db (kM ) → − D/[1] (kM ) such that ιM (F ) ≃ ιM (F )[1] and which satisfies Corollary IX.1.9 below (the proof of this result is also inspired by [31]). In this section we assume k = Z/2Z.

Quick reminder on localization We recall quickly some facts about quotients of triangulated categories. We follow the exposition of [28, §1.6] or [29, §10.2]. Let D be a triangulated category. A multiplicative system S in D is a family of morphisms such that (S1) any isomorphism is in S , (S2) S is stable by composition of morphisms, (S3) for given f : X → − Y and s : X → − X ′ , with s ∈ S , there exist t : Y → − Y ′ and ′ ′ g: X → − Y , with t ∈ S , such that g ◦ s = t ◦ f , and the same holds with all X arrows reversed (visualized by

f

s

Y

X t

g

and

f

s

Y t

)

g

X′ Y′ X′ Y′ (S4) For f, g : X ⇒ Y , there exists s : W → − X in S such that f ◦ s = g ◦ s if and only if there exists t : Y → − Z in S such that t ◦ f = t ◦ g. The localization of D by S is then the category DS with the same objects as D and with morphisms HomDS (X, Y ) = {(X ′ , s, f ); s : X ′ → − X, f : X ′ → − Y with s ∈ S , f any morphism }/ ∼, where the equivalence relation ∼ is generated by:

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CHAPTER IX.1. DEFINITION OF TRIANGULATED ORBIT CATEGORIES

s ′

X′

f

′′

(X , s, f ) ∼ (X , t, g) if there exists a commutative diagram X

Y. t

g

X ′′ The composition is defined using the property (S3). The localization comes with a functor QS : D → − DS such that, for any s ∈ S , QS (s) is an isomorphism. Moreover the pair (DS , QS ) is universal with respect to this property. (Up to now we did not use the triangulated structure.) We now define a distinguished triangle in DS as a triangle A → − B → − C → − A[1] which is isomorphic in DS to the image by QS of a distinguished triangle of D. This turns DS into a triangulated category. In the framework of triangulated categories, a convenient way to obtain a multiplicative system is to start with a full triangulated subcategory N of D which is saturated in the following sense: for any isomorphism X ≃ Y in D, if X ∈ N , then Y ∈ N . We then define SN as the family of morphisms s in D such that the cone of s belongs to N . We can check that SN is a multiplicative system and consider the localization DSN . Then, for any N ∈ N , QSN (N ) ≃ 0; moreover (DSN , QSN ) is universal with respect to this property. We call DSN the quotient of D by N and denote it by D/N . The localization is used to define the derived category. If C is an abelian category, we let C(C ) be the category of complexes of objects of C and Cb (C ) its full subcategory of bounded complexes. We define K(C ) as the category with the same objects as C(C ) and morphisms HomK(C ) (X, Y ) = HomC(C ) (X, Y )/ ∼ where ∼ is the homotopy equivalence of morphisms. We define Kb (C ) as the full subcategory of K(C ) of bounded complexes. These categories K(C ), Kb (C ) are triangulated. We recall that u : X → − Y is a quasiisomorphism if H n (u) : H n (X) → − H n (Y ) is an isomorphism for all n ∈ Z. Then D(C ) = (K(C ))qis and Db (C ) = (Kb (C ))qis , where qis denotes the family of quasiisomorphisms. We have a natural functor Db (C ) → − D(C ) which identifies Db (C ) as b a full subcategory of D(C ). It is classical that D (C ) is also equivalent to the full subcategory of D(C ): (IX.1.1)

{X ∈ D(C ); H n (X) ≃ 0 for n ≪ 0 and n ≫ 0}.

We use similar constructions to define the derived categories of complexes bounded from below, D+ (C ), or above, D− (C ). If C ′ is another abelian category, any additive functor F : C → − C ′ induces a functor, ′ F1 : K(C ) → − K(C ), compatible with the triangulated structures. If F is exact, then F1 sends quasi-isomorphisms to quasi-isomorphisms and Q ◦ F1 sends quasi-isomorphisms to isomorphisms, where Q : K(C ′ ) → − D(C ′ ) is the localization functor. By the universal property, Q ◦ F1 factorizes through a functor (IX.1.2)

F : D(C ) → − D(C ′ )

(compatible with the triangulated structures). In the same way F induces functors, all denoted F , D∗ (C ) → − D∗ (C ′ ), where ∗ stands for b, + or −.

ASTÉRISQUE 440

165

DEFINITION OF THE ORBIT CATEGORY

In general, if F is only additive, Q ◦ F1 does not factorizes through D(C ) but, under some assumptions (for example, C has enough injectives), we can define a “best possible approximation” to a factorization, which is called the right derived functor of F , RF : D+ (C ) → − D+ (C ′ ).

(IX.1.3)

If F is left exact, we have F ≃ H 0 ◦ RF ◦ ι, where ι is the embedding of C as the full subcategory of D+ (C ) of complexes concentrated in degree 0 (if F is not left exact, F and RF are not related in a useful way). If F is exact, then F ≃ RF . If C has enough projectives, we can also derive functors on the left. However we are interested in categories of sheaves, which do not have enough projectives. For a specific functor F , we can still derive F on the left, if C has enough F -projective objects. We are interested in the case of the tensor product of sheaves; the ⊗-projective sheaves L

are the flat sheaves and there are enough flat sheaves. So we can define ⊗ on the bounded from above derived categories of sheaves: L

⊗kM : D− (kM ) × D− (kM ) → − D− (kM ),

(IX.1.4)

where M is any manifold and k any ring.

Definition of the orbit category We recall that k = Z/2Z and we set K = k[X]/⟨X 2 ⟩. We let ε be the image of X in K. Hence K = k[ε] with ε2 = 0. Let M be a manifold. We denote as usual by Mod(kM ) (resp. Mod(KM )) the category of sheaves of k-modules (resp. K-modules) on M . We set Db (kM ) = Db (Mod(kM )) and Db (KM ) = Db (Mod(KM )), in the sense of (IX.1.1). The natural ring morphisms k → − K and K → − k induce two pairs of adjoint functors (eM , rM ) and (EM , RM ), where eM , EM are scalar extensions and rM , RM restrictions of scalars (or rather, their derived versions, see the discussion around (IX.1.2)-(IX.1.4) for a quick reminder): eM

Db (kM ) ⇄ Db (KM ), rM

eM (F ) = KM ⊗kM F,

Db (KM ) ←−− Db (kM ),

RM (F ) = F,

RM

EM

D− (KM ) ⇄ D− (kM ), RM

rM (G) = G,

L

EM (G) = kM ⊗KM G, RM (F ) = F.

We will sometimes use the fact that rM is conservative, that is, for a morphism u : F → − G in Db (KM ), if rM (u) is an isomorphism, so is u. Indeed rM is the derived functor of an exact functor and the assertion follows from the case of the module categories Mod(KM ), Mod(kM ), where it is clear. The ring morphism K → − k gives a basis of Homk (K, k) (which is a free K-module). ∼ It induces KM −− → RHom kM (KM , kM ) and then a canonical isomorphism, for all

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F ∈ Db (kM ), KM ⊗kM F ≃ RHom kM (KM , F )

(IX.1.5)

and we deduce the isomorphisms: HomDb (KM ) (G, eM (F )) ≃ HomDb (KM ) (G, RHom kM (KM , F )) (IX.1.6)

≃ HomDb (kM ) (G ⊗KM KM , F ) ≃ HomDb (kM ) (rM (G), F ).

Definition IX.1.1. — We let E(KM ) be the full triangulated subcategory of Db (KM ) generated by the image of eM , that is, by the objects of the form KM ⊗kM F with F ∈ Db (kM ). We denote by D/[1] (kM ) the quotient category Db (KM )/E(KM ). We let QM : Db (KM ) → − D/[1] (kM ) be the quotient functor and we set ιM = QM ◦ RM : Db (kM ) → − D/[1] (kM ). An object of E(KM ) is obtained by taking iterated cones of objects in eM (Db (kM )). There are objects in E(KM ) which are not of the form eM (F ). For example, when M is a point, the objects of Db (k) are split (sum of their cohomology) and so are the objects of e{pt} (Db (k)), but the object Lp,q ∈ E(K) defined in (IX.1.17) below is not split. Let E′ (KM ) be the subcategory of Db (KM ) formed by the P such that QM (P ) ≃ 0. ∼ Then E(KM ) ⊂ E′ (KM ) and we also have D/[1] (kM ) −− → Db (KM )/E′ (KM ). We do ′ not know whether E (KM ) = E(KM ). A general result (see [29] Ex. 10.11) says that P ∈ E′ (KM ) if and only if P ⊕ P [1] ∈ E(KM ). Notation IX.1.2. — If the context is clear, we will not write the functor QM or RM , that is, for F ∈ Db (kM ) we often write F instead of RM (F ), and, for G ∈ Db (KM ), we often write G instead of QM (G). In particular for a locally closed subset Z ⊂ M , we consider kZ = RM (kZ ) ∈ Db (KM ) and kZ = QM (kZ ) ∈ D/[1] (kM ). The exact sequence of K-modules 0 → − k→ − K→ − k→ − 0 induces a morphism (IX.1.7)

in Db (KM )

sM : kM → − kM [1]

and a distinguished triangle, for any F ∈ Db (kM ), s

⊗id

M F RM (F ) → − eM (F ) → − RM (F ) −− −−−→ RM (F )[1]. ∼ We thus obtain an isomorphism sM ⊗ idF : RM (F ) −− → RM (F )[1] in D/[1] (kM ), for b any F ∈ D (kM ). This would work for any field k. In characteristic 2 we can generalize this isomorphism to any F ∈ Db (KM ) (see Remark IX.1.4).

(IX.1.8)

ASTÉRISQUE 440

INTERNAL TENSOR PRODUCT AND HOMOMORPHISM

167

Internal tensor product and homomorphism For two K-modules E1 , E2 we define E1 ⊗εk E2 ∈ Mod(K) as follows. The underlying k-vector space is E1 ⊗k E2 and ε acts by ε · (x ⊗ y) = (εx) ⊗ y + x ⊗ (εy). Since the characteristic is 2, we can check that ε2 acts by 0 and this defines an object of Mod(K) that we denote E1 ⊗εk E2 . − Mod(K). For We obtain in this way a bifunctor ⊗εk : Mod(K) × Mod(K) → F1 , F2 ∈ Mod(KM ), we define F1 ⊗εkM F2 ∈ Mod(KM ) as the sheaf associated with the presheaf U 7→ F1 (U )⊗εk F2 (U ). We remark that rM (F1 ⊗εkM F2 ) ≃ rM (F1 )⊗kM rM (F2 ), where rM is seen here as a functor Mod(KM ) → − Mod(kM ), and it follows easily that ⊗εkM is an exact functor. Hence it induces a functor on the derived category (see the discussion around (IX.1.2)), denoted in the same way: (IX.1.9)

− Db (KM ). ⊗εkM : Db (KM ) × Db (KM ) →

For any F, G ∈ Db (KM ) we have canonical isomorphisms (IX.1.10)

kM ⊗εkM F ≃ F ⊗εkM kM ≃ F

in Db (KM ),

(IX.1.11)

rM (F ⊗εkM G) ≃ rM (F ) ⊗kM rM (G)

in Db (kM ).

Using (IX.1.10) and the exact sequence 0 → − k → − K → − k → − 0, we obtain as in (IX.1.7)–(IX.1.8) a morphism sM (F ) : F → − F [1], for any F ∈ Db (KM ), and a distinguished triangle (IX.1.12)

sM (F )

F → − KM ⊗εkM F → − F −−−−→ F [1].

Using the adjunction (eM , rM ) and (IX.1.11) we have the isomorphism, for any F ∈ Db (kM ) and G ∈ Db (KM ), (IX.1.13)

HomDb (KM ) (eM (F ⊗kM rM (G)), eM (F ) ⊗εkM G) ≃ HomDb (kM ) (F ⊗kM rM (G), (rM eM (F )) ⊗kM rM (G)).

By adjunction we have a morphism aF : F → − rM eM (F ). The inverse image of aF ⊗ idrM (G) by (IX.1.13) gives a canonical morphism, for any F ∈ Db (kM ) and G ∈ Db (KM ), (IX.1.14)

eM (F ⊗kM rM (G)) → − eM (F ) ⊗εkM G.

Lemma IX.1.3. — Let F ∈ Db (kM ) and G ∈ Db (KM ). Then the morphism (IX.1.14) is an isomorphism. In the same way G⊗εkM eM (F ) ≃ eM (rM (G)⊗kM F ). In particular, for F, G ∈ Db (KM ) such that F or G belongs to E(KM ), we have F ⊗εkM G ∈ E(KM ) and ⊗εkM induces a functor ⊗εkM : D/[1] (kM ) × D/[1] (kM ) → − D/[1] (kM ). Proof. — (i) Let us denote by uF,G the morphism (IX.1.14). Using the distinguished +1

triangle τ≤i F → − F → − τ>i F −−→ and the similar one for G we can argue by induction on the length of F and G to prove that uF,G is an isomorphism. Then we are reduced

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to the case where F and G are concentrated in degree 0. Writing K = k ⊕ εk we have eM (F ⊗kM rM (G)) = (F ⊗ G) ⊕ ε(F ⊗ G) and eM (F ) ⊗εkM G = (F ⊕ εF ) ⊗ G. For sections f and g of F and G we have uF,G (f ⊗ g) = f ⊗ g, uF,G (ε(f ⊗ g)) = ε · (f ⊗ g) = (εf ) ⊗ g + f ⊗ (εg) and we can check directly that uF,G is an isomorphism with inverse u−1 F,G ((f0 + εf1 ) ⊗ g) = (f0 ⊗ g + f1 ⊗ (εg)) + ε(f1 ⊗ g). (ii) For any F ∈ E(KM ), there exist a sequence F1 , F2 , . . . , Fn = F ∈ Db (KM ) and +1 distinguished triangles Fi → − Fi′ → − Fi′′ −−→ in Db (KM ), i = 1, . . . , n, such that Fi′ , Fi′′ belong to eM (Db (kM )) ∪ {F1 , F2 , . . . , Fi−1 }. Then (i) and an induction on n give F ⊗εkM G ∈ E(KM ). Remark IX.1.4. — An easy case of Lemma IX.1.3 is F = kM in (IX.1.14). We obtain ∼ eM rM (G) −− → KM ⊗εkM G, for any G ∈ Db (KM ). Hence the distinguished triangle (IX.1.12) becomes (IX.1.15) Applying QM

sM (F )

for any F ∈ Db (KM ). ∼ to this triangle gives an isomorphism sM (F ) : F −− → F [1] in D/[1] (kM ). F → − eM rM (F ) → − F −−−−→ F [1]

We can define an adjoint Hom ε to ⊗ε by a similar construction. For F1 , F2 ∈ Mod(KM ), we define Hom ε (F1 , F2 ) ∈ Mod(KM ) as the sheaf of k-vector spaces Hom k (F1 , F2 ) with the action of ε given by (ε · φ)(x) = εφ(x) + φ(εx), where φ is a section of Hom k (F1 , F2 ) over an open set U and x a section of F1 over a subset V of U . Then we see that Hom ε is right adjoint to ⊗ε , hence left exact. We check also that its derived functor RHom ε is right adjoint to ⊗ε in Db (KM ) and that, for any F, G ∈ Db (KM ), (IX.1.16)

rM (RHom ε (F, G)) ≃ RHom (rM (F ), rM (G)).

We have the similar result as Lemma IX.1.3. Lemma IX.1.5. — Let F, G ∈ Db (KM ). We assume that F or G belongs to E(KM ). Then RHom ε (F, G) ∈ E(KM ). The induced functor RHom ε : D/[1] (kM )op × D/[1] (kM ) → − D/[1] (kM ) is right adjoint to ⊗εkM .

Morphisms in the triangulated orbit category We prove the Formula (IX.1.20) which describes the morphisms in D/[1] (kM ).

ASTÉRISQUE 440

169

MORPHISMS IN THE TRIANGULATED ORBIT CATEGORY

Lemma IX.1.6. — Let F, P ∈ Db (KM ). We assume that P ∈ E(KM ). Then RHom(P, F ) and RHom(F, P ) belong to Db (K). Proof. — Since E(KM ) is generated by eM (Db (kM )), the same argument as in (ii) of the proof of Lemma IX.1.3 implies that we can assume P = eM (Q), for some Q ∈ Db (kM ). Then RHomK (P, F ) is isomorphic to RHomk (Q, rM (F )) and is bounded. Using (IX.1.6) the same proof gives that RHom(F, P ) is bounded. We define the following objects Lp,q of Db (K), for any two integers p ≤ q, by (IX.1.17)

ε

ε

ε

Lp,q = 0 → − K− →K− → ··· − →K→ − 0,

where the first K is in degree p and the last one in degree q. Then Lp,q ∈ E(K) and there is a distinguished triangle in Db (K), (IX.1.18)

sp,q

k[−p] → − Lp,q → − k[−q] −−→ k[−p + 1],

with sp,q = sq−p+1 {pt} [−q], where s{pt} is (IX.1.7). b p,q For F ∈ D (KM ) we define sp,q ⊗ idF : F [−q] → − F [−p + 1]. We deduce M (F ) := s b the triangle, for any F ∈ D (KM ) and any n ≥ 1, s1,n (F )

+1

ε − F [−n] −−M−−−→ F −−→ . L1,n M ⊗kM F → +1

Lemma IX.1.7. — We consider a distinguished triangle P → − F′ → − F −−→ in Db (KM ) and we assume that P ∈ E(KM ). Then there exist n ∈ N and a morphism of triangles ε L1,n M ⊗kM F

F [−n]

P

F′

s1,n M (F )

F

F

+1

+1

.

Proof. — We set for short sn = s1,n M (F ). We consider the diagram F [−n] a

P

F′

sn

F

sn

F

w

P [1] .

We have w ◦ sn ∈ Hom(F [−n], P [1]). Since P ∈ E(KM ) this group vanishes for n big enough, by Lemma IX.1.6. In this case we have w ◦ sn = 0 and there exists a morphism a as in the diagram making the square commute. Then we can extend the square to a commutative diagram as in the lemma. For F ∈ Db (KM ) we have sn,n − F [−n + 1]. Then {F [−n], sn,n M (F ) : F [−n] → M (F )}n∈N gives a projective system. For G ∈ Db (KM ) we define (IX.1.19)

lim HomDb (KM ) (F [−n], G) → − HomD/[1] (kM ) (F, G), −→

n∈N

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−1 by sending φn : F [−n] → − G to φn ◦ (s1,n . This is well defined since s1,n M (F )) M (F ) 1,n 1,n−1 n,n becomes invertible in D/[1] (kM ) and sM (F ) = sM (F ) ◦ sM (F ).

Proposition IX.1.8. — Let F, G ∈ Db (KM ). Then the inductive limit in the left hand side of (IX.1.19) stabilizes and the morphism (IX.1.19) is an isomorphism. More precisely, if H i (F ) = H i (G) = 0 for all i outside an interval [a, b], then ∼ (IX.1.20) Hom b (F [−n], G) −− → HomD (k ) (F, G), D (KM )

/[1]

M

for all n > b − a + dim M + 1. Proof. — (i) We prove that the limit stabilizes. We choose a ≤ b such that H i (F ) = H i (G) = 0 for all i outside [a, b]. By (IX.1.15) we have the distinguished triangle HomDb (KM ) (eM rM (F )[−n], G) → − HomDb (KM ) (F [−n], G) (IX.1.21) s′n +1 −→ HomDb (KM ) (F [−n − 1], G) −−→, for all n ∈ Z. By adjunction we have HomDb (KM ) (eM rM (F )[−n], G) ≃ HomDb (kM ) (rM (F )[−n], rM (G)) and this is zero for n > b − a + dim M + 1 (recall that the flabby dimension of a manifold M is dim M + 1 and that injective over a field is the same as flabby). It follows that the morphism s′n in (IX.1.21) is an isomorphism for n > b−a+dim M +1. (ii) We prove that (IX.1.19) is an isomorphism. We recall that HomD/[1] (kM ) (F, G) ≃

lim −→

i : F ′→ −F ′

HomDb (KM ) (F ′ , G),

where the limit runs over the morphisms i : F → − F whose cone belongs to E(KM ) (and a morphism u : F ′ → − G is send to u ◦ i−1 in D/[1] (kM )). By Lemma IX.1.7 we can restrict to the family of morphisms s1,n − F for n ∈ N. This gives M (F ) : F [−n] → the result. Corollary IX.1.9. — Let F, G ∈ Db (kM ). Let ιM = QM ◦ RM . We have M HomD/[1] (kM ) (ιM (F ), ιM (G)) ≃ HomDb (kM ) (F [−n], G). n∈Z

Proof. — By Proposition IX.1.8 the left hand side of the formula is isomorphic to HomDb (KM ) (RM (F )[−n0 ], RM (G)) ≃ HomD− (kM ) (EM RM (F )[−n0 ], G), for any big enough n0 ∈ N. Using the resolution of k as a K-module given by L ε ε ··· → − K− →K− →K→ − k→ − 0, we see that EM RM (F ) ≃ i∈N F [i]. The result follows easily. This last result says in particular that ιM is faithful.

ASTÉRISQUE 440

DIRECT AND INVERSE IMAGES

171

We define ι0M : Mod(kM ) → − D/[1] (kM ) as the composition of ιM and the embedb ding Mod(kM ) → − D (kM ) which sends a sheaf to a complex concentrated in degree 0. For F ∈ D/[1] (kM ) we let h0M (F ) be the sheaf associated with the presheaf U 7→ HomD/[1] (kU ) (kU , F | ). This defines a functor h0M : D/[1] (kM ) → − Mod(kM ). U

Corollary IX.1.10. — For all F ∈ Mod(kM ) we have h0M (ι0M (F )) ≃ F . When M is a point, the functors ι0 and h0 are mutually inverse equivalences of categories between Mod(k) and D/[1] (k). L n Proof. — (i) We have HomD/[1] (kU ) (kU , ιM (F )| ) ≃ n∈Z H (U ; F ) by CorolU n n lary IX.1.9. The sheaf associated with U 7→ H (U ; F ) is H F and we obtain h0M (ιM (F )) ≃ F when F is in degree 0. (ii) When M is a point, Corollary IX.1.9 says that ι0 is fully faithful. Let us prove that it is essentially surjective. Any G ∈ Mod(K) can be written as an extension 0 → − R(F ) → − G → − R(F ′ ) → − 0 with F, F ′ ∈ Mod(k). It follows that any object in D/[1] (k) is obtained from objects in ι0 (Mod(k)) by taking iterated cones. Hence to prove that ι0 is essentially surjective, it is enough to see that if we have a distinguished u +1 triangle ι0 (F ) − → ι0 (F ′ ) → − G −−→ in D/[1] (k), with F, F ′ ∈ Mod(k), then G ≃ ι0 (F ′′ ) for some F ′′ ∈ Mod(k). Since ι0 is fully faithful, we have u ∈ HomMod(k) (F, F ′ ). Then G ≃ coker(u) ⊕ ker(u)[1] ≃ coker(u) ⊕ ker(u) and the result follows.

Direct sums We recall that we denote by QM the quotient functor Db (KM ) → − D/[1] (kM ). Lemma IX.1.11. — Let I be a small set and {Fi }i∈I a family in Db (KM ). We assume k that there exist = 0 for all k outside L [a, b] and all L two integers a ≤ b such that H (Fi )L i ∈ I. Then i∈I QM (Fi ) exists in D/[1] (kM ) and i∈I QM (Fi ) ≃ QM ( i∈I Fi ). L Proof. — By the hypothesis on the degrees the sum i∈I Fi exists in Db (KM ) and L we have H k ( i∈I Fi ) = 0 for k outside [a, b]. Let G ∈ Db (KM ) and let a′ ≤ a, b′ ≥ b be such that H k (G) = 0 for k outside [a′ , b′ ]. We set n = b′ − a′ + dim M + 2. If F = Fi L for some i ∈ I, or F = i∈I Fi , we have ∼ Hom b (F [−n], G) −− → HomD (k ) (QM (F ), QM (G)) D (KM )

/[1]

M

by Proposition IX.1.8. Now the lemma follows from the universal property of the sum.

Direct and inverse images Let f : M → − N be a morphism of manifolds. We have functors Rf∗ , Rf! , f −1 and f , between Db (KM ) and Db (KN ). Indeed, the functors Rf∗ , Rf! : D(KM ) → − D(KN ) commute with the functors rN : D(KN ) → − D(kN ) and rM . We remark that, for !

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G ∈ D(KN ), if rN (G) ∈ Db (kN ), then G ∈ Db (KN ). Hence Rf∗ and Rf! induce functors Db (KM ) → − Db (KN ) between the bounded categories. −1 The case of f is clear since it is an exact functor. To prove the existence of f ! , right adjoint to Rf! , it is enough, by factorizing through the graph embedding, to consider the cases where f is an embedding or f is a submersion. The usual formulas work in our case. If f is an embedding, then f ! (·) = f −1 RHom ε (kM , ·) is adjoint to Rf! . If f is a submersion, then f ! (·) = f −1 (·) ⊗εkM ωM |N . We also remark that f −1 and f ! commute with rN and rM . Lemma IX.1.12. — Let f : M → − N be a morphism of manifolds. Then the functors Rf∗ , Rf! , f −1 and f ! , between Db (KM ) and Db (KN ), preserves the categories E(KM ) and E(KN ). They induce pairs of adjoint functors (that we denote in the same way) between D/[1] (kM ) and D/[1] (kN ). Proof. — We only consider the case of Rf∗ , the other cases being similar. For F ∈ Db (kM ) we have a natural morphism u : KN ⊗kN Rf∗ F → − Rf∗ (KM ⊗kM F ). 2 Since rN (KN ) ≃ kN we see easily that rN (u) is an isomorphism. Since rN is conservative, u is an isomorphism and we obtain Rf∗ (KM ⊗kM F ) ∈ E(KN ). It follows as in (ii) of the proof of Lemma IX.1.3 that Rf∗ (E(KM )) ⊂ E(KN ), as required. For F ∈ D/[1] (kM ) and j : U → − M the inclusion of a locally closed subset, we use the standard notations F | = j −1 F , FU = Rj! j −1 F , RΓU (F ) = Rj∗ j −1 F , U RΓ(U ; F ) = RaU ∗ (F | ) ∈ D/[1] (k), where aU is U → − {pt}, and RΓc (U ; F ) = U RaU ! (F | ) ∈ D/[1] (k). We have the same formulas as in Db (kM ): U

FU ≃ F ⊗εkM kU ,

RΓU (F ) ≃ RHom ε (kU , F ).

We also define RHomε (F, G) = RΓ(M ; RHom ε (F, G)) ∈ D/[1] (k). The adjunctions ε ε (a−1 M , RaM ∗ ) and (⊗kM , RHom ) give HomD/[1] (kM ) (F, G) ≃ HomD/[1] (k) (k, RHomε (F, G))

(IX.1.22)

≃ HomD/[1] (kM ) (kM , RHom ε (F, G)).

Let N be a submanifold of M . We recall that Sato’s microlocalization is a functor µN : Db (kM ) → − Db (kTN∗ M ). It is defined by composing direct and inverse images functors and Lemma IX.1.12 implies that it induces functors, denoted in the same way: (IX.1.23)

µN : Db (KM ) → − Db (KTN∗ M ),

(IX.1.24)

µN : D/[1] (kM ) → − D/[1] (kTN∗ M ).

Definition IX.1.13. — Let q1 , q2 : M × M → − M be the projections. We identify ∗ ∗ T∆ (M × M ) with T M through the first projection. For F, G ∈ D/[1] (kM ) we define M as in (I.3.1) µhomε (F, G) = µ∆M (RHom ε (q2−1 F, q1! G))

∈

D/[1] (kT ∗ M ).

The following result follows from the analogous one in Db (KM ).

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DIRECT AND INVERSE IMAGES

Lemma IX.1.14. — Let {Fi }i∈I a small family in Db (KM ) satisfying the hypotheses of Lemma IX.1.11. Let f : M ′ → − M and g : M → − M ′′ be morphisms of manifolds and let G ∈ D/[1] (kM ). Then we have canonical isomorphisms L L f −1 ( i∈I QM (Fi )) ≃ i∈I f −1 QM (Fi ), L L Rg! ( i∈I QM (Fi )) ≃ i∈I Rg! QM (Fi ), L L ( i∈I QM (Fi )) ⊗εkM G ≃ i∈I (QM (Fi )) ⊗εkM G). Lemma IX.1.15. — Let U be an open subset of M . Let F ∈ Db (KM ) and F ′ ∈ Db (KU ). We assume that there exists an isomorphism F | ≃ F ′ in D/[1] (kU ). Then there exists U F1 ∈ Db (KM ) such that F1 | ≃ F ′ in Db (KU ) and F1 ≃ F in D/[1] (kM ). U

Proof. — We let j : U → − M be the inclusion and we set Z = M \ U . Let u : F | → − F′ U be an isomorphism in D/[1] (kU ). By Proposition IX.1.8 there exist n ∈ Z and a morphism F [−n] → − F ′ in Db (KM ) which represents u. Defining P by the distinguished +1 triangle F | [−n] → − F′ → − P −−→ we have QU (P ) ≃ 0. We apply j! to this triangle U and get (IX.1.26) below; we also consider the excision triangle (IX.1.25) and the triangle (IX.1.27) built on the composition FZ [−n − 1] → − FU [−n] → − j! F ′ : (IX.1.25) (IX.1.26) (IX.1.27)

a

+1

FZ [−n − 1] − → FU [−n] → − F [−n] −−→, b

+1

FU [−n] → − j! F ′ → − j! P −−→, b◦a

+1

FZ [−n − 1] −−→ j! F ′ → − F1 −−→ . +1

Then the octahedron axiom gives the triangle F [−n] → − F1 → − j! P −−→. We have QM (j! P ) ≃ j! QM (P ) ≃ 0, hence F1 ≃ F [−n] ≃ F in D/[1] (kM ). Applying j −1 to the triangle (IX.1.27) gives F ′ ≃ F1 | , as required. U

Definition IX.1.16. — For F ∈ D/[1] (kM ) we define supporb (F ) ⊂ M as the complement of the union of the open subsets U ⊂ M such that F | ≃ 0. U

For an open subset U ⊂ M we have F | ≃ 0 in D/[1] (kU ) if and only if FU ≃ 0 U in D/[1] (kM ). By the Mayer-Vietoris triangle we deduce that, for a finite covering Sn U = i=1 Ui , we have F | ≃ 0 if and only if F | ≃ 0 for all i. For an increasing Ui S∞ U countable union U = i=1 Ui we have an exact sequence 0→ −

∞ M i=1

id−t

kUi −−−→

∞ M

s

kUi − → kU → − 0

i=1

in Mod(kM ), where t is the sum of the morphisms ti : kUi → − kUi+1 induced by the inclusions Ui ⊂ Ui+1 and s the sum of the morphisms si : kUi → − kU (the exactness is easily checked in the stalks). Turning this sequence into a distinguished triangle

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in D(kM ), then in D/[1] (kM ), and applying F ⊗εkM − we obtain a similar triangle (IX.1.28)

∞ M i=1

FUi → −

∞ M

+1

FUi → − FU −−→

i=1

and we deduce as in the finite case that F | ≃ 0 if F | ≃ 0 for all i. We obtain U Ui finally, for any F ∈ D/[1] (kM ), ∼ (IX.1.29) F| ≃ 0 and F −− → Fsupporb (F ) . M \supporb (F )

ASTÉRISQUE 440

CHAPTER IX.2 MICROSUPPORT IN THE TRIANGULATED ORBIT CATEGORIES

We define the microsupport of objects of D/[1] (kM ) and check that it satisfies the same properties as the usual microsupport. We recall that the microsupport is invariant by restriction of scalars, that is, for F ∈ D(KM ), we have SS(rM (F )) = SS(F ). We deduce that Theorem I.2.13, about L

the microsupports of F ⊗ G and RHom (F, G), for F, G ∈ D(KM ), is still true if we L

replace ⊗ and RHom by ⊗εkM and RHom ε , because of (IX.1.11) and (IX.1.16). IX.2.1. Definition and first properties We define the microsupport SSorb (F ) of an object of F ∈ D/[1] (kM ) from the microsupports of its representatives in Db (KM ). We prove in Proposition IX.2.3 that, for a given x0 ∈ M and F ∈ D/[1] (kM ), we can find a representative F ′ ∈ Db (KM ) with Tx∗0 M ∩ SS(F ′ ) contained in an arbitrary neighborhood of Tx∗0 M ∩ SSorb (F ) . Definition IX.2.1. — Let F ∈ D/[1] (kM ). We define SSorb (F ) ⊂ T ∗ M by SSorb (F ) = T ′ ′ b ′ F ′ SS(F ) where F runs over the objects of D (KM ) such that F ≃ F in D/[1] (kM ). orb ˙ orb (F ) = SS (F ) ∩ T˙ ∗ M . We set SS We remark that SSorb (F ) is a closed conic subset of T ∗ M . We deduce from Lemma IX.1.15 that SSorb (F ) is a local notion, that is, for U ⊂ M open, we have (IX.2.1)

SSorb (F | ) = SSorb (F ) ∩ T ∗ U. U

orb

In other words p = (x; ξ) ̸∈ SS (F ) if and only if there exist a neighborhood U of x and F ′ ∈ Db (KU ) such that F | ≃ F ′ in D/[1] (kU ) and p ̸∈ SS(F ′ ). We also U ∗ ∗ have supporb (F ) = TM M ∩ SSorb (F ). Indeed we have TM M ∩ SSorb (F ) ⊂ supporb (F ) by (IX.2.1). Conversely, if (x; 0) ̸∈ SSorb (F ), then F has a representative F ′ ∈ Db (KM ) such that (x; 0) ̸∈ SS(F ′ ). Hence F ′ , and thus F , vanishes in some neighborhood of x. Lemma IX.2.2. — Let F ∈ D/[1] (kM ) and F ′ , F ′′ ∈ Db (KM ) such that QM (F ′ ) ≃ QM (F ′′ ) ≃ F . Let x0 ∈ M and let A ⊂ T˙x∗0 M be an open contractible cone

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with a smooth boundary such that (A \ {x0 }) ∩ SS(F ′′ ) = ∅. Let C ⊂ T˙x∗0 M be a conic neighborhood of SS(F ′ ) ∩ ∂A. Then there exists G ∈ Db (KM ) such that QM (G) ≃ F and SS(G) ∩ Tx∗0 M ⊂ (SS(F ′ ) \ A) ∪ C. Proof. — Let U be a chart around x0 such that T ∗ U ≃ U × Tx∗0 M and (U × A) ∩ SS(F ′′ ) = ∅. We apply Proposition III.3.1 with B = SS(F ′ ) ∩ Tx∗0 M and B ′ = C. We obtain a neighborhood W of x0 and a functor R : D(KU ) → − D(KW ) together with a morphism of functors (−)| → − R. This functor R is a composition of W usual sheaf operations and, by the results of the previous section, it induces a functor that we denote by the same letter R : D/[1] (KU ) → − D/[1] (KW ). By (ii) of Propo∼ ∼ sition III.3.1 we have F ′′ | −− → R(F ′′ | ). Hence F | −− → R(F | ) and it follows W U W U ′ ′ that QW (R(F | )) ≃ R(QU (F | )) ≃ F | . By (i) of Proposition III.3.1 the sheaf U U W R(F ′ | ), defined on W , satisfies the conclusion of the lemma. By Lemma IX.1.15 we U can extend R(F ′ | ) into G, defined on M . U

Proposition IX.2.3. — Let F ∈ D/[1] (kM ). Let x0 ∈ M be given and let B ⊂ T˙x∗0 M be a closed conic subset such that SSorb (F ) ∩ B = ∅. Then there exists F ′ ∈ Db (KM ) such that QM (F ′ ) ≃ F and SS(F ′ ) ∩ B = ∅. Proof. — (i) We first prove the following claim to reduce the problem to Lemma IX.2.2: Let A1 , . . . , An be open cones and S a closed cone in Tx∗0 M . Let C be Sn a conic neighborhood of S ∩ ∂( i=1 Ai ). Then there exist conic subsets C1 , . . . , Cn of Tx∗0 M such that, defining inductively S0 = S and Si = (Si−1S\ Ai ) ∪ Ci , we have: Ci is a neighborhood of Si−1 ∩ ∂Ai and n Sn ⊂ (S \ i=1 Ai ) ∪ C. We prove the claim by induction on n. For n = 1 we take C1 = C and the claim is Sn clear. Let us assume weShave proved it for n − 1. Since (S ∩ ∂A1 ∩ ∂( i=2 Ai )) ⊂ C n and (S ∩ ∂A1S) ⊂ C ∪ i=2 Ai , we can choose Sn a neighborhood C1 of S ∩ ∂A1 such n that (C1 ∩ ∂( i=2 Ai )) ⊂ C andSC1 ⊂ C ∪ i=2 Ai . We set S1 = (S \ A1 ) ∪ C1 . Then n C is a neighborhood of S1 ∩ ∂( i=2 Ai ) and we can apply the induction hypothesis with A2 , . . . , An , S = S1 and C. S Sn n The claim follows (using (S1 \ i=2 Ai ) ∪ C ⊂ (S \ i=1 Ai ) ∪ C)). (ii) For any p ∈ B we can find F p ∈ Db (KM ) representing F such that p ̸∈ SS(F p ). We can find an open convex cone Ap around p such that (Ap \ {x0 }) ∩ SS(F p ) = ∅. We Sn can choose finitely many such cones, say A1 , . . . , An , such that B ⊂ i=1 Ai and we let F1 , . . . , Fn ∈ Db (KM ) be the corresponding sheaves. We also choose an arbitrary representative Sn F0 of F and set S = SS(F0 ). Finally we choose a neighborhood C of S ∩ ∂( i=1 Ai ) such that C ∩ B = ∅. We let C1 , . . . , Cn be the subsets of Tx∗0 M given by the claim in (i). Now we define Gi ∈ Db (KM ), i = 0, . . . , n inductively: we start with G0 = F0 , and using Lemma IX.2.2, with F ′ = Gi−1 , F ′′ = Fi , A = Ai , C = Ci , we set Gi = G (G given by the lemma). Then Gi satisfies SS(Gi ) ⊂ Si , for each i. In particular the sheaf Gn satisfies the conclusion of the proposition.

ASTÉRISQUE 440

IX.2.2. FUNCTORIAL BEHAVIOR

177

+1

Proposition IX.2.4. — Let F → − F′ → − F ′′ −−→ be a distinguished triangle in D/[1] (kM ). orb orb orb ′′ Then SS (F ) ⊂ SS (F ) ∪ SS (F ′ ). Proof. — Let p = (x0 ; ξ0 ) ∈ T ∗ M be given. We assume that p ̸∈ SSorb (F )∪SSorb (F ′ ). Let u : F → − F ′ be the morphism of the triangle. By definition we can find ′ b F0 , F0 ∈ D (KM ) such that QM (F0 ) ≃ F , QM (F0′ ) ≃ F ′ and p ̸∈ SS(F0 ), p ̸∈ SS(F0′ ). By Proposition IX.1.8 there exist n ∈ Z and a morphism u0 : F0 [n] → − F0′ such ′′ ′′ that QM (u0 ) = u. Then the cone of u0 , say F0 , represents F and p ̸∈ SS(F0′′ ) by the triangular inequality for the usual microsupport. The result follows. IX.2.2. Functorial behavior We prove that SSorb (·) satisfies the same properties as SS(·) with respect to the usual sheaf operations. Proposition IX.2.5. — Let f : M → − N be a morphism of manifolds. Let G ∈ D/[1] (kN ). We assume that f is non-characteristic for SSorb (G). Then SSorb (f −1 G) ∪ SSorb (f ! G) ⊂ fd fπ−1 SSorb (G). Proof. — (i) The cases of f −1 and f ! are similar and we only consider f −1 . We can write f = p ◦ i where i : M → − M × N is the graph embedding and p : M × N → − N is the projection. Since the result is compatible with the composition it is enough to consider the case of an embedding and a submersion separately. (ii) We assume that f is a submersion. Let x ∈ M and set y = f (x). Then \ \ SSorb (f −1 G) ∩ Tx∗ M = SS(F ′ ) ∩ Tx∗ M ⊂ SS(f −1 G′ ) ∩ Tx∗ M, F′

G′

where F ′ runs over the objects of Db (KM ) such that F ′ ≃ f −1 G in D/[1] (kM ) and G′ over the objects of Db (KN ) such that G′ ≃ G in D/[1] (kN ). Now the result follows from Theorem I.2.8 and the fact that t fx′ : Ty∗ N → − Tx∗ M is injective. (iii) We assume that f is an embedding. Let (x0 ; ξ0 ) ∈ T ∗ M be such that (x0 ; ξ0 ) ̸∈ fd (SSorb (G) ∩ Tx∗0 N ). Let l be the half line R≥0 · (x0 ; ξ0 ). Since f is non-characteristic for SSorb (G), we have fd−1 (l) ∩ SSorb (G) ⊂ {x0 }. By Proposition IX.2.3 there exist a neighborhood V of x0 and G′ ∈ Db (KV ) such that G′ ≃ G in D/[1] (kV ) and fd−1 (l) ∩ SS(G′ ) ⊂ {x0 }. Then f −1 G′ ≃ f −1 G in D/[1] (kM ∩V ) and (x0 ; ξ0 ) ̸∈ fd (SS(G′ ) ∩ Tx∗0 N ). This proves (x0 ; ξ0 ) ̸∈ SSorb (f −1 G), hence the inclusion of the proposition. Proposition IX.2.6. — Let f : M → − N be a morphism of manifolds. Let F ∈ D/[1] (kM ). We assume that f is proper on supporb (F ). Then SSorb (Rf! F ) ⊂ fπ fd−1 SSorb (F ). Proof. — (i) As in the proof of Proposition IX.2.5 we can reduce the problem to the cases where f is an embedding or a projection. The case of an embedding is similar to the part (ii) of the proof of Proposition IX.2.5.

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(ii) We assume now that M = M ′ × N and f is the projection. Let q = (y; η) ∈ T ∗ N be such that q ̸∈ fπ fd−1 SSorb (F ). Let us prove that q ̸∈ SSorb (Rf! F ). Since f is proper on supporb (F ), we can assume, up to restricting to a neighborhood of y, that supporb (F ) ⊂ C × N , for some compact set C ⊂ M ′ . We can then find an open convex cone A in Ty∗ N containing η and a neighborhood Ω of C, such that, for any x ∈ Ω there exists G ∈ Db (kM ) representing F with SS(G) ∩ (Tx∗ M ′ × A) = ∅. As in the proof of Lemma IX.2.2 we apply Proposition III.3.1 to obtain two neighborhoods W ⊂ U of y in N and a functor R+ : Db (KM ′ ×U ) → − Db (KM ′ ×W ), of the + b form R (H) = K ◦ H, where K ∈ D (kW ×U ) is given by the proposition, which satisfies (a) R+ induces a functor on D/[1] (kM ′ ×U ), (b) for G ∈ Db (KM ′ ×U ) and x ∈ M ′ such that SS(G) ∩ (Tx∗ M ′ × A) = ∅ we have R+ (G) ≃ G| around {x} × W , W ∗ ′ (c) for any G ∈ Db (KM ′ ×U ) we have SS(R+ (G)) ∩ (TM ′ M × {p}) = ∅. Then (a) and (b) imply R+ (F ) ≃ F | . Now we choose a representative F ′ ∈ Db (kM ) W of F . Then R+ (F ′ ) is another representative of F and we deduce the result from the condition (c) and Proposition I.2.4. Proposition IX.2.7. — Let F, G ∈ D/[1] (kM ). ∗ (i) We assume that SSorb (F ) ∩ SSorb (G)a ⊂ TM M . Then orb orb orb ε SS (F ⊗kM G) ⊂ SS (F ) + SS (G). ∗ M . Then (ii) We assume that SSorb (F ) ∩ SSorb (G) ⊂ TM ε orb orb orb a SS (RHom (F, G)) ⊂ SS (F ) + SS (G).

Proof. — Let us prove (i). Let x0 ∈ M and let A, B ⊂ Tx∗0 M be conic neighborhoods of SSorb (F )∩Tx∗0 M , SSorb (G)∩Tx∗0 M such that A∩B a ⊂ {x0 }. By Proposition IX.2.3 we can find representatives F ′ , G′ ∈ Db (KM ) of F, G such that SS(F ′ ) ∩ Tx∗0 M ⊂ A, SS(G′ ) ∩ Tx∗0 M ⊂ B. Since microsupports are closed we have SS(F ′ ) ∩ SS(G′ )a ⊂ TU∗ U for some neighborhood U of x0 . Then Theorem I.2.13 gives SS(F ′ ⊗εkM G′ ) ∩ Tx∗0 M ⊂ A + B. Since A and B are arbitrarily close to our microsupports we deduce (i). The proof of (ii) is the same. IX.2.3. Microsupport in the zero section In Proposition IX.2.10 we give a special case of Proposition I.2.9 for SSorb . Lemma IX.2.8. — Let C = [a, b]d be S a compact cube in Rd and let {Ui }i∈I be a family d of open subsets of R such that C ⊂ i∈I Ui . Then there exists a finite family of open subsets {Vn }, n = 1, . . . , N , such that (i) for each n = 1, . . . , N there exists i ∈ I such that Vn ⊂ Ui , SN (ii) C ⊂ n=1 Vn ,

ASTÉRISQUE 440

179

IX.2.3. MICROSUPPORT IN THE ZERO SECTION

(iii) (

Sn

k=1

Vk ) ∩ Vn+1 is contractible, for each n = 1, . . . , N − 1.

Proof. — For x ∈ Rd and ε > 0 we set Cxε = x + ]−ε, ε[d . We can choose ε > 0 such that, for any x ∈ C, there exists i ∈ I satisfying Cxε ⊂ Ui (ε is a Lebesgue number of the covering). We let xn , n = 1, . . . , N , be the points of the lattice C ∩ (εZ)d ordered by the lexicographic order of their coordinates. Then the family Vn = Cxεn , n = 1, . . . , N , satisfies the required properties. Lemma IX.2.9. — Let M be a manifold and U , V ⊂ M two open subsets. Let F ∈ D/[1] (kM ). We assume that U ∩ V is contractible and that there exist A, A′ ∈ Mod(k) such that F | ≃ AU and F | ≃ A′V . Then A ≃ A′ and U V F| ≃ AU ∪V . U ∪V

Proof. — We set W = U ∩ V and X = U ∪ V . Taking the stalks at some x ∈ W gives u +1 A ≃ A′ . The Mayer-Vietoris triangle yields in our case AW − → AU ⊕ AV → − FX −−→, where u is of the form (1, v) for some isomorphism v ∈ HomD/[1] (kM ) (AW , AW ). By Corollary IX.1.9 we have M HomD/[1] (kM ) (AV , AV ) ≃ HomDb (kM ) (AV , AV [i]) i∈Z

≃ Hom(A, A) ⊗

M

H i (V ; k).

i∈N

In the same way HomD/[1] (kM ) (AW , AW ) ≃ Hom(A, A) since W is contractible. Hence v can be extended as an isomorphism to V and we can define the commutative square (C) below: AW

(1,v)

(C) AW

(1,1)

AU ⊕ AV ≀



1 0 0 v −1

AU ⊕ AV

FX

+1



AX

+1

.

We extend this square to an isomorphism of triangles and obtain FX ≃ AX , as required. Proposition IX.2.10. — Let E = Rd and F ∈ D/[1] (kE ). We SSorb (F ) ⊂ TE∗ E. Then there exists A ∈ Mod(k) such that F ≃ AE .

assume

that

Proof. — (i) By Proposition IX.2.3, for any x ∈ E there exists a representative of F , say F x , in Db (KE ) such that SS(F x ) ∩ Tx∗ E ⊂ {x}. Since microsupports are closed there exists an open neighborhood of x, say Ux , such that SS(F x ) ∩ T˙ ∗ Ux = ∅, that is, F x | is constant. In other words, there exists B x ∈ Db (K) such that F x | ≃ BUx x . Ux Ux By Corollary IX.1.10 there exists Ax ∈ Mod(k) such that B x ≃ Ax in D/[1] (k) and we have F | ≃ AxUx in D/[1] (kUx ). Ux

(ii) We set In = ]−n, n[d for n ∈ N \ {0}. The family {Ux }x∈In covers In . Hence, by Lemmas IX.2.8 and IX.2.9 there exists A ∈ Mod(k) such that FIn ≃ AIn for

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all n ∈ N \ {0}. We can assume that these isomorphisms are compatible with the morphisms in : (·)In → − (·)In+1 . We obtain the commutative square (D) below between triangles deduced as in (IX.1.28) ⊕n FIn

u

(D)

≀

⊕n AIn

u

⊕n FIn

F

+1

≀

⊕ n AI n

AE

+1

,

where the n-th-component of u is id − in . We extend this square to an isomorphism of triangles and we see that F ≃ AE . We deduce a version of the Corollary I.2.16 for the orbit category. We state a particular case on the real line (the only case we will use), but the statement of Corollary I.2.16 follows by applying this case to Rϕ∗ F and using the triangle +1 k]−∞,a[ → − k]−∞,b[ → − k[a,b[ −−→. Corollary IX.2.11. — Let F ∈ D/[1] (kR ) and let a < b be given. We assume that SSorb (F ) ∩ πR−1 ([a, b[) is contained in Tτ∗≤0 R = {(t; τ ) ∈ TR∗ R; τ ≤ 0}. Then RHomε (k[a,b[ , F ) ≃ 0. Proof. — We set G = Rs∗ RHom ε (q2−1 k]0,∞[ , q1! RHom ε (k[a,b[ , F )), where 2 q1 , q2 , s : R → − R are the two projections and the sum. Using the bounds in Propositions IX.2.5, IX.2.7 and IX.2.6) we obtain SSorb (RHom ε (k[a,b[ , F )) ⊂ {(t; τ ); a ≤ t < b, τ ≤ 0} ∪ Tb∗ R, SSorb (G) ⊂ {(t; 0); t ̸= b} ∪ {(b; τ ); τ ≥ 0}. We deduce that G| is constant by Proposition IX.2.10. ]−∞,b[ Let it : R → − R2 , t′ 7→ (t′ , t − t′ ) be the inclusion of s−1 (t). Then q1 ◦ it = id and q2 ◦ it is the reflexion along 2t . Hence Proposition I.1.1-(h-j) gives RΓ{t} G ≃ RHomε (k]−∞,t[ , RHom ε (k[a,b[ , F )) ≃ RHomε (k]−∞,t[ ⊗ε k[a,b[ , F ) ≃ RHomε (k[a,c[ , F ), where c = min{b, t}. In particular RΓ{t} G ≃ 0 for t < a and hence G| vanishes. ]−∞,b[ ε We also obtain RΓ{b} G ≃ RHom (k[a,b[ , F ). ˜ ∈ D(KR ) with (b; −1) ̸∈ SS(G). ˜ The definition of the We can represent G by G ∼ ˜ b [−1] −− ˜ b . The same isomorphism holds microsupport implies (RΓ]−∞,b[ G) → (RΓ{b} G) in D/[1] (kR ) and we deduce RΓ{b} G ≃ 0 because G| vanishes. ]−∞,b[

ASTÉRISQUE 440

PART X THE KASHIWARA-SCHAPIRA STACK Let M be a manifold and Λ a locally closed conic Lagrangian submanifold of T˙ ∗ M . In this part we define the Kashiwara-Schapira stack µSh(kΛ ) and its orbit category version. It is obtained by quotienting the category of sheaves on M by subcategories defined by microsupport conditions. These categories are introduced by KashiwaraSchapira in [28]. For a conic subset S of T ∗ M we denote by Db (kM ; S) the quotient of Db (kM ) by the subcategory of sheaves F with SS(F ) ∩ S = ∅. Then µSh(kΛ ) is the stack on Λ associated with the prestack whose value over Λ0 ⊂ Λ is the subcategory of Db (kM ; Λ0 ) generated by the F such that there exists a neighborhood Ω of Λ0 in T ∗ M with SS(F ) ∩ Ω ⊂ Λ0 . We remark that we lose the triangulated structure in the stackification process. An important result of [28] says that the Hom sheaf in µSh(kΛ ) is given by H 0 µhom (see Corollary X.1.6). It is then possible to describe an object of µSh(kΛ ) by local data (see Remark X.1.7). We check in Lemma X.2.5 that, locally on M , there exist simple sheaves with microsupport Λ: if B is a small enough ball in M and Λ is in generic position, there exist simple sheaves on B with microsupport Λ ∩ T ∗ B. We 1 sh 2 × define two classes µsh 1 (Λ) ∈ H (Λ; Z) and µ2 (Λ) ∈ H (Λ; k ) which are obstructions for the existence of a global object of µSh(kΛ ) (that is, an object of µSh(kΛ )(Λ)). In the last two sections we make the link between these classes and the usual Maslov class µ1 (Λ) and another obstruction class µgf 2 (Λ) (an obstruction class to trivialize the Gauss map of Λ). In fact it would not be difficult to deduce µsh 1 (Λ) = µ1 (Λ) directly from Proposition I.4.1 and the description of the Maslov class by Čech cohomology (for example in [20]). However the class µsh 2 (Λ) requires more work; we only prove the gf useful implication that the vanishing of µsh 2 (Λ) implies the vanishing of µ2 (Λ). When we work with sheaves of k-modules we only have the obstruction sh classes µsh 1 (Λ) and µ2 (Λ) for the existence of a global section of µSh(kΛ ). If we work with sheaves of spectra, there are infinitely many of them. Of course this requires to work with dg-categories or infinity categories (see [47, 48] or [36]). In this framework the stack µSh(kΛ ) has the structure of a stable category, like a category of sheaves (whereas the triangulated structures are not suited for stackification).

182

This is explained in [23] where the higher classes µsh i (Λ) are described (they do not gf coincide with the obstruction classes µi (Λ) – see also [24]). In [3] it is proved that these classes vanish when Λ is a Lagrangian embedding in T ∗ M in the homotopy class of the base. Notation X.0.1. — For a conic subset S of T ∗ M we recall the categories DS (kM ), D[S] (kM ) and D(S) (kM ) of Notation I.2.2. We also denote by D(kM ; S) the quotient of D(kM ) by DT ∗ M \S (kM ) (see the reminder on localization in Section IX.1). We use similar notations for the (locally) bounded derived categories: D∗S (kM ), ∗ D[S] (kM ), D∗(S) (kM ) and D∗ (kM ; S), where ∗ = b or ∗ = lb. We also define D/[1],S (kM ), D/[1],[S] (kM ), D/[1],(S) (kM ) and D/[1] (kM ; S) in the same way, replacing D by D/[1] and SS by SSorb . As recalled in Section IX.1 the objects of Db (kM ; S) are those of Db (kM ). A morphism u : F → − G in Db (kM ; S) is represented by a triple (F ′ , s, u′ ) where F ′ ∈ Db (kM ) ′ and s, u are morphisms (X.0.1)

s

u′

F ← − F ′ −→ G

such that the L defined (up to isomorphism) by the distinguished triangle s +1 F′ − →F → − L −−→ satisfies S ∩ SS(L) = ∅. Two such triples (Fi′ , si , u′i ), i = 1, 2, represent the same morphism if there exists a third triple (F ′ , s, u′ ) and two morphisms vi : F ′ → − Fi′ such that s = si ◦ vi , i = 1, 2, and u′1 ◦ v1 = u′2 ◦ v2 . The notion of stack used here is that of “sheaf of categories”. We refer for example to [29, §19]. A prestack C on a topological space X consists of the data of a category C (U ), for each open subset U of X, restriction functors rV,U : C (U ) → − C (V ), for V ⊂ U , and isomorphisms of functors rW,V ◦ rV,U ≃ rW,U , for W ⊂ V ⊂ U , satisfying compatibility conditions. A stack is a prestack satisfying some gluing conditions. In particular, if C is a stack and A, B ∈ C (U ), then the presheaf V 7→ HomC (V ) (A| , B | ) is a sheaf on U . MoreV V S over, if U = i∈I Ui and Ai ∈ C (Ui ) are given objects with compatible isomorphisms between their restrictions on the intersections Ui ∩ Uj , then these objects glue into an object of C (U ). For any given prestack we can construct its associated stack, similar to the associated sheaf of a presheaf.

ASTÉRISQUE 440

CHAPTER X.1 DEFINITION OF THE KASHIWARA-SCHAPIRA STACK

We use the categories associated with a subset of T ∗ M introduced in Notation X.0.1.

Definition X.1.1. — Let Λ ⊂ T ∗ M be a locally closed conic subset. We define a prestack µSh0Λ on Λ as follows. Over an open subset Λ0 of Λ the objects of µSh0Λ (Λ0 ) are those of Db(Λ0 ) (kM ). For F, G ∈ µSh0Λ (Λ0 ) we set HomµSh0Λ (Λ0 ) (F, G) := HomDb (kM ;Λ0 ) (F, G). We define the Kashiwara-Schapira stack of Λ as the stack associated with µSh0Λ . We denote it by µSh(kΛ ). For Λ0 ⊂ Λ we usually write abusively µSh(kΛ0 ) instead of µSh(kΛ )(Λ0 ). We denote by mΛ : Db(Λ) (kM ) → − µSh(kΛ ) the obvious functor. However, for F ∈ Db(Λ) (kM ), we often write F instead of mΛ (F ) if there is no risk of ambiguity.

Several results in the next sections give links between µSh(kΛ ) and stacks of the following type.

Definition X.1.2. — Let X be a topological space. We let DL0 (kX ) be the subprestack of U 7→ Db (kU ), U open in X, formed by the F ∈ Db (kU ) with locally constant cohomologically sheaves. We let DL(kX ) be the stack associated with DL0 (kX ). We denote by Loc(kX ) the substack of Mod(kX ) formed by the locally constant sheaves.

Remark X.1.3. — We remark that DL(kX ) is only a stack of additive categories (the triangulated structure is of course lost in the “stackification”). However − Mod(kU ) induce functors of stacks the cohomological functors H i : Db (kU ) → − Loc(kX ) and the natural embedding Mod(kU ) ,→ Db (kU ) induces H i : DL(kX ) → i : Loc(kX ) → − DL(kX ). We have H 0 ◦ i ≃ idLoc(kX ) . Hence i is faithful and Loc(kX ) is a subcategory of DL(kX ).

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Link with microlocalization We recall now some results of [28] which explain the link between the localized categories Db (kM ; S), for S ⊂ T ∗ M , and the microlocalization, making these categories easier to describe. First we recall some properties of the functor µhom. For U ⊂ M and F, G ∈ D(kU ), the isomorphism RHom (F, G) ≃ RπM ∗ µhom(F, G) of (I.3.2) implies 0 ∗ ∼ Hom(F, G) −−→ H (T U ; µhom(F, G)); for u : F → − G we denote by uµ ∈ H 0 (T ∗ U ; µhom(F, G))

(X.1.1)

the image of u by this isomorphism (this notation has been introduced in (VIII.3.6)). For F, G, H ∈ Db (kM ) we have a composition morphism (see [28, Cor. 4.4.10]) L

µhom(F, G) ⊗ µhom(G, H) → − µhom(F, H).

(X.1.2)

(With the notations of Definition I.3.4, it is induced by a natural morphism L

L

µ∆M (A) ⊗ µ∆M (B) → − µ∆M (A ⊗ B) RHom (q2−1 F, q1! G).)

and

the

composition

morphism

for

Notation X.1.4. — For F, G, H ∈ Db (kM ), S ⊂ T ∗ M and sections of µhom(−, −), say a ∈ H i (S; µhom(F, G)| ), b ∈ H j (S; µhom(G, H)| ), we denote by S

S

µ

b ◦ a ∈ H i+j (S; µhom(F, H)| ) S

the image of a ⊗ b by the morphism induced by (X.1.2) on sections. µ

By construction ◦ is compatible with the usual composition morphism for RHom µ through the isomorphism (I.3.2): for u : F → − G, v : G → − H we have (v ◦ u)µ = v µ ◦ uµ . µ In the same way ◦ is also compatible with the functoriality of µhom as follows. A morphism v : G → − H induces (X.1.3)

µhom(F, v) : µhom(F, G) → − µhom(F, H)

and we have, for a section a of µhom(F, G), (X.1.4)

µ

µhom(F, v)(a) = v µ ◦ a.

We come back to Db (kM ; S) for S ⊂ T ∗ M . Let u : F → − G be a morphism in Db (kM ; S), represented by a triple (F ′ , s, u′ ) as in (X.0.1). Let L be defined (up s +1 to isomorphism) by the distinguished triangle F ′ − → F → − L −−→. Then L satisfies S ∩ SS(L) = ∅. By (I.3.6) we have µhom(L, G)| ≃ 0 and µhom(s, G) as in (X.1.3) S gives an isomorphism ∼ µhom(s, G) : µhom(F, G)| −− → µhom(F ′ , G)| . S S

Hence the triple (F ′ , s, u′ ) yields a section (µhom(s, G))−1 ((u′ )µ ) of H 0 (S; µhom(F, G)| ). S We can check that this section only depends on u and not its representative (F ′ , s, u′ ). Thus we obtain a well-defined morphism (X.1.5)

ASTÉRISQUE 440

HomDb (kM ;S) (F, G) → − H 0 (S; µhom(F, G)| ). S

LINK WITH MICROLOCALIZATION

185

Theorem X.1.5 (Thm. 6.1.2 of [28]). — If S = R>0 · p for some p ∈ T ∗ M , then (X.1.5) is an isomorphism. ∗ When p ∈ M ≃ TM M we have S = {p} and the statement has the following meaning: we remark that Db (kM ; {p}) is the localization of Db (kM ) by the subcategory of sheaves which are zero in some neighborhood of p. We can then see that HomDb (kM ;{p}) (F, G) identifies with H 0 (RHom (F, G))p . We also have µhom(F, G)| ∗ ≃ RHom (F, G) (use (I.3.2) and the fact that µhom is conic), hence TM M (µhom(F, G))p ≃ (RHom (F, G))p . Let F, G ∈ Db (kM ) be given. It follows from Theorem (X.1.5) that the sheaf associated with the presheaf Ω 7→ HomDb (kM ;R>0 ·Ω) (F, G), where Ω runs over the open subsets of T ∗ M , is H 0 µhom(F, G) (here R>0 · Ω is the image of R>0 × Ω → − T ∗M , (a, p) 7→ a · p). We obtain an alternative definition of µSh(kΛ ):

Corollary X.1.6. — Let Λ ⊂ T ∗ M be as in Definition X.1.1. We define a prestack µSh1Λ on Λ as follows. Over an open subset Λ0 of Λ the objects of µSh1Λ (Λ0 ) are those of Db(Λ0 ) (kM ). For F, G ∈ µSh1Λ (Λ0 ) we set HomµSh1Λ (Λ0 ) (F, G) := H 0 (Λ0 ; µhom(F, G)| ). The composition is induced by (X.1.2). Then, the natuΛ0

ral functor of prestacks µSh0Λ → − µSh1Λ induces an isomorphism on the associated stacks. Remark X.1.7. — By Corollary X.1.6 an object of µSh(kΛ ) is determined by the data of an open covering {Λi }i∈I of Λ, objects Fi ∈ Db(Λi ) (kM ), for any i ∈ I, and sections uji ∈ H 0 (Λij ; µhom(Fi , Fj )| ), for any i, j ∈ I, such that Λij

(i) uii is induced by idFi , for any i ∈ I, µ (ii) ukj ◦ uji = uki , for any i, j, k ∈ I. For a complex of sheaves A, the sheaf associated with the presheaf U 7→ H 0 (U ; A) is H 0 A. Hence, for F, G ∈ Db(Λ) (kM ), we find that the homomorphism sheaf Hom µSh(kΛ ) (mΛ (F ), mΛ (G)) is H 0 (µhom(F, G))| . In particular, for an open subΛ set Λ0 ⊂ Λ, (X.1.6)

Hom(mΛ (F )| , mΛ (G)| ) ≃ H 0 (Λ0 ; H 0 (µhom(F, G))). Λ0

Λ0

Remark X.1.8. — We will only consider µSh(kΛ ) when Λ is conic Lagrangian submanifold of T˙ ∗ M . In this case, for F, G ∈ Db(Λ) (kM ) we know by Corollary I.3.7 that µhom(F, G) has locally constant cohomology sheaves on Λ. Moreover, for a given p = (x; ξ) ∈ Λ we have µhom(F, G)p ≃ RHom((RΓ{φ0 ≥0} (F ))x , (RΓ{φ0 ≥0} (G))x ), where φ0 is such that Λ and Λφ0 intersect transversely at p (see (I.4.6)). Hence, if F and G are pure along Λ, µhom(F, G) is concentrated in one degree. If this degree is 0, the right hand side of (X.1.6) coincides with H 0 (Λ0 ; µhom(F, G)). If F and G are simple, then µhom(F, G) is moreover of rank one.

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Remark X.1.9. — For a conic subset Ω of T ∗ M , we recall that a morphism a : F → − G in D(kM ) is an isomorphism on Ω if SS(C(a)) ∩ Ω = ∅, where C(a) is given by a +1 the distinguished triangle F − → G → − C(a) −−→. In particular a induces an isob morphism in D (kM ; Ω). We assume that Λ is a conic Lagrangian submanifold and F, G ∈ Db(Λ) (kM ). Let Λ0 be an open subset of Λ and let a : F → − G be an isomorphism on Λ0 . Then the morphism mΛ (a)| : mΛ (F )| → − mΛ (G)| is an isomorphism. It Λ0

Λ0

Λ0

follows from Remark X.1.8 that there exists b ∈ H 0 (Λ0 ; H 0 (µhom(G, F ))) such that µ

aµ ◦ b = idµG ,

µ

b ◦ aµ = idµF .

The functor µSh0,op × µSh0Λ → − Db (kΛ ), (F, G) 7→ µhom(F, G) induces a functor of Λ stacks (X.1.7)

ASTÉRISQUE 440

µhom : µSh(kΛ )op × µSh(kΛ ) → − DL(kΛ ).

CHAPTER X.2 SIMPLE SHEAVES

In this section we assume that Λ is a locally closed conic Lagrangian submanifold of T˙ ∗ M . We have seen the notion of pure and simple sheaves along Λ in Section I.4. We give here some additional properties. It is easy to describe the simple sheaves along a Lagrangian submanifold at a generic point. They are given in the following example. Example X.2.1. — We consider the hypersurface S = Rn−1 × {0} in M = Rn . We let Λ = {(x, 0; 0, ξn ); ξn > 0} be the “positive” half part of TS∗ M . We set Z = Rn−1 × R≥0 . The sheaf kZ is simple along Λ. More generally, by Example I.2.10, the simple sheaves F along Λ fit in a distinguished triangle +1

′ EM → − kZ [i] → − F −−→,

for some integer i and some E ′ ∈ D(k). Let F ∈ Db[Λ] (kM ). Then, there exists L ∈ Db (k) such that the image of F in the quotient category Db (kM ; T˙ ∗ M ) is isomorphic to LZ = LM ⊗ kZ . The pure sheaves correspond to the case where L is concentrated in one degree and free. The simple sheaves correspond to the case where L ≃ k[i] for some degree i ∈ Z. For any p ∈ Λ we can find a homogeneous Hamiltonian isotopy that sends a neighborhood of p in Λ to the conormal bundle of a smooth hypersurface. Then Theorem II.1.1 reduces the general case to Example X.2.1 (since this is a local statement in T ∗ M we can also use Theorem 7.2.1 of [28]). We deduce: Lemma X.2.2. — Let p = (x; ξ) be a given point of Λ. Then there exists a neighborhood Λ0 of p in Λ such that (i) there exists F ∈ Db(Λ0 ) (kM ) which is simple along Λ0 , (ii) for any G ∈ Db(Λ0 ) (kM ) there exist a neighborhood Ω of Λ0 in T ∗ M and an L ∼ isomorphism F ⊗ LM −− → G in Db (kM ; Ω), where L ∈ Db (k) is given by L = µhom(F, G)p .

The subset Λ0 of the lemma is contractible by construction if it is obtained from the Λ of Example X.2.1 by a homogeneous Hamiltonian isotopy. Conversely if the

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CHAPTER X.2. SIMPLE SHEAVES

condition (ii) of the lemma is satisfied, we must have π1 (Λ0 ) = 0 and H i (Λ0 ; k) ≃ 0 for all i > 0. Definition X.2.3. — Let Λ ⊂ T˙ ∗ M be a locally closed conic Lagrangian submanifold. We let µShp (kΛ ) (resp. µShs (kΛ )) be the substack of µSh(kΛ ) formed by the pure (resp. simple) sheaves along Λ. Lemma X.2.2 implies the following result. Proposition X.2.4. — Let Λ ⊂ T˙ ∗ M be a locally closed conic Lagrangian submanifold. We assume that there exists a simple sheaf F ∈ µShs (kΛ ). Then the functor µhom defined in (X.1.7) induces an equivalence of stacks ∼ µhom(F, ·) : µSh(kΛ ) −− → DL(kΛ ), G 7→ µhom(F, G). By Lemma X.2.2 we can find a simple sheaf with microsupport Λ locally around a given point p ∈ Λ. When Λ is in a good position we can improve this result as follows. Lemma X.2.5. — Let M be a manifold and let Λ ⊂ T˙ ∗ M be a locally closed conic Lagrangian submanifold such that the projection Λ/R>0 → − M has finite fibers. Let p = (x; ξ) ∈ Λ. Then there exist a neighborhood U of x and F ∈ Db (kU ) such ˙ that SS(F ) = Λ ∩ T ∗ U and F is simple along Λ ∩ T ∗ U . Proof. — (i) By hypotheses Λ ∩ Tx∗ M consists of finitely many half-lines, say R>0 · pi , with pi = (x; ξi ), i = 1, . . . , n. Up to a restriction to a neighborhood of x we can assume that the pi belong to distinct connected components of Λ, say Λi , i = 1, . . . , n. If Fi is simple along Λi , then the direct sum ⊕i Fi is simple along Λ. Hence we can assume that Λ ∩ Tx∗ M = R>0 · p for some p = (x; ξ). (ii) By Lemma X.2.2 there exists a neighborhood Ω of p in T ∗ M and F0 ∈ Db (kM ) such that SS(F0 )∩Ω = Λ and F0 is simple along Λ at p. For a neighborhood V of x we choose a trivialization T ∗ V ≃ V × Tx∗ M . Up to shrinking V we can find two disjoint ˙ 0 ) ⊂ V × (A ⊔ A′ ) closed conic contractible subsets A, A′ ⊂ T˙x∗ M such that SS(F ∗ ˙ ˙ and SS(F0 ) ∩ (V × A) = Λ ∩ T V . By Proposition III.3.2 there exists a distinguished +1 triangle F ⊕ F1 → − F0 | → − L −−→ in D(kU ) on some smaller neighborhood U of x U ˙ ˙ 0 ) ∩ (U × A), SS(F ˙ 1 ) = SS(F ˙ 0 ) ∩ (U × A′ ) and L is constant. such that SS(F ) = SS(F ˙ Then SS(F ) = Λ ∩ T ∗ U and F is simple.

ASTÉRISQUE 440

CHAPTER X.3 OBSTRUCTION CLASSES

In this section we see that there are two obstructions to the existence of a global simple object in µSh(kΛ ) (by this we mean an object of µShs (kΛ )(Λ)). They are 1 sh 2 × × classes µsh is the group of units 1 (Λ) ∈ H (Λ; Z) and µ2 (Λ) ∈ H (Λ; k ), where k in k (when k = Z/2Z this last class is automatically zero). On the other hand, it is proved in [17] that Λ has a local generating function if and only if the stable Gauss map g : Λ → − U/O is homotopic to the constant map v : Λ → − U/O which sends a point to the vertical fiber of T ∗ M . Here U/O = limn U (n)/O(n) is the (stable) Lagrangian Grassmannian (some choices −→ are needed to define the Gauss map, but it is well-defined up to homotopy). The i obstruction classes to find such a homotopy are classes µgf i (L) ∈ H (Λ; πi (U/O)) for i = 1, . . . , dim M (“gf” stands for generating function). The first class is the Maslov class µgf 1 (Λ) = µ1 (Λ). We will see in §X.6 that, for i = 1, 2, the vanishing gf of µsh i (Λ) implies the vanishing of µi (Λ). sh To define the classes µi (Λ) we recall how we can describe an object of µShs (kΛ )(Λ). By Remark X.1.7 a global simple object of µSh(kΛ ) is determined by the data of an open covering {Λi }i∈I of Λ, objects Fi ∈ Db(Λi ) (kM ), for all i ∈ I, which are simple along Λi , and sections uji ∈ H 0 (Λij ; µhom(Fi , Fj )| ), for any i, j ∈ I, such that Λij

(i) uii is induced by idFi , for any i ∈ I, µ (ii) ukj ◦ uji = uki , for any i, j, k ∈ I. We try to find such a set of data. First we choose a covering {Λi }i∈I of Λ by small open subsets. We assume that the Λi ’s and all intersections Λij , Λijk are contractible. We have seen in the previous section that, if the Λi ’s are small enough, we can choose Fi ∈ Db(Λi ) (kM ), for all i ∈ I, which are simple along Λi . Moreover, for each i ∈ I, if Fi′ ∈ Db(Λi ) (kM ) is another simple sheaf along Λi we have mΛi (Fi′ ) ≃ mΛi (Fi )[d] for some shift d. The sheaf µhom(Fi , Fj )| is constant on Λij , free of rank one. Hence there exist Λij isomorphisms φji : µhom(Fi , Fj )| ≃ kΛij [−dij ], for some integers dij . They inΛij

duce φji ∈ H 0 (Λij ; µhom(Fi , Fj )[dij ]). In view of (X.1.6) the φji ’s give isomorphisms

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in µShs (kΛ )(Λij ): vji : mΛi (Fi )|

Λij

∼ −− → mΛj (Fj )|

Λij

[dij ].

We deduce that the Čech cochain {dij }i,j∈I is a cocyle and defines (X.3.1)

1 µsh 1 (Λ) = [{dij }] ∈ H (Λ; Z).

By the remark that mΛi (Fi ) is well defined up to shift, this class only depends on Λ. If there exists a global simple object F in µSh(kΛ ), for some coefficient ring k, then we can choose Fi ’s which represent F | and this implies dij = 0 for all i, j and thus Λi

µsh 1 (Λ) = 0. Let us assume that µsh 1 (Λ) = 0. Then we can write dij = dj − di for some family of integers di , i ∈ I. We set Fi′ = Fi [di ] and obtain isomorphisms ∼ wji : mΛi (Fi′ )| −− → mΛj (Fj′ )| . Λij Λij For i, j, k ∈ I we define an automorphism cijk of mΛi (Fi′ )|

Λijk

by cijk = wik ◦wkj ◦wij .

Since ′

′

≃ H 0 µhom(Fi′ , Fi′ ) ≃ kΛi , we can canonically identify the automorphisms group of mΛi (Fi′ )| Hom (mΛi (Fi ), mΛi (Fi ))

Λijk

with

k× ⊂ H 0 (Λijk ; kΛijk ) ≃ k. Hence the cijk ’s give a Čech cochain with coefficient in k× . It is easy to see that it is a cocycle and defines (X.3.2)

2 × µsh 2 (Λ) = [{cijk }] ∈ H (Λ; k ).

The isomorphism H 0 µhom(Fi′ , Fi′ ) ≃ kΛi also implies that wji is well defined up to sh multiplication by a unit. It follows that µsh 2 (Λ) only depends on Λ. If µ2 (Λ) = 0, then we can write {cijk } as the boundary of a 2-cochain, say {bij }, with bji ∈ k× . Defining ′ ′ ∼ wji = b−1 − → mΛj (Fj′ )| ji wji : mΛi (Fi )|Λij − Λij ′ ′ ′ we have wik ◦ wkj = wij and the mΛi (Fi′ ) glue into a global simple object of µSh(kΛ ).

ASTÉRISQUE 440

CHAPTER X.4 THE KASHIWARA-SCHAPIRA STACK FOR ORBIT CATEGORIES

In this section we set k = Z/2Z. We have defined the usual sheaf operations for the triangulated orbit categories D/[1] (kM ) and we can define a Kashiwara-Schapira stack in this situation. We give quickly the analogs of the results obtained in the previous sections. For a conic subset S of T ∗ M we recall the categories D/[1],S (kM ), D/[1],[S] (kM ), D/[1],(S) (kM ) and D/[1] (kM ; S) (see Notation X.0.1). Let Λ ⊂ T˙ ∗ M be a locally closed conic Lagrangian submanifold. We define a stack µSh/[1] (kΛ ) on Λ as in Definition X.1.1, again replacing Db by D/[1] . It comes with a functor m/[1],Λ : D/[1],(Λ) (kM ) → − µSh/[1] (kΛ ). We say that F ∈ D/[1] (kM ) is simple along Λ if SSorb (F ) ∩ T˙ ∗ M ⊂ Λ and, for any p ∈ Λ, there exists F ′ ∈ Db (kM ) such that ιM (F ′ ) ≃ F , SS(F ′ ) = Λ in a neighborhood of p and F ′ is simple along Λ at p. As in Section X.1 we can define the substack µShs/[1] (kΛ ) of µSh/[1] (kΛ ) associated with the simple sheaves. For Ω ⊂ T ∗ M , we have a morphism similar to (X.1.5) HomD/[1] (kM ;Ω) (F, G) → − HomD/[1] (kΩ ) (kΩ , µhomε (F, G)| ) Ω

and, as in Theorem X.1.5, it is an isomorphism if Ω = {p} for some p ∈ T ∗ M . We remark that Proposition IX.2.10 implies in particular that, if B is homeomorphic to a ball and L ∈ D/[1] (kB ) is locally of the form kU , then there exists an isomorphism u : L ≃ kB . Moreover HomD/[1] (kB ) (kB , kB ) ≃ k (and k = Z/2Z), hence u is unique. For simple sheaves this gives: Lemma X.4.1. — Let Λ ⊂ T˙ ∗ M be a locally closed conic Lagrangian submanifold. We assume that Λ is contractible. Let F, F ′ ∈ D/[1] (kM ) be two simple sheaves along Λ and let Ω be a neighborhood of Λ such that SSorb (F ) ∩ Ω = SSorb (F ′ ) ∩ Ω = Λ. Then we have a unique isomorphism µhomε (F, F ′ )| ≃ kΛ in D/[1] (kΩ ). Ω

By Lemma X.4.1 there exists a unique simple sheaf in µSh/[1] (kΛ0 ) for any contractible open subset Λ0 ⊂ Λ, up to a unique isomorphism. In other words µShs/[1] (kΛ ) has locally a unique object with the identity as unique isomorphism. Hence gluing is trivial. Since µShs/[1] (kΛ ) is a stack it follows that it has a unique global object.

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We recall that Loc(kX ) is the substack of Mod(kX ) formed by the locally constant sheaves. Definition X.4.2. — Let X be a manifold. We let OL0 (kX ) be the subprestack of U 7→ D/[1] (kU ), U open in X, formed by the F ∈ D/[1] (kU ) such that SSorb (F ) ⊂ TU∗ U . We let OL(kX ) be the stack associated with OL0 (kX ). By Proposition IX.2.10 the condition SSorb (F ) ⊂ TU∗ U is equivalent to: F is locally isomorphic to AU for some A ∈ Mod(k). We recall the functors ι0X : Mod(kX ) → − D/[1] (kX ) and h0X : D/[1] (kX ) → − Mod(kX ) defined before Corollary IX.1.10. They induce functors of stacks iX : Loc(kX ) → − OL(kX ) and hX : OL(kX ) → − Loc(kX ). Lemma X.4.3. — The functors iX and hX are mutually inverse equivalences of stacks. Proof. — We have seen in Corollary IX.1.10 that hX ◦ iX ≃ idLoc(kX ) . Hence it is enough to see that iX is locally an equivalence, that is, essentially surjective and fully faithful. Let U ⊂ X be an open subset homeomorphic to a ball. By Proposition IX.2.10 the functor iU is essentially surjective and, for F, G ∈ Loc(kU ), Corollary IX.1.9 gives M HomD/[1] (kU ) (iU (F ), iU (G)) ≃ HomDb (kU ) (F [−n], G) n∈Z

≃ HomLoc(kU ) (F, G), which proves that iU is fully faithful. As we remarked after Lemma X.4.1 µShs/[1] (kΛ ) has a unique global object. As in (X.1.7) the functor µhomε induces a functors of stacks µhomε : (µSh/[1] (kΛ ))op × µSh/[1] (kΛ ) → − OL(kΛ ) ≃ Loc(kΛ ). We have an analog of Proposition X.2.4 in the orbit category case. Proposition X.4.4. — The stack µShs/[1] (kΛ ) has a unique object, say F0 , defined over Λ. Moreover the functor µhomε (F0 , −) induces an equivalence of stacks ∼ µSh/[1] (kΛ ) −− → Loc(kΛ ).

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In this section and the next one we see the link between the classes introgf 1 sh 2 × 1 duced in §X.3, µsh 1 (Λ) ∈ H (Λ; Z), µ2 (Λ) ∈ H (Λ; k ) and µ1 (Λ) ∈ H (Λ; Z), gf 2 µ2 (Λ) ∈ H (Λ; Z/2Z). We only prove the useful implication that the vanishing gf of µsh i (Λ) (for a ring k with 2 ̸= 0) implies the vanishing of µi (Λ) but a little more work would show that they coincide. In the definition of the microsupport of a sheaf F we consider whether some local cohomology group vanishes, namely (RΓ{x; ϕ(x)≥ϕ(x0 )} (F ))x0 for some function ϕ. A natural question is then how this group depends on ϕ and not only on ξ0 = dϕ(x0 ). In general it really depends on ϕ, but it is proved in Proposition 7.5.3 of [28] that it is independent of ϕ (up to a shift dϕ ) if we assume that Λ = SS(F ) is a Lagrangian submanifold near (x0 ; ξ0 ) and that Γdϕ is transverse to Λ at (x0 ; ξ0 ) (see Proposition I.4.1). Moreover the shift dϕ is related with the Maslov index of the three Lagrangian subspaces of T(x0 ;ξ0 ) T ∗ M given by λ = T(x0 ;ξ0 ) Λ, λϕ = T(x0 ;ξ0 ) Γdϕ and λ0 = T(x0 ;ξ0 ) (π −1 (x0 )). In particular (RΓ{x; ϕ(x)≥ϕ(x0 )} (F ))x0 only depends on λϕ which is a Lagrangian subspace of T(x0 ;ξ0 ) T ∗ M transverse both to λ and λ0 . In this section we precise a little bit this result and prove that there exists a locally constant sheaf on some open subset of the Lagrangian Grassmannian of Λ ×T ∗ M T T ∗ M whose stalks at λϕ is (RΓ{x; ϕ(x)≥ϕ(x0 )} (F ))x0 . We will also see in §X.6 that it has a non trivial monodromy. We first introduce some notations. In this section M is a manifold of dimension n and Λ is a locally closed conic Lagrangian submanifold of T˙ ∗ M . We recall the notations (I.4.2): for a given point p = (x; ξ) ∈ Λ we have the following Lagrangian subspaces of Tp (T ∗ M ) λ0 (p) = Tp (Tx∗ M ),

λΛ (p) = Tp Λ

and, for a function φ : M → − R, we set Λφ = Γdφ = {(x; dφ(x)); x ∈ M } and λφ (p) = Tp Λφ . We let (X.5.1)

σT ∗ M : LM → − T ∗ M,

0 σT0 ∗ M : LM → − T ∗M

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be respectively the fiber bundle of Lagrangian Grassmannian of T ∗ M and the subbundle whose fiber over p ∈ T ∗ M is the set of Lagrangian subspaces of Tp T ∗ M which 0 are transverse to λ0 (p). Then LM is an open subset of LM . For a given p ∈ T ∗ M we set V = TπM (p) M and we identify Tp T ∗ M with V × V ∗ . We use coordinates (ν; η) 0 on Tp T ∗ M . Then we can see that any l ∈ (LM )p is of the form l = {(ν; η) ∈ Tp T ∗ M ; η = A · ν},

(X.5.2)

0 where A : V → − V ∗ is a symmetric matrix. This identifies the fiber (LM )p with the space of n × n-symmetric matrices. For a function φ defined on a product X × Y and for a given x ∈ X we use the general notation φx = φ| . {x}×Y 0 Lemma X.5.1. — There exists a function ψ : LM ×M → − R of class C ∞ such that, for 0 any l ∈ LM with σT ∗ M (l) = (x; ξ),

ψl (x) = 0,

dψl (x) = ξ,

λψl (σT ∗ M (l)) = l.

Proof. — (i) We first assume that M is the vector space V = Rn . We identify T ∗ M 0 )p is identified with the space of and M × V ∗ . For p = (x; ξ) ∈ M × V ∗ the fiber (LM 0 quadratic forms on V through (X.5.2). For l ∈ (LM )p we let ql be the corresponding quadratic form. Now we define ψ0 by ψ0 (l, y) = ⟨y − x, ξ⟩ +

1 2

where (x; ξ) = σT0 ∗ M (l).

ql (y − x),

We can check that ψ0 satisfies the conclusion of the lemma. (ii) In general we choose an embedding i : M ,→ RN . For a given p′ = (x; ξ ′ ) ∈ M ×RN T ∗ RN the subspace Tp′ (M ×RN T ∗ RN ) of Tp′ T ∗ RN is coisotropic. The symplectic reduction of Tp′ T ∗ RN by Tp′ (M ×RN T ∗ RN ) is canonically identified with Tp T ∗ M , where p = id (p′ ). The symplectic reduction sends Lagrangian subspaces to Lagrangian subspaces and we − LM,p . The restriction of rp′ to the set of Lagrangian deduce a map, say rp′ : LRN ,p′ → subspaces which are transverse to Tp′ (M ×RN T ∗ RN ) is an actual morphism of 0 manifolds. In particular it induces a morphism rp0′ : LR0N ,p′ → − LM,p . We can see 0 ′ ∗ N that rp′ is onto and is a submersion. When p runs over M ×RN T R we obtain a surjective morphism of bundles, say r: LR0N |

r

M ×RN T ∗ RN

M × R N T ∗ RN

id

0 LM

T ∗ M.

We can see that r is a fiber bundle, with fiber an affine space. Hence we can find a 0 0 section, say j : LM → − LR0N . For (l, x) ∈ LM × M we set ψ(l, x) = ψ0 (j(l), i(x)), where ψ0 is defined in (i). Then ψ satisfies the conclusion of the lemma.

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195

We come back to the Lagrangian submanifold Λ of T˙ ∗ M . We let 0 UΛ ⊂ LM |

(X.5.3)

Λ

0 ∗ be the subset of LM |Λ consisting of Lagrangian subspaces of Tp T M which are transverse to λΛ (p). We define σΛ = σT ∗ M | , τM = πM | ◦ σΛ and iΛ : UΛ ,→ UΛ × M , UΛ Λ l 7→ (l, τM (l)):

UΛ (X.5.4)

σΛ

iΛ

UΛ × M

τM

Λ πM |

Λ

q2

M.

0 We note that UΛ is not a fiber bundle over Λ but only an open subset of LM |Λ . −1 However, for a given p ∈ Λ, we will use the notation UΛ,p = σΛ (p). 0 Definition X.5.2. — Let ψ : LM ×M → − R be a function satisfying the conclusions of Lemma X.5.1 and let φ : UΛ × M → − R be its restriction to UΛ × M . For F ∈ Db(Λ) (kM ) we define mφ (F ) ∈ Db (kUΛ ) by −1 mφ (F ) = i−1 Λ (RΓφ−1 ([0,+∞[) (q2 F )),

where q2 : UΛ × M → − M is the projection. Proposition X.5.3. — Let F ∈ Db(Λ) (kM ). Let φ : UΛ × M → − R be as in Definition X.5.2. Then the object mφ (F ) ∈ Db (kUΛ ) has locally constant cohomology sheaves and its stalks are (mφ (F ))l ≃ (RΓφ−1 ([0,+∞[) (F ))x , l

for any l ∈ UΛ and x = τM (l). Proof. — (i) We first prove that mφ (F ) is locally constant. For this we give another expression of mφ (F ). We define G ∈ Db (kT ∗ (UΛ ×M ) ) by G = µhom(kφ−1 ([0,+∞[) , q2−1 F ). We use the notations in (X.5.4) and we define IΛ = im(iΛ ) ⊂ UΛ × M and JΛ ⊂ T˙ ∗ (UΛ × M ), JΛ = {(l, x; 0, λξ); (x; ξ) = σΛ (l), λ > 0}. We remark that JΛ is a fiber bundle over IΛ with fiber R>0 . We prove in (ii) and (iii) below that there exists a neighborhood V of IΛ in UΛ × M such that (a) supp(G) ∩ T˙ ∗ V ⊂ JΛ , (b) SS(G| ˙ ∗ ) ⊂ TJ∗Λ T ∗ (UΛ × M ), T V (c) (RΓφ−1 ([0,+∞[) (q2−1 F ))IΛ ≃ Rπ˙ V ∗ (G|T˙ ∗ V ). By Proposition I.2.9 the properties (a-b) imply that G| ˙ ∗ has support in JΛ and T V is locally constant along JΛ . Since JΛ is a fiber bundle over IΛ , we deduce by (c) that (RΓφ−1 ([0,+∞[) (q2−1 F ))IΛ is locally constant on IΛ , hence mφ (F ) is locally constant on UΛ . (ii) We prove (i-a) and (i-b). By Proposition I.3.5 and Lemma X.5.5 below we have: for Λ1 , Λ2 two conic Lagrangian submanifolds of a cotangent bundle T˙ ∗ X

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with a clean intersection Ξ = Λ1 ∩ Λ2 and for Fi ∈ Db[Λi ] (kX ), i = 1, 2, we have supp(µhom(F1 , F2 )| ˙ ∗ ) ⊂ Ξ and SS(µhom(F1 , F2 )| ˙ ∗ ) ⊂ TΞ∗ T ∗ X. T X T X ˙ φ−1 ([0,+∞[) ) = Λ′ , where By Example I.2.3 (iii) we have SS(k φ

Λ′φ

= {(l, x; λ · dφ(l, x)); (l, x) ∈ UΛ × M, λ > 0, φ(l, x) = 0}.

˙ −1 F ) = T ∗ UΛ × Λ. It is then enough to find a neighborhood V We also have SS(q 2 UΛ of IΛ in UΛ ×M such that T ∗ V ∩Λ′φ and T ∗ V ∩(TU∗Λ UΛ ×Λ) have a clean intersection, which is JΛ . Let us first prove that Λφ (= Γdφ ) is transverse to T ∗ UΛ × Λ. For l0 ∈ UΛ , with σΛ (l0 ) = (x0 ; ξ0 ), we know that Λφl0 is transverse to Λ at the point (x0 ; ξ0 ) and ξ0 = dφl0 (x0 ). Hence we can find a neighborhood Vl0 of x0 in M such that Λφl0 ∩ Λ ∩ T ∗ Vl0 = {(x0 ; ξ0 )}. Since Λφl0 is the projection of (Tl∗0 UΛ × T ∗ M ) ∩ Λφ to T ∗ M , it follows that, in Tl∗0 UΛ × T ∗ Vl0 , the submanifolds (Tl∗0 UΛ × T ∗ Vl0 ) ∩ Λφ ∂φ and Tl∗0 UΛ × (T ∗ Vl0 ∩ Λ) are transverse at the point (l0 , x0 ; ∂φ ∂l , ∂x ) (and this is the onlyFintersection point). We can make Vl0 move nicely enough with l0 so that V = l∈UΛ {l} × Vl is a neighborhood of IΛ in UΛ × M . Then T ∗ V ∩ Λφ is transverse to T ∗ V ∩ (T ∗ UΛ × Λ), with intersection    ∂φ ∂φ 1 JΛ = l, x; , ; l ∈ Uλ , x = τM (l) . ∂l ∂x Let us prove that JΛ = R>0 · JΛ1 . For l0 ∈ Uλ and x0 = τM (l0 ) we have ∂φ (x0 ; ∂φ ∂x (l0 , x0 )) = σΛ (l0 ) by the definition of φ. It remains to see that ∂l (l0 , x0 ) = 0. We recall that φ(l, τM (l)) = 0 for all l ∈ UΛ . Differentiating this relation we obtain ∂φ ∂l (l0 , x0 ) + ξ0 ◦ dτM (l0 ) = 0, for all l0 ∈ Uλ and (x0 ; ξ0 ) = σΛ (l0 ) (here we view − M factorizes ξ0 = ∂φ ∂x (l0 , x0 ) as a map from Tx0 M to R). Since the map τM : UΛ → through σΛ : UΛ → − Λ ⊂ T ∗ M , we have ξ0 ◦ dτM (l0 ) = ξ0 ◦ dπM (x0 ; ξ0 ) ◦ dσΛ (l0 ) = αM ◦ dσΛ (l0 ), where αM is the Liouville 1-form on T ∗ M . Now Λ is conic Lagrangian, hence the pull-back of αM to Λ vanishes and we obtain ξ0 ◦ dτM (l0 ) = 0, hence ∂φ ∂l (l0 , x0 ) = 0, as required. It follows from this discussion that A = T ∗ V ∩ R>0 · Λφ is transverse to B = T ∗ V ∩ (T ∗ UΛ × Λ) with intersection JΛ . Now JΛ is contained in A1 = {φ = 0} and in B1 = TU∗Λ UΛ × Λ. We deduce that A ∩ A1 and B ∩ B1 have a clean intersection which is still JΛ . Since A ∩ A1 = T ∗ V ∩ Λ′φ and B ∩ B1 = T ∗ V ∩ (TU∗Λ UΛ × Λ), this concludes the proof of (i-a) and (i-b). (iii) Now we prove the claim (c) of (i). Sato’s triangle (I.3.5) gives +1

(D′ (kφ−1 ([0,+∞[) ) ⊗ q2−1 F )IΛ → − (RΓφ−1 ([0,+∞[) (q2−1 F ))IΛ → − Rπ˙ UΛ ×M ∗ (G)IΛ −−→ . By definition dφ does not vanish in a neighborhood of IΛ . Hence φ−1 (0) is a smooth hypersurface near IΛ and D′ (kφ−1 ([0,+∞[) ) ≃ kφ−1 (]0,+∞[) . Since IΛ ⊂ φ−1 (0), the first

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term of the above triangle is zero. By (i-a) the support of Rπ˙ V ∗ (G|T˙ ∗ V ) is already contained in IΛ . So we can forget the subscript IΛ in the third term and we obtain (i-c). (iv) We prove the last assertion of the proposition. Let l0 ∈ UΛ be given and (x0 ; ξ0 ) = σ(l0 ). Since Λ′φl is transverse to Λ at (x; ξ) = σ(l), by Lemma X.5.6 below we can find neighborhoods U of l0 and W of x0 and a homogeneous Hamiltonian isotopy of T˙ ∗ W parameterized by U , say Ψ : U × T˙ ∗ W → − T˙ ∗ W , such ∗ ∗ ′ ∗ ′ ∗ that Ψl (Λ) ∩ T W = Λ ∩ T W and Ψl (Λφl ) ∩ T W = Λφl ∩ T W , for all l ∈ U . We 0 −1 ′ set Λ+ = Λ′φ ∪ (TU∗ U × Λ) and Λ+ l = Λφl ∪ Λ. Then kφ−1 ([0,+∞[) and q2 F belong + to D[Λ+ ] (kU ×W ) and Ψl (Λ+ l0 ) = Λl . By Proposition II.2.3 (ii) we deduce RHom (kφ−1 ([0,+∞[) , q2−1 F )|

{l0 }×W

∼ −− → RHom (kφ−1 ([0,+∞[) |

{l0 }×W

, q2−1 F |

{l0 }×W

)

≃ RHom (kφ−1 ([0,+∞[) , F )| . l0

W

Taking the germs at x0 ∈ W we obtain the required isomorphism. Let X be a manifold and Y, Z two submanifolds of X. We recall that Y and Z have a clean intersection if W = Y ∩ Z is a submanifold of X and T W = T Y ∩ T Z. This means that we can find local coordinates (x, y, z, w) such that Y = {x = z = 0} and Z = {x = y = 0}. Using these coordinates the following lemma is easy. Lemma X.5.4. — Let X be a manifold and Y, Z two submanifolds of X which have a clean intersection. We set W = Y ∩ Z. Then C(Y, Z) = W ×X T Y + W ×X T Z. Lemma X.5.5. — Let X be a manifold and Λ1 , Λ2 be two Lagrangian submanifolds of T˙ ∗ X. Let F1 ∈ Db(Λ1 ) (kX ) and F2 ∈ Db(Λ2 ) (kX ). We assume that Λ1 and Λ2 have a clean intersection and we set Ξ = Λ1 ∩ Λ2 . Then there exists a neighborhood U of Ξ in T ∗ X such that SS(µhom(F1 , F2 )| ) ⊂ TΞ∗ T ∗ X, that is, µhom(F1 , F2 )| is U U supported on Ξ and has locally constant cohomology sheaves on Ξ. Proof. — We have SS(µhom(F1 , F2 )) ⊂ (H −1 (C(SS(F2 ), SS(F1 ))))a by the bound (I.3.7). Let Ui be a neighborhood of Λi such that SS(Fi ) ∩ Ui ⊂ Λi , i = 1, 2. Then U = U1 ∩ U2 is a neighborhood of Ξ and we have H −1 (C(SS(F2 ), SS(F1 ))) ∩ T ∗ U ⊂ H −1 (C(Λ2 , Λ1 )). Since Λi is Lagrangian we have H −1 (T Λi ) = TΛ∗i T ∗ X, for i = 1, 2. In particular H (Ξ ×T ∗ X T Λi ) ⊂ TΞ∗ T ∗ X and the result follows from Lemma X.5.4. −1

Lemma X.5.6. — Let B be a neighborhood of 0 in RN . Let φ : B × Rn → − R be a family of functions. (i) We assume that Γdφ0 is transverse to the zero-section TR∗n Rn of T ∗ Rn and Γdφ0 ∩ TR∗n Rn = {0}. Then there exist neighborhoods B ′ of 0 in RN and V of 0 in Rn and a family of Hamiltonian isotopies of T ∗ Rn parameterized by B ′ , say Ψ : B ′ × T ∗ Rn → − T ∗ Rn , such that Ψb (TR∗n Rn ) = TR∗n Rn and Ψb (Γdφ0 ) ∩ T ∗ V = ∗ Γdφb ∩ T V , for all b ∈ B ′ .

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(ii) Let Λ ⊂ T˙ ∗ Rn be a closed conic Lagrangian submanifold. We assume that Γdφ0 is transverse to Λ with Γdφ0 ∩ Λ = {(0; ξ0 )}, Γdφb ∩ Λ = {(xb ; ξb )} and φb (xb ) = 0 for all b ∈ B. Then there exist neighborhoods B ′ of 0 in RN and V of 0 in Rn and a family of homogeneous Hamiltonian isotopies of T˙ ∗ Rn parameterized by B ′ , say Ψ : B ′ × T˙ ∗ Rn → − T˙ ∗ Rn , such ∗ ∗ ′ ∗ ′ ∗ ˙ ˙ ˙ ˙ that Ψb (Λ) ∩ T V = Λ ∩ T V and Ψb (Λ0 ) ∩ T V = Λb ∩ T V , for all b ∈ B ′ , where Λ′b = {(x; λ · dφb (x)); λ > 0, φb (x) = 0}. Proof. — (i) The transversality hypothesis implies that dφb viewed as a function from Rn to (Rn )∗ is invertible near 0, for b small enough. We set θb = (dφb )−1 and view θb as a 1-form on (Rn )∗ defined in some neighborhood of 0. Since the graph of θb is Lagrangian, it is a closed 1-form and we can write θb = dhb near 0. We consider hb (ξ) as a Hamiltonian function on T ∗ Rn . By construction its Hamiltonian vector field is P Xhb (x; ξ) = i (θb )i (ξ)∂xi and the time 1 of its flow satisfies ϕ1hb ({0} × (Rn )∗ ) = Γdφb near 0. Moreover ϕthb (Rn × {0}) ⊂ Rn × {0}. Now we set Ψb = ϕ1hb ◦ (ϕ1h0 )−1 . (ii) We can find a homogeneous Hamiltonian isotopy Φ arbitrarily close to id and a neighborhood W of Φ(0; ξ0 ) such that T ∗ W ∩ Φ(Λ) is half of the conormal bundle of a smooth hypersurface X and T ∗ W ∩ Φ(Γdϕb ) is still the graph of a function for b close enough to 0. We take coordinates on W such that X = Rn−1 × {0}, Φ(0; ξ0 ) = (0, 0; 0, 1) and we write Φ(xb ; ξb ) = (yb , 0; 0, ηb ). Then Φ(Λ′b ) is the conormal bundle of a hypersurface which is the graph of a function φ′b : Rn−1 → − R. Moreover Γdφ′b is transverse to TR∗n−1 Rn−1 . Part (i) of the proof gives a family Ψ′ of Hamiltonian isotopies of T ∗ Rn−1 such that Ψ′b (TR∗n−1 Rn−1 ) = TR∗n−1 Rn−1 and Ψ′b (Γdφ′0 ) ∩ T ∗ V = Γdφ′b ∩T ∗ V . We lift Ψ′ into a family Ψ′′ of homogeneous Hamiltonian isotopies of T˙ ∗ Rn and we set Ψb = Φ−1 ◦ Ψ′′b ◦ Φ. Remark X.5.7. — The assignment F 7→ mφ (F ) of Definition X.5.2 is a functor mφ : Db(Λ) (kM ) → − Db (kUΛ ). By Proposition X.5.3 it factorizes through DL0 (kUΛ ) (see Definition X.1.2). Let us check that mφ induces a functor of stacks mφ : µSh(kΛ ) → − DL(kUΛ ). We recall that µSh(kΛ ) is associated with some prestack µSh0Λ ; the objects of µSh0Λ (Λ0 ) are those of Db(Λ0 ) (kM ) and the Hom set between F and G is HomDb (kM ;Λ0 ) (F, G) (see Definition X.1.1). Now we remark that the definition of mφ (F ) applies to any F ∈ Db (kM ) and gives in fact mφ : Db (kM ) → − Db (kUΛ ). By the definition of the microsupport we see that mφ (F ) ≃ 0 if SS(F ) ∩ Λ = ∅, that is, mφ (F ) ≃ 0 if F ∈ DbT ∗ M \Λ (kM ). By the definition of Db (kM ; Λ) this means that mφ factorizes through Db (kM ; Λ) (see the reminder on localization in Section IX.1). Hence mφ induces a functor from µSh0Λ (Λ) to Db (kUΛ ), still with image in DL0 (kUΛ ). We can replace Λ by any of its open subset Λ0 and obtain in this way a functor of prestacks from µSh0Λ to DL0 (kUΛ ). Passing to the associated stacks it induces a functor of stacks mφ : µSh(kΛ ) → − DL(kUΛ ).

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We keep the notations of Definition X.5.2 and Remark X.5.7. In particular we have a choice of function φ : UΛ ×M → − R which gives a functor mφ : Db(Λ) (kM ) → − Db (kUΛ ). For F ∈ Db(Λ) (kM ) we know that mφ (F ) is a locally constant object on UΛ . Here we describe the monodromy of its restriction to a fiber UΛ,p of σΛ : UΛ → − Λ. We first recall well-known results on locally constant sheaves and introduce some ∗ X, that is, notations. Let X be a manifold and L ∈ Db (kX ) such that SS(L) ⊂ TX L has locally constant cohomology sheaves. Then any path γ : [0, 1] → − X induces an isomorphism ∼ ∼ −− RΓ([0, 1]; γ −1 L) −− → Lγ(1) . (X.6.1) Mγ (L) : Lγ(0) ← Moreover, Mγ (L) only depends on the homotopy class of γ with fixed ends and Mγ (L)◦ Mγ ′ (L) = Mγ◦γ ′ (L) if γ, γ ′ are composable. In particular, if we fix a base point x0 ∈ X, we obtain the monodromy morphism M (L) : π1 (X; x0 ) → − Iso(Lx0 ) (X.6.2) γ 7→ Mγ (L), where Iso(Lx0 ) is the group of isomorphisms of Lx0 in Db (k). Now we go back to the situation of Section X.5. For a given p ∈ Λ we −1 set UΛ,p = σΛ (p). This is the open subset of Lagrangian Grassmannian mani∗ fold L (Tp T M ) formed by the Lagrangian subspaces of Tp Λ which are transverse to λ0 (p) and λΛ (p). Let us describe its connected components and their fundamental groups. Let (V, ω) be a symplectic vector space of dimension 2n. P Let l1 , l2 be two Lagrangian subspaces. We can assume that V = R2n with ω = ei ∧ fi in the canonical base (e1 , . . . , en , f1 , . . . , fn ) and l1 = ⟨e1 , . . . , en ⟩, l2 = ⟨e1 , . . . , ek , fk+1 , . . . , fn ⟩. Let U (l1 ) ⊂ L (V ) be the open subset of Lagrangian subspaces which are transverse to l1 . Then U (l1 ) is diffeomorphic to Symn , the space of symmetric matrices of
size n × n, through M 7→ lM := {(M y, y); y = (y1 , . . . , yn )}. Writing M = tAB B , with A of C size k × k, we see that lM is also transverse to l2 if and only if C is invertible. Hence

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U (l1 ) ∩ U (l2 ) is diffeomorphic to Rd × Sym0n−k where d is some integer and Sym0n−k is the subset of invertible matrices in Symn−k . Let us recall the topology of Symp,q n the subset of Symn of matrices with p positive p,q eigenvalues and q negative eigenvalues, p + q = n. The action of GL+ n on Symn , p,q + p,q t A · M = AM A, gives Symn ≃ GLn / SO(p, q). In particular Symn is connected. Now a maximal compact subgroup of SO(p, q) is K = S(O(p) × O(q)) and SO(p, q) is diffeomorphic to K × Rd for some d. We deduce an exact sequence of fundamental groups: π1 (K) → − π1 (GL+ − π1 (Symp,q − π0 (K) → − π0 (GL+ n) → n )→ n ) = {1}. We recall that π1 (SO(n)) = π1 (GL+ n ) is Z for n = 2 and Z/2Z for n ≥ 3. We will only need the case where n is big. In particular we can assume n ≥ 3 and p or q ≥ 2. Since K contains SO(p) × SO(q) it follows that the map π1 (K) → − π1 (GL+ n ) is surjective. p,q ∼ Hence π1 (Symn ) −−→ π0 (K). If p and q are both ≥ 1, then O(p) × O(q) has four components and π0 (S(O(p)×O(q))) = Z/2Z. If p or q vanishes, then SO(p, q) = SO(n) p,q and Symp,q n is contractible. In conclusion, for n ≥ 3 we have π1 (Symn ) ≃ Z/2Z if p,q p ̸= 0 and q ̸= 0 and π1 (Symn ) ≃ 0 if p = 0 or q = 0. −1 Let us write down the above results for UΛ,p = σΛ (p). We set n = dim M and k = dim(λ0 (p) ∩ λΛ (p)). We assume n − k ≥ 3. Then UΛ,p is topologically equivalent Fn−k to Sym0n−k = p=0 Symp,n−k−p , which has n − k + 1 components, two of them being n−k contractible and the other ones having π1 = Z/2Z. The inertia index τTp T ∗ M introduced in (I.4.3) gives a function on

τp : UΛ,p → − Z,

l 7→ τTp T ∗ M (λ0 (p), λΛ (p), l)

which is constant on each component of UΛ,p . In the coordinates chosen above for l1 , l2
and with l = lM , M = tAB B we can see that τp (lM ) = sgn(C), using Lemma A.3.3 C of [28] (here sgn(C) is p+ − p− , where p+ and p− are the numbers of positive and negative eigenvalues of C). Since C is an invertible symmetric matrix of size (n−k), we obtain that the values of τp are {−n + k, −n + k + 2, . . . , n − k}. Hence τp distinguishes the components of UΛ,p and we can index them as follows: (X.6.3)

i UΛ,p is the connected component of UΛ,p where τp = i.

We have to be careful that the components of UΛ cannot be indexed in this way: the function τp is locally constant on UΛ,p but the function l 7→ τTσ(l) T ∗ M (λ0 (σ(l)), λΛ (σ(l)), l) is not locally constant on UΛ . For example, when dim(λ0 (p) ∩ λΛ (p)) changes by 1 (which happens when p moves from a generic point to a cusp), the parity of the possible values of τp also changes. However we have the following result.

Lemma X.6.1. — The function δ : UΛ ×Λ UΛ → − Z, (l, l′ ) 7→ τp (l) − τp (l′ ), where ′ p = σ(l) = σ(l ), is locally constant on UΛ ×Λ UΛ .

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Proof. — We recall that τ satisfies a cocycle relation (see for example [28], Thm A.3.2) τ (l1 , l2 , l3 ) − τ (l2 , l3 , l4 ) + τ (l3 , l4 , l1 ) − τ (l4 , l1 , l2 ) = 0 and that τ (l1 , l2 , l3 ) is constant when (l1 , l2 , l3 ) moves but the dimensions of li ∩ lj , i, j = 1, 2, 3, and l1 ∩ l2 ∩ l3 do not change. The function δ is locally constant on UΛ,p × UΛ,p for a given p because so are τp (l) and τp (l′ ). When p moves we choose a local trivialization of T ∗ M and we consider l, l′ , λ0 (p), λΛ (p) as subspaces of a fixed symplectic space. Then τp (l) − τp (l′ ) = τ (l, l′ , λ0 (p)) − τ (l, l′ , λΛ (p)) is constant for l, l′ fixed and λ0 (p), λΛ (p) remaining transverse to l, l′ . j i Proposition X.6.2. — Let F ∈ Db(Λ) (kM ). For p ∈ Λ let UΛ,p and UΛ,p be two components of UΛ,p (see (X.6.3)). Then j i we have mφ (F )l ≃ mφ (F )l′ [(i − j)/2], (a) for l ∈ UΛ,p and l′ ∈ UΛ,p i (b) if π1 (UΛ,p ) = Z/2Z, the monodromy of mφ (F )| i along the non trivial loop is UΛ,p

the multiplication by −1. i ) = Z/2Z if |i| is not maximal, By the discussion before (X.6.3) we have π1 (UΛ,p that is, |i| = ̸ dim M − dim(λ0 (p) ∩ λΛ (p)).

Proof. — The point (a) is already proved in [28] and stated here as Proposition I.4.1. It will be recovered in the course of the proof of (b). (i) Since mφ (F ) is locally constant on UΛ and j − i is well-defined in a neighborhood j i × UΛ,p by Lemma X.6.1, we can assume for the proof of (a) that p is a generic of UΛ,p i point of Λ. This also works for the proof of (b) since any loop in UΛ,p can be deformed i′ into a loop in a nearby fiber UΛ,q . Hence we assume that Λ = TN∗ M in a neighborhood of p, for some submanifold N ⊂ M . By Example I.4.5 we know that kN is simple. By Lemma X.2.2 there exists a neighborhood Ω of p in T ∗ M such that F is isomorphic to kN ⊗ LM ≃ ZN ⊗Z LM in Db (kM ; Ω), for some L ∈ Db (k). Hence mφ (F ) ≃ mφ (ZN ) ⊗Z LUΛ and we can assume that k = Z and F = ZN . (ii) We take coordinates (x1 , . . . , xn ) so that N = {x1 = · · · = xk = 0} and 0 p = (0; 1, 0). We identify (LM )p with a space of matrices as in (X.5.2). Then UΛ,p is the space of symmetric matrices A such that det(Ak ) ̸= 0, where Ak is the matrix i obtained from A by deleting the k first lines and columns. The component UΛ,p is dei fined by sgn(Ak ) = i. We can choose a base point B ∈ UΛ,p represented by a diagonal matrix B with entries 0, 1 or −1. More precisely the diagonal consists of k 0’s, α 1’s i and β −1’s. We have α − β = i, hence 2α = i + n − k. Since π1 (UΛ,q ) ̸= 0 we also have α, β ≥ 1.

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We choose a, b such that Baa = 1 and Bbb = −1. For θ ∈ [0, 2π], we define the matrix B(θ) which is equal to B except ! ! Baa (θ) Bab (θ) cos(θ) sin(θ) = . Bba (θ) Bbb (θ) sin(θ) − cos(θ) i Then γ : θ 7→ B(θ) defines a non trivial loop in UΛ,p and we want to prove that the monodromy of mφ (ZN ) around γ is the multiplication by −1. Since mφ (ZN ) has stalk Z up to some shift, we only have to check that this monodromy is not trivial.

(iii) We define φ : [0, 2π] × M → − R by φ(θ, x) = x1 + x B(θ) t x. Then Cθ = {φθ ≥ 0} ∩ N is a quadratic cone and the pair (N, Cθ ) is homotopically equivalent to the pair (N, Vθ ), where Vθ is the subspace Vθ = ⟨eθ , ep ; p = k + 1, . . . , k + α, p ̸= a⟩, of dimension α with eθ = (0, cos( θ2 ), 0, sin( θ2 ), 0) and ep = (0, 1, 0). The stalk a

b

p

i is of mφ (ZN ) at B(θ) ∈ UΛ,p

mφ (ZN )B(θ) ≃ RΓ{φθ ≥0} (ZN ) ≃ (HVn−k−α (ZN ))0 [k + α − n] θ and a non zero germ sθ ∈ mφ (ZN )B(θ) gives a choice of relative orientation of Vθ . In particular a non zero section of mφ (ZN ) defined on some neighborhood of γ would induce relative orientations of all Vθ together for θ ∈ [0, 2π], varying continuously with θ and coinciding for θ = 0 and θ = 2π, which is impossible. Hence the monodromy of mφ (ZN ) is not 1, which proves (b). The part (a) follows from the fact that mφ (ZN )B(θ) is concentrated in cohomological degree n − k − α = (n − k)/2 − i/2. Let M be a manifold and L ⊂ T ∗ M a closed Lagrangian submanifold. We set L = LM | . We have already defined the Gauss map g : L → − L as g(p) = λΛ (p). L We have also considered the tangent to the vertical fiber and we see it now as a map v : L → − L , p 7→ λ0 (p). We can define the same maps after stabilization. For an integer N we let VN be the symplectic vector space VN = CN and we let l0N = RN , l1N = iRN be two Lagrangian subspaces. We define LN → − L the fiber bundle whose fiber at p ∈ L is L (Tp T ∗ M ⊕VN ). We extend g and v into two sections gN , vN : L → − LN defined by gN (p) = λΛ (p) ⊕ l1N and vN (p) = λ0 (p) ⊕ l0N . Taking the limit for N → − ∞, we set L∞ = limN LN and obtain the sections g∞ and v∞ of L∞ . As recalled in §X.3 −→ it is proved in [17] that Λ has a local generating function if and only if the sections g∞ and v∞ are homotopic. In this situation it is classical to consider the obstructions classes to find a homotopy of sections between g∞ and v∞ . The i-th class belongs to H i (L; πi (U/O)); we denote it by µgf i (L) (more precisely, these classes are defined inductively and we need the vanishing of the first i − 1 classes to define the i-th class). Let us assume that L has a triangulation and denote by Sk (L) the k-skeleton of L. Then µgf i = 0 for i = 1, . . . , k, if and only if there exists a homotopy between g∞ | and v∞ | . Sk Sk

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The question of finding an isotopy between g and v is related to finding a section of the map UL → − L, where UL is the open subset of LM | introduced in (X.5.3). L Indeed, for p ∈ L and l ∈ UL,p the open subset U (l) of LM,p consisting of Lagrangian subspaces transverse to l is an affine chart of LM,p isomorphic to a space of symmetric matrices. It has a natural structure of affine space. Since g(p) and v(p) belong to U (l), we obtain an isotopy hl : t 7→ tg(p) + (1 − t)v(p), t ∈ [0, 1]. Hence any section of the map σ : UL → − L over a subset S of L gives an isotopy between g | and v | . S S For an integer N we set ΞN = T˙R∗N RN +1 and ξN = (0, 0; 0, 1) ∈ ΞN . For F ∈ D(kM ) we consider F ⊠ kRN ∈ D(kM ×RN +1 ). Since F ⊠ kRN ≃ i∗ p−1 F , where p : M × RN → − M is the projection and i : M × RN → − M × RN +1 the inclusion, we ˙ ˙ have SS(F ⊠ kRN ) = SS(F ) × ΞN . If SS(F ) = Λ, then SS(F ⊠ kRN ) contains Λ × ΞN as an open set and Λ × ΞN is a neighborhood of Λ ≃ Λ × {ξN } which retracts to Λ. We can identify L (M × RN )| with the Lagrangian Grassmannian LN of the Λ×{ξN } stabilization of T ∗ M introduced above. Summing up the discussion we obtain the following result. We assume that Λ is triangulated. If the map σ : UΛ×ΞN → − Λ × ΞN has a section over the k-skeleton gf of Λ × {ξN }, then µi (Λ) = 0 for i ≤ k. Corollary X.6.3. — Let Λ ⊂ T˙ ∗ M be a closed conic Lagrangian submanifold. We assume that there exists F ∈ µSh(kΛ )(Λ) which is simple along Λ. Then µgf 1 (Λ) = 0. Moreover, if k = Z, then µgf (Λ) = 0. 2 Proof. — (i) We choose a triangulation of Λ. By the discussion before the corollary it is enough to prove that, for N big enough, the map σ : UΛ×ΞN → − Λ × ΞN has a section over the k-skeleton of Λ × {ξN }, for k = 1, 2. We consider the connected components of UΛ×ΞN | , that we denote by UaN , Λ×{ξN } a ∈ AN . We let Ua , a ∈ A, be the connected components of UΛ . To prove the vanishing − Λ is of µ1 it is enough to see that there exists N and UaN such that σ | N : UaN → Ua surjective with connected fibers (then it is possible to find a section on the 1-skeleton, since UaN is an open subset of a fiber bundle over Λ). (ii) By Remark X.5.7 we can define mφ (F ) ∈ DL(kΛ ). Since F is simple, mφ (F ) is (locally) concentrated in one degree with germs k in this degree. Hence, for any a ∈ A there exists da = da (F ) ∈ Z such that mφ (F )| [da ] is locally constant with germs k. Ua Let a, a′ ∈ A and p ∈ Λ. We recall that the connected components of UΛ,p are i distinguished by the inertia index and that we denote by UΛ,p the component with i i′ index i. We assume that UΛ,p is a component of Ua ∩ UΛ,p and UΛ,p is a component of Ua′ ∩UΛ,p . By Proposition X.6.2-(a) this implies i−i′ = 2(da −da′ ). We thus obtain (ii-a) Ua cannot intersect UΛ,p in more than one connected component, (ii-b) if da (F ) = da′ (F ) and σ(Ua ) ∩ σ(Ua′ ) ̸= ∅, then Ua = Ua′ . Applying (ii-a) to F ⊠ kRN (writing abusively kRN for mΞN (kRN )) we obtain that σ | N has connected fibers, for any component UaN introduced in (i). U a

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(iii) We denote by VbN , b ∈ BN , the components of UΞN and by db (kRN ) (as in (ii)) the cohomological degree such that mφN (kRN ) has germs k in degree db (kRN ), for a function φN : UΞN × RN +1 → − R similar to φ. We can see directly on the formula of Proposition X.5.3 that db (kRN ) takes the values 0, 1, . . . , N (more precisely ∗ H{f ≥0} kRN is concentrated in degree N −i when f (x) = xN +1 +q(x1 , . . . , xN ) and q is a non degenerate quadratic form with i negative eigenvalues). We can then identify BN with {0, . . . , N } and we get db (kRN ) = b. We remark that the product of two Lagrangian subspaces gives a natural inclusion of UΛ ×UΞN in UΛ×ΞN . For a ∈ A and b ∈ BN the component UcN of UΛ×ΞN containing Ua × VbN satisfies dc (F ⊠ kRN ) = da (F ) + db (kRN ) = da (F ) + b. We set d′ = max{da (F ); a ∈ A}, d′′ = min{da (F ); a ∈ A} and we choose N bigger N than d′ − d′′ . For a ∈ A we let Uc(a) , c(a) ∈ AN , be the component which contains N N Ua × Vd′ −da (F ) . Then dc(a) (F ⊠ kRN ) = d′ , for all a ∈ A. Since σ(Uc(a) ) contains N σ(Ua ) × ΞN , the open subsets σ(Uc(a) ) cover Λ × ΞN . By (ii-b) we deduce that these N components Uc(a) are in fact a single component. By the final remark of (i) this proves that µ1 = 0. (iv) Now we assume k = Z and prove µgf 2 = 0. −1 N N Let us set U0N = Uc(a) ∩ σΛ×Ξ (Λ × {ξN }) (Uc(a) is the component of UΛ×ΞN N N found in (iii)). Hence σ := σΛ×ΞN | N : U0 → − Λ is surjective with connected fibers. U0

We recall that the components of UΛ,p have fundamental group Z/2Z except the two with extremal inertia index. Hence, up to taking the product with V12 (the component of UΞ2 with index 1—see (iii)), we can assume that the fibers of σ | N have fundamental U0

group Z/2Z. Let us recall how µgf 2 is defined. We assume that our triangulation is fine enough so that π1 (σ −1 (T )) = π1 (σ −1 (p)) = Z/2Z for any triangle T and p ∈ T . We choose a section of σ on the 1-skeleton, say i : S1 (Λ) → − U0N . Then µgf 2 is the obstruction to extend it to the 2-skeleton and is defined as follows. The boundary of each triangle T gives a loop i(∂T ) in σ −1 (T ), hence an element c(T ) ∈ π1 (σ −1 (T )) = Z/2Z. Then i can be extended to the 2-skeleton if and only if the chain c : T 7→ c(T ) vanishes. It is easy to see that c is a cocycle and, by definition µgf 2 = [c]. If [c] = 0, we write c = ∂b where b is a 1-chain and we can modify i by b and obtain a new section i′ : S1 (Λ) → − U0N which can be extended to S2 (Λ). Now we see how we can use our sheaf to define i such that the chain c vanishes. By Proposition X.6.2-(b) the sheaf G = mφ (F ⊠ ZRN )| N is locally constant with U0

germs Z and its restriction to the fibers has monodromy −1. We first define i on S0 (Λ) arbitrarily. For each p ∈ S0 (Λ) we also choose a generator up of Gi(p) ≃ Z. We have two such generators up and −up . Let E ⊂ S1 (Λ) be an edge with boundaries p, p′ . The fundamental group of σ −1 (E) is π1 (σ −1 (E)) = Z/2Z. Hence we have two sections j, j ′ of σ over E up to homotopy such that j(p) = j ′ (p) = i(p) and j(p′ ) = j ′ (p′ ) = i(p′ ). Then j −1 (G) is a

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constant sheaf on E and we have a canonical isomorphism ∼ ∼ aj : Gi(p) ≃ (j −1 G)p ← −− Γ(E; j −1 G) −− → (j −1 G)p′ ≃ Gi(p′ ) and a similar one aj ′ . Since G has monodromy −1, we have aj = −aj ′ . Hence we can choose one section j or j ′ , that we call i, such that ai (up ) = up′ . We do this for all edges and we obtain i : S1 (Λ) → − U0N . With this definition the monodromy of G along i(∂T ) is 1, for any triangle T . Using again π1 (σ −1 (T )) = Z/2Z and the fact that G has monodromy −1 along the non trivial loop, we deduce that i(∂T ) is a trivial loop. Hence we can extend i to S2 (Λ). This proves µgf 2 = 0.

SOCIÉTÉ MATHÉMATIQUE DE FRANCE 2023

PART XI CONVOLUTION AND MICROLOCALIZATION Let N be a manifold and Λ ⊂ T˙ ∗ N a locally closed conic Lagrangian submanifold. As explained in Remark X.1.7 the objects of µShs (kΛ ) are described by simple sheaves along a covering {Λi }i∈I of Λ, say Fi ∈ Db(Λi ) (kN ), and gluing “isomorphisms” uji ∈ H 0 (Λij ; µhom(Fi , Fj )| ). For a given object of µShs (kΛ ) we want to find Λij

lb a representative in Dlb (Λ) (kN ), or better, D[Λ] (kN ). For this we would like to glue the Fi ’s in the category Dlb (kN ) instead of µShs (kΛ ). A first step for this is to find other representatives of the mΛi (Fi )’s for which the uji arise from morphisms in Dlb (kN ). In this part we introduce a functor, Ψ, which gives an answer to this question (see Proposition XI.3.7 and Corollary XI.3.8 below). This functor is a variation on Tamarkin’s projector of §III.5. To define Ψ we need to choose a direction on N and we assume that N is a product N = M × R. For an open subset U of N we define ΨU : D(kU ) → − D(kU ×]0,+∞[ ) with the following properties: setting for ε > 0, we have, for F, G ∈ D(kU ) with a microsupport ΨεU (F ) = ΨU (F )| U ×{ε} contained in T ∗ M × {τ ≥ 0} (τ the fiber coordinate of T ∗ R—see (XI.0.1) below), ε ˙ ˙ ˙ (i) SS(Ψ U (F )) = SS(F ) ∪ Tε (SS(F )), where Tε is the translation by ε along R, ε ˙ (ii) ΨU (F ) is isomorphic to F along SS(F ) in the sense that we have a triangle +1 ε ˙ ) = ∅, ΨU (F ) → − F → − H −−→ with SS(H) ∩ SS(F ∗ ˙ (iii) if F ≃ G in D(kU ; T U ), then ΨU (F ) ≃ ΨU (G), (iv) H 0 (T˙ ∗ V ; µhom(F, G)) ≃ limε→0 Hom(ΨεU (F )| , ΨεU (G)| ), if V is a relatively V V −→ compact open subset of U .

The property (iv) will be used in the next part to glue representatives of objects of µShs (kΛ ) as follows. For W ⊂ M × R we set ΛW = Λ ∩ T ∗ W . We are given s F ∈ µSh (kΛ )(ΛW ), a covering W = W1 ∪ W2 and Fi ∈ D(kWi ) representing F | . ΛWi

We set U = W1 ∩ W2 . We then have isomorphisms mΛU (F1 | ) ≃ F | U

ΛU

≃ mΛU (F2 | ), U

208

hence a section of H 0 µhom(F1 , F2 ) over T˙ ∗ U . Using the property (iv) we deduce an isomorphism ΨεU (F1 ) ≃ ΨεU (F2 ) and we can glue ΨεU (F1 ) and ΨεU (F2 ) into a representative of F over W . With this procedure we can construct sheaves representing objects of µShs (kΛ )(Λ0 ) when Λ0 ⊂ Λ is of the type Λ ∩ T ∗ U0 , U0 open in M × R. Unfortunately we will need to consider more general open subsets Λ0 in §XII.2. Our Λ0 will not be a union of subsets Λ ∩ T ∗ Wi , but only a union of connected components of Λ ∩ T ∗ Wi . This means that πM ×R (Λ0 ) ∩ πM ×R (∂Λ0 ) is a priori non empty. Let p = (x; ξ) ∈ ∂Λ0 , Ξ a neighborhood of p in Λ and U = πM ×R (Ξ), U0 = πM ×R (Ξ ∩ Λ0 ). Near x we will have to consider sheaves of the form F = RΓU0 (F ′ ) with F ′ ∈ D(kU ), SS(F ′ ) = Ξ. For sheaves of this kind the above formula (iv) may still hold but the microsupport of F is bigger than Ξ ∩ T ∗ U0 . If G = RΓU1 (G′ ) is another sheaf of the same type, what we really need in (iv) is not H 0 (T˙ ∗ V ; µhom(F, G)) (for some V ⊂ U ) but H 0 (T˙ ∗ V ∩ Λ0 ; µhom(F, G)| ). In fact, for generic open subsets U0 , U1 (in the sense Λ0 that the conormal bundles of their boundaries are well positioned with respect to Λ), these two groups are isomorphic. We check this in §XI.4, where we introduce a category of sheaves which are locally of the form ΨU (RΓU0 (F ′ )). We introduce some notations. We set for short R>0 = ]0, +∞[ and R≥0 = [0, +∞[. We usually endow the factor R in M × R with the coordinate t. We will need an extra parameter, usually denoted u, running over R≥0 or R>0 . The associated coordinates in the cotangent bundles are (t; τ ) for T ∗ R and (u; υ) for T ∗ R>0 . We set Tτ∗≥0 R = {(t, τ ) ∈ T ∗ R; τ ≥ 0} and we define Tτ∗>0 R similarly. For a manifold M and an open subset U ⊂ M × R we define Tτ∗≥0 U = (T ∗ M × Tτ∗≥0 R) ∩ T ∗ U, Tτ∗≤0 U = (Tτ∗≥0 U )a , (XI.0.1) Tτ∗>0 U = (T ∗ M × Tτ∗>0 R) ∩ T ∗ U, Tτ∗0 U )a . Definition XI.0.1. — Let U be an open subset of M × R. For ∗ = ∅, b or lb, we let D∗τ >0 (kU ) (resp. D∗τ ≥0 (kU )) be the full subcategory of D∗ (kU ) of sheaves F satis˙ fying SS(F ) ⊂ Tτ∗>0 U (resp. SS(F ) ⊂ Tτ∗≥0 U ).

ASTÉRISQUE 440

CHAPTER XI.1 THE FUNCTOR Ψ

The convolution product is a variant of the “composition of kernels” considered in [28] (denoted by ◦—see (I.5.1)). It is used in [45] to study the localization of D(kM ×R ) by the objects with microsupport in Tτ∗≤0 (M ×R), in a framework similar to the present one. Namely, Tamarkin proves that the functor F 7→ k[0,+∞[ ⋆ F is a projector from D(kM ×R ) to the left orthogonal of the subcategory DTτ∗≤0 (M ×R) (kM ×R ) of objects with microsupport in Tτ∗≤0 (M × R) (see [22] for a survey). We will use a variant of Tamarkin’s definition. We will use the product ⋆ in the following special situation. We define the subsets of R × R>0 : γ = {(t, u); 0 ≤ t < u}, (XI.1.1) λ0 = {0} × R>0 , λ1 = {(t, u) ∈ R × R>0 ; t = u}. Definition XI.1.1. — Let M be a manifold and let U ⊂ M × R be an open subset. We define Uγ ⊂ M × R × R>0 by Uγ = {(x, t, u) ∈ M × R × R>0 ; {x} × [t − u, t] ⊂ U }. For F ∈ D(kU ) and G ∈ D(kR×R>0 ) with supp(G) ⊂ γ, we define G ⋆ F ∈ D(kUγ ) by L

(XI.1.2)

G ⋆ F = (Rs! (F ⊠ G))|

Uγ

,

where s = sU : U × R × R>0 → − M × R × R>0 is the sum s(x, t1 , t2 , u) = (x, t1 + t2 , u) (in general U is understood and we write s for sU ). We define the functor ΨU : D(kU ) → − D(kUγ ) by (XI.1.3)

ΨU (F ) = kγ ⋆ F = (Rs! (F ⊠ k{(t,u); 0≤t0 ) = (U ∩ (M ′ × R))γ . In particular UγF ∩ ({x} × R × R>0 ) = (U ∩ ({x}
× R))γ , for any point x ∈ M , and we can write Uγ = x∈M {x} × (U ∩ ({x} × R))γ .

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F F For a disjoint union U = i∈I Ui we have Uγ = i∈I Ui,γ . Hence we can reduce the description of Uγ to the case where M is a point and U = ]a, b[ is an interval of R. Then we have (]a, b[)γ = {(t, u) ∈ R × R>0 ; a + u < t < b}.

(XI.1.4)

(ii) We have s−1 (Uγ ) ∩ (M × R × γ) ⊂ U × γ. Since supp(G) ⊂ γ, it follows that L

we also have G ⋆ F = (RsM ×R ! (F ′ ⊠ G))| that F ′ | = F .

Uγ

where F ′ ∈ D(kM ×R ) is any object such

U

(iii) For the same reason the restriction of s to s−1 (Uγ ) ∩ (M × R × γ) is a proper map. Hence we can replace Rs! by Rs∗ in (XI.1.2). Example XI.1.3. — In Fig. XI.1.1 we give the easiest examples of ΨU (F ) when M is a point and U = R. The three pictures describe ΨU (Fi ) respectively for F1 = k[0,+∞[ , F2 = (k]−∞,0[ )[1], F3 = k[0,1[ . We have ΨU (F1 ) ≃ ΨU (F2 ) ≃ kW , where W = {(t, u); 0 ≤ t < u} ⊂ R × R>0 , and ΨU (F3 ) is concentrated in degrees 0, 1, with H 0 ΨU (F3 ) = kW0 , H 1 ΨU (F3 ) = kW1 , for some locally closed subsets W0 , W1 of R × R>0 (however ΨU (F3 ) ̸≃ kW0 ⊕ kW1 [−1]). We have sketched W , W0 , W1 in the pictures. +1 We remark that we have a distinguished triangle kR → − F1 → − F2 −−→ and Lemma XI.1.5 below implies ΨU (kR ) ≃ 0, hence ΨU (F1 ) ≃ ΨU (F2 ) as we have said. ˙ 1 ) = SS(F ˙ 2 ) = Λ0 Setting Λx0 = {(x0 ; τ ), τ > 0} ⊂ T˙ ∗ R, for x0 ∈ R, we also have SS(F and mΛ0 (F1 ) ≃ mΛ0 (F2 ). ˙ 3) = Similar things happen for F3 and F4 := F1 ⊕ k[1,+∞[ [−1]. We have SS(F ˙ SS(F4 ) = Λ0 ⊔Λ1 and mΛ0 ⊔Λ1 (F3 ) ≃ mΛ0 ⊔Λ1 (F4 ). However ΨU (F3 ) ̸≃ ΨU (F4 ); indeed ΨU is additive and the result for F1 gives here ΨU (F4 ) ≃ kW ⊕ kW ′ [−1], where W ′ = {(t, u); 1 ≤ t < 1 + u}. What remains true is that, for ε > 0 small enough (here ε ≤ 1), ΨU (F3 )| ≃ ΨU (F4 )| . R×]0,ε[ R×]0,ε[ In Corollary XI.3.8 we will see: for a closed conic Lagrangian submanifold Λ ⊂ Tτ∗>0 U and F, G ∈ D[Λ] (kU ), if mΛ (F ) ≃ mΛ (G), then ΨU (F )| ≃ ΨU (G)| , for V V some open subset V of Uγ such that (U × {0}) ⊔ V is open in U × R≥0 . So ΨU gives canonical representatives of objects of µSh(kΛ ), but with microsupport “doubled” (see Remark XI.1.8—the problem is addressed in §XII.3). We define the projections q : M × R × R>0 → − M × R, (XI.1.5) r : M × R × R>0 → − M × R,

(x, t, u) 7→ (x, t), (x, t, u) 7→ (x, t − u)

and, for an open subset U of M × R, we denote by (XI.1.6)

qU , rU : Uγ → − U

−1 the restrictions of q, r to Uγ . Using the notations (XI.1.1) we have kλ0 ⋆ F ≃ qU (F ) −1 and kλ1 ⋆ F ≃ rU (F ), for any F ∈ D(kU ). The closed inclusions λ0 ⊂ γ, λ1 ⊂ γ and

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CHAPTER XI.1. THE FUNCTOR Ψ

kW

kW

•

k[0,∞[

kW0

(k]−∞,0[ )[1]

kW1 [−1] u t

•

t

k[0,1[ Figure XI.1.1.

λ1 ⊂ γ \ λ0 give excision distinguished triangles b′′

a

+1

kInt(γ) −→ kγ − → kλ0 −−→,

+1

b

kλ1 [−1] → − kγ → − kγ −−→,

b′

+1

kλ1 [−1] − → kInt(γ) → − kγ\λ0 −−→ and we have b = b′′ ◦ b′ . The convolution − ⋆ F turns the morphisms a, b, b′ , b′′ into morphisms of functors: (XI.1.7)

−1 α(F ) : ΨU (F ) → − qU (F ),

−1 β(F ) : rU (F )[−1] → − ΨU (F ),

−1 and β ′ (F ) : rU (F )[−1] → − kInt(γ) ⋆ F , β ′′ (F ) : kInt(γ) ⋆ F → − ΨU (F ). ′′ We have β(F ) = β (F ) ◦ β ′ (F ) and two distinguished triangles β ′′ (F )

α(F )

+1

(XI.1.8)

−1 kInt(γ) ⋆ F −−−−−→ ΨU (F ) −−−−−→ qU (F ) −−→,

(XI.1.9)

−1 rU (F )[−1] −−−−−→ kInt(γ) ⋆ F → − kγ\λ0 ⋆ F −−→ .

β ′ (F )

+1

Lemma XI.1.4. — For F ∈ Dτ ≥0 (kU ) the morphism β ′ (F ) is an isomorphism and (XI.1.8) gives the distinguished triangle (XI.1.10)

β(F )

α(F )

+1

−1 −1 rU (F )[−1] −−−−−→ ΨU (F ) −−−−−→ qU (F ) −−→ .

Proof. — Since β(F ) = β ′′ (F ) ◦ β ′ (F ) the second part of the lemma follows from the claim that β ′ (F ) is an isomorphism. In view of (XI.1.9) we only have to prove

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that kγ\λ0 ⋆ F ≃ 0. For (x, t, u) ∈ Uγ we have the Cartesian square i(x,t,u)

R

M × R2 × R>0 s

{(x, t, u)}

M × R × R>0 ,

where i(x,t,u) (t′ ) = (x, t′ , t−t′ , u). Writing i(x,t,u) = (ix ×i(t,u) )◦δ, with δ(t′ ) = (t′ , t′ ), ix (t′ ) = (x, t′ ) and i(t,u) (t′ ) = (t − t′ , u), we deduce by the base change formula that −1 (kγ\λ0 ⋆ F )(x,t,u) ≃ RΓc (R; i−1 x F ⊗ i(t,u) kγ\λ0 )

≃ RΓc (R; i−1 x F ⊗ k[t−u,t[ ). ∗ Then G = i−1 x F ⊗ k[t−u,t[ has compact support and satisfies SS(G) ⊂ Tτ ≥0 R by Theorems I.2.8 and I.2.13. Using Corollary I.2.16 we deduce that RΓc (R; G) ≃ 0. It follows that kγ\λ0 ⋆ F ≃ 0.

Let U be an open subset of M × R and V an open subset of U . Let X be a submanifold of M and U ′ = U ∩ (X × R). We have ΨV (F | ) ≃ (ΨU (F ))| ,

(XI.1.11)

V

Vγ

ΨU ′ (F | ′ ) ≃ (ΨU (F ))|

(XI.1.12)

U

Uγ′

,

where the first isomorphism follows from supports estimates as in Remark XI.1.2 (ii) and the second one follows from the base change formula. In the next lemma we use an analog of the convolution for sets. For A ⊂ M × R and B ⊂ R × R>0 we define B ⋆ A ⊂ M × R × R>0 by B ⋆ A = sM ×R (A × B),

(XI.1.13)

where sM ×R : M × R2 × R>0 → − M × R × R>0 is the sum as in Definition XI.1.1. Lemma XI.1.5. — Let F ∈ D(kU ) and let V ⊂ U be an open subset. We assume that F|

(XI.1.14)

V ∩({x}×R)

is locally constant, for any x ∈ M .

≃ 0. As a special case, if SS(F | ) ⊂ TV∗ V , then ΨU (F )| ≃ 0. In V Vγ ˙ particular supp(ΨU (F )) ⊂ (γ ⋆ π˙ U (SS(F ))) ∩ Uγ . F Proof. — We set Vx = V ∩ ({x} × R). Then Vγ = x∈M ({x} × (Vx )γ ) and we have to prove ΨU (F )| ≃ 0, for all x ∈ M . By (XI.1.12) we have {x}×(Vx )γ ΨU (F )| ≃ ΨVx (F | ). The set Vx is a disjoint union of open intervals Then ΨU (F )|

Vγ

{x}×(Vx )γ

Vx

of R and F | is constant on each of these intervals. A direct computation gives Vx ΨVx (F | ) ≃ 0 and we obtain the result. Vx

Lemma XI.1.6. — Let F ∈ D(kU ). ! ∼ (i) We have RqU ! qU (F ) −− → F and RrU ! (ΨU (F )) ≃ 0.

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−1 (ii) If F ∈ Dτ ≥0 (kU ), then RqU ! rU (F ) satisfies (XI.1.14) (with V = U ). In partic−1 ular ΨU (RqU ! rU (F )) ≃ 0. (iii) We assume that U = M × R, that F ∈ Dτ ≥0 (kU ) and that supp(F ) ⊂ M × [a, +∞[ −1 for some a ∈ R. Then RqU ! rU (F ) ≃ 0. ! Proof. — (i) The first morphism is the adjunction for (RqU ! , qU ). It is an isomorphism because the fibers of qU are intervals. Let us prove the second isomorphism. By the base change formula (see (XI.1.12)) we may as well assume that M is a point. Then U is an open subset of R and we can restrict to one component of U . Hence we assume U is an interval. We define r′ : R2 × R>0 → − R2 , (t1 , t2 , u) 7→ (t1 , t2 − u) and ′ 2 s:R → − R, (t1 , t2 ) 7→ (t1 + t2 ). We have the commutative diagram

R2 × R>0

r′

R2 s′

s

R × R>0

r

R.

We let j : U → − R be the inclusion and we set F ′ = j! F . Then ΨU (F ) ≃ (kγ ⋆ F ′ )|

Uγ

.

Let q2 : R × R × R>0 → − R × R>0 be the projection on the last two factors. We have RrU ! (ΨU (F )) ≃ Rr! (Rs! (F ′ ⊠ kγ ) ⊗ kUγ ) ≃ R(r ◦ s)! ((F ′ ⊠ kR×R>0 ) ⊗ kq−1 γ∩s−1 Uγ ) 2

≃ R(s′ ◦ r′ )! (r′−1 (F ′ ⊠ kR ) ⊗ kq−1 γ∩s−1 Uγ ) 2

≃ Rs′ ! ((F ′ ⊠ kR ) ⊗ Rr′ ! (kq−1 γ∩s−1 Uγ )). 2

Hence it is enough to prove that Rr′ ! (kq−1 γ∩s−1 Uγ ) ≃ 0. We write U = ]a, b[. Then 2 we have Uγ = {(t, u) ∈ R × R>0 ; a + u < t < b}. For any (t1 , t2 ) ∈ R2 the fiber r′−1 (t1 , t2 ) ∩ (q2−1 γ ∩ s−1 Uγ ) is identified with {u > 0; (t1 ,t2 + u, u) ∈ q2−1 γ ∩ s−1 Uγ } = {u > 0; 0 ≤ t2 + u < u and (t1 + t2 + u, u) ∈ Uγ } = {u > 0; −t2 ≤ u and u < b − t1 − t2 }, where we assume t2 < 0 and a < t1 + t2 (otherwise the fiber is empty). Since −t2 > 0 we see that the fiber is either empty or a half closed interval. This implies Rr′ ! (kq−1 γ∩s−1 Uγ ) ≃ 0, as required. 2

(ii) We choose x ∈ M and we set Ux = U ∩ ({x} × R). By the base change formula −1 −1 we have (RqU ! rU (F ))| ≃ RqUx ! rU (F | ). Hence we can assume that M is a point x Ux

Ux

−1 and that U is an interval. It is enough to prove that (RqU ! rU (F ))|

]a,b[

is constant

for any a, b ∈ U . We set V = U ∩ ]−∞, a[, Z = U \ V . In view of the distinguished +1 −1 triangle FV → − F → − FZ −−→ it is enough to see that G1 = (RqU ! rU (FV ))| and ]a,b[

−1 G2 = (RqU ! rU (FZ ))|

]a,b[

are constant.

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−1 −1 The maps qU and rU restricted to W = qU (]a, b[) ∩ rU (V ) identify W with ]a, b[ × V and we deduce, by the base change formula, that G1 is constant with stalks RΓc (V ; F | ). V Let j : U → − R be the inclusion and let q ′ , r′ : R2 → − R be the maps (t, u) 7→ t, (t, u) 7→ t−u. We set F ′ = Rj! (FZ ). Then we have G2 ≃ (Rq ′ ! (r′−1 (F ′ )⊗kR×R>0 ))| ]a,b[

and SS(F ′ ) ⊂ Tτ∗≥0 R. Using Theorems I.2.13 and I.2.8 we obtain, with coordinates (t, u; τ, υ) on T ∗ R2 :

SS((r

′−1

SS(r′−1 (F ′ )|

U ×R

(F ) ⊗ kR×R>0 )|

U ×R

′

) ⊂ {(t, u; τ, −τ ); τ ≥ 0}, ) ⊂ {(t, u; τ, −τ + υ); τ ≥ 0, υ ≤ 0}.

This last set intersects T ∗ R × TR∗ R along TR∗2 R2 . Since q ′ is proper on r′−1 (Z) ∩ (R × R≥0 ) it follows from Proposition I.2.4 that SS(G2 ) is contained in the zero section. Hence G2 is constant. −1 (iii) By (ii) RqU ! rU (F ) is constant on the fibers {x} × R for all x ∈ M . By the hypothesis its restriction to M × {a − 1} vanishes. Hence it is zero.

Lemma XI.1.7. — We let j : U × R>0 → − U × R be the inclusion. We set A = (U × {0}) ×U ×R T ∗ (U × R). We use the maps qU , rU of (XI.1.6) and the notations (I.1.1). Let F ∈ Dτ ≥0 (kU ). Then −1 −1 SS(ΨU (F )) ⊂ qU,d qU,π (SS(F )) ∪ rU,d rU,π (SS(F )),

SS(Rj! ΨU (F )) ∩ A ⊂ {(x, t, 0; ξ, τ, υ); (x, t; ξ, τ ) ∈ SS(F ), −τ ≤ υ ≤ 0}. ∼ In particular Rj! ΨU (F ) −− → Rj∗ ΨU (F ). Remark XI.1.8. — If we restrict to a “slice” Uγε = Uγ ∩ (U × {ε}), we ob∗ ε ˙ ˙ ˙ tain SS(Ψ U (F )| ε ) ⊂ (SS(F ) ∪ Tε (SS(F ))) ∩ T Uγ , where Tε is the translation Uγ

Tε (x, t; ξ, τ ) = (x, t + c; ξ, τ ). Proof. — (i) The first inclusion follows from the triangle (XI.1.10) and the triangular inequality for the microsupport. To prove the second inclusion we consider γ as a subset of R2 rather than R × R>0 . We also consider the sum s, (x, t1 , t2 , u) 7→ (x, t1 + t2 , u), as a map from U × R3 to M × R2 . Then by the base change formula we have Rj! ΨU (F ) ≃ Rs! (F ⊠ kγ ). Setting V = (U ×]−∞, 0])∪Uγ we see that s is proper as a map from s−1 (V )∩(U ×γ) to V . Hence we can use Proposition I.2.4 to bound SS(Rs! (F ⊠ kγ )| ). We see that V ∗ we only need to know SS(F ⊠ kγ ) above U × R × {0}. Now SS(kγ ) ∩ T(t,0) R2 is empty for t ̸= 0 and is the cone {−τ ≤ υ ≤ 0} for t = 0. Hence SS(F ⊠ kγ ) ∩ {u = 0} ⊂ {(x, t1 , 0, 0; ξ, τ1 , τ2 , υ); (x, t1 ; ξ, τ1 ) ∈ SS(F ), −τ2 ≤ υ ≤ 0}. We conclude with Proposition I.2.4.

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∼ (ii) We deduce Rj! ΨU (F ) −− → Rj∗ ΨU (F ). Let us set Z = U × ]−∞, 0] and W = U × ]0, +∞[. Then Rj∗ ΨU (F ) ≃ RΓW (Rj! ΨU (F )) and, by the excision distinguished triangle, we are reduced to prove that RΓZ (Rj! ΨU (F )) ≃ 0. Since the support of Rj! ΨU (F ) is contained in W , it is enough to prove (RΓZ (Rj! ΨU (F )))z ≃ 0 for any z ∈ U × {0}. Going back to the definition of the microsupport, this follows from (z, 0; 0, −1) ̸∈ SS(Rj! ΨU (F )). Proposition XI.1.9. — Let F, G ∈ Dτ ≥0 (kU ). We set Ω = Tτ∗>0 Uγ for short. Then we have a natural decomposition −1 −1 −1 −1 µhom(ΨU (F ), ΨU (G))| ≃ µhom(qU (F ), qU (G))| ⊕ µhom(rU (F ), rU (G))| Ω

Ω

Ω

such that the corresponding projection from µhom(ΨU (F ), ΨU (G))| to Ω −1 −1 −1 µhom(qU (F ), qU (G))| coincides with α2 ◦ α1 , where α1 , α2 are respectively Ω induced by the morphisms α(G) and α(F ) of (XI.1.7) as follows: α

1 −1 µhom(ΨU (F ), ΨU (G))| −→ µhom(ΨU (F ), qU (G))| Ω Ω −1 −1 ∼ ← −− µhom(qU (F ), qU (G))| . Ω α2

Proof. — (i) We set for short h(A) = µhom(ΨU (F ), A)| for a sheaf A on Uγ . The Ω triangle (XI.1.10) induces a distinguished triangle α

γ1

1 −1 −1 −1 h(rU (G))[−1] → − h(ΨU (G)) −→ h(qU (G)) −→ h(rU (G))

on Ω. We recall the bound supp µhom(F1 , F2 ) ⊂ SS(F1 ) ∩ SS(F2 ). By Theorem I.2.8 −1 −1 the microsupports of qU (G) and rU (F ) are respectively contained in {(x, t, u; ξ, τ, 0)} and {(x, t, u; ξ, τ, −τ )}. Hence their intersection is contained in {τ = 0}. Since we work −1 −1 on Ω = Tτ∗>0 Uγ it follows that the supports of h(qU (G)) and h(rU (G)) are disjoint, −1 −1 hence that γ1 = 0. This proves that h(ΨU (G)) ≃ h(qU (G)) ⊕ h(rU (G)). −1 −1 −1 (ii) Now we check that α2 : µhom(qU (F ), qU (G))| → − h(qU (G)) is an isoΩ morphism. Using the triangle (XI.1.10) again, we see that the cone of α2 is −1 −1 −1 −1 µhom(rU (F ), qU (G))| whose support is contained in SS(rU (F )) ∩ SS(qU (G)). Ω The same bound as in (i) shows that this set is contained in {τ = 0} and does not meet Ω. Hence the cone of α2 vanishes and α2 is an isomorphism. −1 −1 −1 ∼ We obtain µhom(rU (F ), rU (G))| −− → h(rU (G)) by swapping qU and rU in the Ω previous argument. This concludes the proof of the proposition.

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CHAPTER XI.2 ADJUNCTION PROPERTIES

a Let U ⊂ M × R be an open subset. We let Drτ!≥0 (kUγ ) be the full subcategory of Dτ ≥0 (kUγ ) consisting of RrU ! -acyclic objects, that is, the objects G such that RrU ! G ≃ 0. This is a triangulated category. By Lemma XI.1.6 (i) the functor a ΨU takes values in Drτ!≥0 (kUγ ). Using Theorem I.2.15 for the embedding Uγ ,→ U × R and Proposition I.2.4 we see that the functor RqU ! sends Dτ ≥0 (kUγ ) into Dτ ≥0 (kU ). −1 Moreover the morphism of functors α : ΨU → − qU in (XI.1.7) and the adjunction −1 ! morphism RqU ! qU ≃ RqU ! qU [1] → − id induce:

(XI.2.1)

bU (F ) : RqU ! ΨU (F )[1] → − F,

for all F ∈ Dτ ≥0 (kU ).

a Lemma XI.2.1. — The functor RqU ! [1] : Drτ!≥0 (kUγ ) → − Dτ ≥0 (kU ) is left adjoint r! a to ΨU : Dτ ≥0 (kU ) → − Dτ ≥0 (kUγ ). In particular we have an adjunction morphism

(XI.2.2)

b′U (G) : G → − ΨU RqU ! (G)[1],

a for all G ∈ Drτ!≥0 (kUγ ).

a Proof. — Since Dτ ≥0 (kU ) and Drτ!≥0 (kUγ ) are full subcategories of D(kU ) and D(kUγ ), it is enough to prove

(XI.2.3)

HomD(kUγ ) (G, ΨU (F )) ≃ HomD(kU ) (RqU ! G[1], F )

a for any F ∈ Dτ ≥0 (kU ) and G ∈ Drτ!≥0 (kUγ ). Since rU is a smooth map with fibers −1 ! homeomorphic to R we have a canonical isomorphism of functors rU ≃ rU [1]; −1 hence an adjunction (RrU ! , rU [1]). The same holds for qU . Applying RHom(G, ·) to (XI.1.10) we obtain the distinguished triangle +1

−1 −1 RHom(G, rU F [−1]) → − RHom(G, ΨU (F )) → − RHom(G, qU F ) −−→ . a −1 The adjunction (RrU ! , rU [1]) and the hypothesis G ∈ Drτ!≥0 (kUγ ) −1 RHom(G, rU F [−1]) ≃ 0. We deduce ∼ RHom(G, ΨU (F )) −− → RHom(G, q −1 F ) ≃ RHom(RqU G[1], F ), U

give

!

which implies (XI.2.3).

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Lemma XI.2.2. — Let F ∈ Dτ ≥0 (kU ). Then the morphisms ΨU (bU (F )) and b′U (ΨU (F )) are mutually inverse isomorphisms: ∼ ΨU (bU (F )) : ΨU RqU ΨU (F )[1] −− → ΨU (F ), !

b′U (ΨU (F )) :

∼ ΨU (F ) −− → ΨU RqU ! ΨU (F )[1].

Proof. — (i) We prove the first isomorphism. We apply RqU ! [1] to the distinguished triangle (XI.1.10). Since qU has fibers isomorphic to R the adjunction −1 ! morphism RqU ! qU (F )[1] ≃ RqU ! qU (F ) → − F is an isomorphism and we obtain the distinguished triangle: (XI.2.4)

bU (F )

+1

L→ − RqU ! ΨU (F )[1] −−−−→ F −−→,

−1 where L = RqU ! rU (F ). By Lemma XI.1.6 (ii) we have ΨU (L) ≃ 0. Hence applying ΨU to (XI.2.4) gives the lemma.

(ii) The composition ΨU (bU (F )) ◦ b′U (ΨU (F )) is the identity morphism of ΨU (F ), by general properties of adjunctions. Hence the lemma follows from (i). Proposition XI.2.3. — We assume that U = M × R, that F ∈ Dτ ≥0 (kU ) and that supp(F ) ⊂ M × [a, +∞[ for some a ∈ R. Then the adjunction morphism bU (F ) : RqU ! ΨU (F )[1] → − F of (XI.2.1) is an isomorphism and for any G ∈ Dτ ≥0 (kU ) we have ∼ Hom(F, G) −− → Hom(ΨU (F ), ΨU (G)). ∼→ F and RqU r−1 (F ) ≃ 0. Proof. — By Lemma XI.1.6 (i) and (iii) we have RqU ! qU! (F ) −− ! U Hence the first part follows from the distinguished triangle (XI.1.10). Then the second part is given by the adjunction (RqU ! , ΨU ) of Lemma XI.2.1.

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In this section we prove Proposition XI.3.7 below which says that the group of homomorphisms from ΨU (F ) to ΨU (G) is isomorphic to the group of sections of µhom(F, G) outside the zero section. We first deduce from Proposition XI.1.9 a morphism from µhom(ΨU (F ), ΨU (G)) to µhom(F, G). Then we use “boundary values” of sheaves on Uγ in the following sense. Let U ⊂ M × R and V ⊂ M × R × R>0 be open subsets satisfying (XI.3.1)

iU (U ) ⊔ jV (V ) is open in M × R × R≥0 ,

where iU , jV are the natural inclusions iU : U → − M × R × R, (XI.3.2) jV : V → − M × R × R.

(x, t) 7→ (x, t, 0),

Then, for G ∈ D(kV ), its boundary value is i−1 U RjV ∗ (G) ∈ D(kU ). We remark that we can shrink V as long as (XI.3.1) is satisfied: Lemma XI.3.1. — Let V ′ ⊂ V be an open subset such that iU (U ) ⊔ jV ′ (V ′ ) is open in M × R × R≥0 . Then the morphism RjV ∗ (G) → − RjV ′ ∗ (G| ′ ), obtained V −1 −1 from jV ′ RjV ∗ (G) ≃ G| ′ by the adjunction (jV ′ , RjV ′ ∗ ), induces an isomorphism V ∼ i−1 − → i−1 U RjV ∗ (G) − U RjV ′ ∗ (G| ′ ). V

Proof. — To check that a given morphism is an isomorphism it is enough to see that it gives an isomorphism on the cohomology of the stalks at any point. For (x, t) ∈ U and k ∈ Z we have H k (i−1 H k (W ∩ V ; G), where W runs U RjV ∗ (G))(x,t) ≃ lim −→W over the neighborhoods of (x, t, 0) in M × R × R≥0 . For a given (x, t) ∈ U we have W ∩ V = W ∩ V ′ when W is small enough and the result follows. Notation XI.3.2. — Because of Lemma XI.3.1 we can forget the subscript V and write the boundary value as i−1 U Rj∗ (G) for G ∈ D(kV ), when V is any open subset of M × R × R>0 satisfying (XI.3.1). Most of the time V will be Uγ or U × R>0 but sometimes it will be convenient to shrink V in the course of a proof.

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>0 >0 Let πU : Tτ∗>0 U → − U , πU : Tτ∗>0 Uγ → − Uγ be the projections. Proposition XI.1.9 γ yields a morphism, for F, G ∈ Dτ ≥0 (kU ),

(XI.3.3)

−1 −1 >0 (µhom(qU (F ), qU (G))| RHom (ΨU (F ), ΨU (G)) → − RπU γ ∗

Tτ∗>0 Uγ

),

which can be written as the composition (XI.3.4)

−1 RHom (ΨU (F ), ΨU (G)) → − RHom (ΨU (F ), qU (G)) −1 ≃ RπUγ ∗ µhom(ΨU (F ), qU (G)) −1 >0 → − RπU (µhom(ΨU (F ), qU (G))| γ

(XI.3.5)

Tτ∗>0 Uγ

∗

−1 −1 >0 ∼ (µhom(qU (F ), qU (G))| ← −− RπU γ ∗

)

Tτ∗>0 Uγ

),

−1 where the first line is induced by α(G) : ΨU (G) → − qU (G), the second line is (I.3.2), ∗ the third line is the restriction to Tτ >0 Uγ and the fourth line is given by Proposition XI.1.9. Now it is easy to describe the boundary value of the right hand side of (XI.3.3) as follows.

Lemma XI.3.3. — For F, G, H ∈ D(kU ) and F ∈ D(kT ∗ U ) we have natural isomorphisms −1 −1 −1 µhom(qU (F ), qU (G)) ≃ qU,d ! qU,π µhom(F, G),

(XI.3.6)

−1 >0 RπU ((qU,d ! qU,π (F ))| γ

(XI.3.7)

∗

Tτ∗>0 Uγ

−1 >0 ) ≃ qU RπU ∗ (F |

Tτ∗>0 U

),

−1 i−1 U Rj∗ qU H ≃ H

(XI.3.8)

which induce (XI.3.9) −1 −1 >0 i−1 U Rj∗ RπUγ (µhom(qU (F ), qU (G))| ∗

Tτ∗>0 Uγ

>0 ) ≃ RπU ∗ (µhom(F, G)|

Tτ∗>0 U

).

Proof. — (i) The behavior of µhom under an inverse image by a submersion is described in [28, Prop. 4.4.7] and gives in our case the isomorphism (XI.3.6). −1 ! (ii) We remark that qU,d ! = qU,d ∗ , since qU,d is an embedding, and qU ≃ qU [−1], −1 ! qU,π ≃ qU,π [−1], since qU is a projection with fibers R. Now (XI.3.7) follows from the >0 >0 ! ! RπU base change formula Ra∗ qU,π ≃ qU ∗ with a = πUγ ◦ qU,d . −1 −1 −1 (iii) We have i−1 − U is the U Rj∗ qU H ≃ iU RΓU ×R>0 (q1 H), where q1 : U × R → ∗ ∗ projection. Since SS(kU ×R>0 ) ⊂ TU U × T R, Theorem I.2.13 gives

RΓU ×R>0 (q1−1 H) ≃ RHom (kU ×R>0 , q1−1 H) ≃ kU ×R≥0 ⊗ q1−1 H −1 −1 −1 and we obtain i−1 U Rj∗ qU H ≃ iU (kU ×R≥0 ⊗ q1 H) ≃ H.

Definition XI.3.4. — For F, G ∈ Dτ ≥0 (kU ) we define the morphism (functorial in F and G) >0 b(F, G) : i−1 − RπU ∗ (µhom(F, G)| U Rj∗ RHom (ΨU (F ), ΨU (G)) →

Tτ∗>0 U

as the composition of (XI.3.3) and (XI.3.9).

ASTÉRISQUE 440

)

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We prove below that b(F, G) is an isomorphism if G ∈ Dτ >0 (kU ). We need some remarks on the µhom functor. We will use Sato’s distinguished triangle (I.3.4) and introduce the following notation. Notation XI.3.5. — Let X be a manifold. Let qX,1 , qX,2 : X × X → − X be the projections and δX : X → − X × X the diagonal embedding. Let F, F ′ ∈ D(kX ). We set (XI.3.10)

Hom

′

−1 −1 −1 (F, F ′ ) := δX RHom (qX,2 F, qX,1 F ′ ).

Then Sato’s distinguished triangle becomes, for F, F ′ ∈ D(kX ), (XI.3.11)

Hom

′

+1

(F, F ′ ) → − RHom (F, F ′ ) → − Rπ˙ X ∗ (µhom(F, F ′ )| ˙ ∗ ) −−→ . T X

Lemma XI.3.6. — (i) Let f : X → − Y be a morphism of manifolds. Let F, F ′ ∈ D(kY ) such that f is non-characteristic for SS(F ) and SS(F ′ ). Then ∼ f −1 Hom ′ (F, F ′ ) −− → Hom ′ (f −1 F, f −1 F ′ ). b SS(F ′ ). (ii) For F, F ′ ∈ D(kX ) we have SS(Hom ′ (F, F ′ )) ⊂ SS(F )a + −1 −1 ′ Proof. — (i) We set G = RHom (qY,2 F, qY,1 F ) (using similar notations as in (XI.3.10)). −1 ′ a Then SS(qY,1 F ) and SS(G) ⊂ SS(F ) × SS(F ′ ) are non-characteristic for f × f . By Theorem I.2.8 we deduce −1 ′ −1 ′ (f × f )! qY,1 F ≃ (f × f )−1 qY,1 F ⊗ ωX×X|Y ×Y , ⊗−1 (f × f )−1 G ≃ (f × f )! G ⊗ ωX×X|Y ×Y .

By Proposition I.1.1-(h) this gives the third isomorphism in the following sequence: −1 −1 ′ f −1 Hom ′ (F, F ′ ) ≃ f −1 δY−1 RHom (qY,2 F, qY,1 F ) −1 −1 −1 ′ ≃ δX (f × f )−1 RHom (qY,2 F, qY,1 F ) ∼ −− → δ −1 RHom ((f × f )−1 q −1 F, (f × f )−1 q −1 F ′ ) X

Y,2

Y,1

−1 −1 −1 −1 −1 ′ ≃ δX RHom (qX,2 f F, qX,1 f F ).

(ii) follows from Remark I.2.14 and Theorem I.2.8. ˙ ˙ Proposition XI.3.7. — Let F, G ∈ Dτ ≥0 (kU ). If SS(F )∩ SS(G) is contained in {τ > 0}, then we have >0 ∼ Rπ˙ U ∗ (µhom(F, G)| ˙ ∗ ) −− → RπU ∗ (µhom(F, G)|T ∗ U ) T U τ >0

and the morphism b(F, G) of Definition XI.3.4 is an isomorphism. Proof. — (i) We recall that supp µhom(F1 , F2 ) ⊂ SS(F1 )∩SS(F2 ). Hence the hypothesis implies that µhom(F, G)| ˙ ∗ is supported in Tτ∗>0 U and this gives the first isomorT U phism. Let us call u and v the morphisms (XI.3.4) and (XI.3.5). To see that b(F, G) is −1 an isomorphism, we prove that the induced morphisms i−1 U Rj∗ (u) and iU Rj∗ (v) are isomorphisms, respectively in (ii) and (iii) below.

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(ii) Let us prove that i−1 U Rj∗ (u) is an isomorphism. By the distinguished trian−1 gle (XI.1.10) its cone is A = i−1 U Rj∗ RHom (ΨU (F ), rU (G)) and we have to prove the vanishing of A. For a given (x, t) ∈ U and k ∈ Z we have −1 H k (A)(x,t) ≃ lim Hom(ΨU (F )| , rU (G)| [k]), W W −→

(XI.3.12)

W

where W runs over the open subsets of M × R × R>0 such that W is a neighborhood of (x, t, 0) in U × [0, +∞[. We may take W = Vγ , where V runs over the open neighborhoods of (x, t) in U . By (XI.1.11) we have ΨU (F )| ≃ ΨV (F | ). We also have Vγ

−1 rU (G)|

Vγ

V

≃ rV−1 (G| ). Since rV! ≃ rV−1 [1], the adjunction (RrV ! , rV! ) gives V

Hom(ΨV (F | ), rV−1 (G| )[k]) ≃ Hom(RrV ! ΨV (F | ), G| [k − 1]). V

V

V

V

By Lemma XI.1.6 we have RrV ! ΨV (F | ) ≃ 0 and we deduce the vanishing of (XI.3.12) V for all (x, t) in U . Hence A ≃ 0, as required. (iii) Now we prove that i−1 U Rj∗ (v) is an isomorphism. Using the hypothesis on the microsupport, we remark as in (i) that the right hand side of (XI.3.5) is isomorphic −1 to Rπ˙ Uγ ∗ (µhom(ΨU (F ), qU (G))| ˙ ∗ ). Hence, by the triangle (XI.3.11), the cone T Uγ

′ −1 −1 of i−1 U Rj∗ (v) is (up to a shift by 1) B := iU Rj∗ Hom (ΨU (F ), qU (G)). Let us prove that B vanishes. Let pU : U × R → − U be the projection. We set U+ = U × R>0 and C = Hom ′ (Rj! ΨU (F ), p−1 U (G)). Then −1 Rj∗ Hom ′ (ΨU (F ), qU (G)) ≃ RΓU+ C.

By Lemma XI.1.7 we have SS(Rj! ΨU (F )) ⊂ {(ξ, τ, υ); −τ ≤ υ ≤ 0}. We also have ˙ SS(p−1 U (G)) ⊂ {(ξ, τ, 0); τ ≥ 0} and we obtain SS(C) ⊂ {υ ≥ 0} by Lemma XI.3.6. ˙ U ) = T ∗ U × {υ < 0}, we deduce RΓU C ≃ D′ (kU ) ⊗ C ≃ C by Since SS(k U

+

+

+

U+

Theorem I.2.13. In particular B ≃ i−1 U C. We also obtain that iU is non-characteristic for SS(Rj! ΨU (F )) and SS(p−1 U (G)). ′ −1 −1 Hence i−1 C ≃ Hom (i Rj Ψ (F ), G) by Lemma XI.3.6. Since i Rj Ψ (F ) ≃ 0, we ! U ! U U U U −1 obtain iU C ≃ 0, as required. For the next result we use the notion of pure sheaves (see Definition I.4.2) along a Lagrangian submanifold Λ ⊂ T˙ ∗ U and the stack µSh(kΛ ) together with the functor mΛ : Db(Λ) (kM ) → − µSh(kΛ ) (see Definition X.1.1). Corollary XI.3.8. — Let Λ ⊂ Tτ∗>0 U be a closed conic Lagrangian submanifold. Let F, G ∈ D[Λ] (kU ) be pure sheaves with the same shift. Then lim Hom(ΨU (F )| , ΨU (G)| ) ≃ Hom(mΛ (F ), mΛ (G)), V V −→ V

where V runs over the open subsets of Uγ such that (U × {0}) ⊔ V is open in M × R × R≥0 . In particular, if mΛ (F ) ≃ mΛ (G), then there exists such an open subset V such that ΨU (F )| ≃ ΨU (G)| . V

ASTÉRISQUE 440

V

CHAPTER XI.3. LINK WITH MICROLOCALIZATION

223

Proof. — (i) We recall that the Hom sheaf in µSh(kΛ ) is induced by H 0 µhom. By the purity hypothesis µhom(F, G) is concentrated in degree 0 and (X.1.6) gives Hom(mΛ (F ), mΛ (G)) ≃ H 0 (Λ; µhom(F, G)). Hence the isomorphism of the corollary follows from Proposition XI.3.7 and the remark that, for any F ′ ∈ D(kUγ ), ′ H 0 (U ; i−1 H k (W ∩ Uγ ; F ′ ), U Rj∗ (F )) ≃ lim −→ W

where W runs over the neighborhoods of U in M × R × R≥0 . ∼ (ii) Let u : mΛ (F ) −− → mΛ (G) be an isomorphism. By (i) there exist V as in the statement of the corollary and a : ΨU (F )| → − ΨU (G)| , b : ΨU (G)| → − ΨU (U )| V V V V −1 representing u, u . Using (i) again, and maybe shrinking V , the relations u◦u−1 = id, u−1 ◦ u = id give a ◦ b = id, b ◦ a = id. Hence a is an isomorphism.

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CHAPTER XI.4 DOUBLED SHEAVES

˙ The isomorphism of Proposition XI.3.7 will be used mainly when SS(F ) = ∗ ˙ ˙ SS(G) = Λ is a smooth Lagrangian submanifold of Tτ >0 (M × R). In this case µhom(F, G)| ˙ ∗ is a locally constant sheaf on Λ. T (M ×R) Unfortunately we will also need to consider the case where F and G have microsupport Λ only in a neighborhood of some open subset Λ0 of Λ. To easily handle this case we actually assume that Λ0 is locally of the form Λ ∩ T ∗ (U × I), for some open set U of M and some open interval I of R, and that F (and G as well) is locally of the form RΓU F ′ for some F ′ with SS(F ′ ) ⊂ Λ. In the proof of Theorem XII.1.1 below we will glue such sheaves F or, rather, their images Ψ(F ). For this reason we define a subcategory of D(kM ×R×R>0 ) of sheaves which are locally of the form Ψ(RΓU F ′ ). We then give the analog of Proposition XI.3.7 for this subcategory (see Proposition XI.4.9 below). Since we deal with sheaves of the form RΓU F ′ , we will face the problem of the commutation of the main functors introduced so far, Ψ and i−1 Rj∗ , with the functors RΓU , RΓUγ ,. . . Lemma XI.4.5 answers this question and its proof requires some conditions on Λ and SS(kU ), essentially to be able to use Theorem I.2.13. This is the aim of the following definition. For T a family of subsetsSUa , a ∈ A, of some set E and for B ⊂ A we set UB = a∈B Ua and U B = a∈B Ua . Definition XI.4.1. — Let Λ ⊂ Tτ∗>0 (M × R) be a conic Lagrangian submanifold such that Λ/R>0 is compact, the map Λ/R>0 → − M has finite fibers. A finite family V = {Va ; a ∈ A} of open subsets of M × R is said to be adapted to Λ if the following conditions hold: −1 (i) for each a ∈ A we have Va = Ua ×Ia and Λ∩T ∗ Va is contained in πM ×R (Ua ×K) for some compact interval K of Ia , ∗ (ii) for all B, B ′ ⊂ A we have D′ (kV B ) ≃ kV B and, setting Λ+ = Λ ∪ TM ×R (M × R),

(XI.4.1)

∗ b SS(kV B′ )a ) ∩ (Λa+ + b Λ+ ) ⊂ TM (SS(kV B ) + ×R (M × R).

Notation XI.4.2. — In this section we use adapted families of open subsets to cut sheaves defined on an open subset W of M × R. To avoid too heavy notations we set

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abusively, for F ∈ D(kW ) and B ⊂ A: RΓV B (F ) := RΓW ∩V B (F ) ∈ D(kW ) and similarly RΓV B ×R>0 (−) := RΓWγ ∩(V B ×R>0 ) (−) for sheaves defined over Wγ or RΓT ∗ V B (−) := RΓT ∗ W ∩T ∗ V B (−) over T ∗ W ,. . . We first check that we have enough such adapted families. Lemma XI.4.3. — Let Λ ⊂ Tτ∗>0 (M S × R) be a conic Lagrangian submanifold such that Λ/R>0 is compact and let Λ = i∈I Λi be a finite covering by conic open subsets. Then there exists a Hamiltonian isotopy Φ, as closed to id as desired, and a finite family {Va }a∈A of open subsets of M × R which is adapted to Λ′ = Φ1 (Λ) such that each connected component of Λ′ ∩ T ∗ Va , for a ∈ A, is contained in Φ1 (Λi ), for some i ∈ I. Moreover we can assume that the family {Va }a∈A is stable by intersection and that Λ′ ∩ T ∗ Va has finitely many connected components, for each a ∈ A. Proof. — (i) We recall Definition I.2.19: a stratification Σ = {Σj , j ∈ J} of M × R S b ΛΣ = ΛΣ , where ΛΣ = j∈J TΣ∗ (M × R). By Proposatisfies the µ-condition if ΛΣ + j sition I.2.20, any sheaf F ∈ D(kM ×R ) which is constructible with respect to Σ satisfies SS(F ) ⊂ ΛΣ . By Proposition I.2.23, there exists an R>0 -homogeneous Hamiltonian isotopy Φ and a stratification Σ of M × R satisfying the µ-condition such that Λ′ := Φ1 (Λ) ⊂ ΛΣ . Now to have (XI.4.1) it is enough that the family {Va }a∈A satisfies (XI.4.2)

−1 ∗ b SS(kV B′ )a ) ∩ ΛΣ ∩ πM (SS(kV B ) + ˙ M ×R (Λ′ )) ⊂ TM ×R (M × R). ×R (π

(ii) We can assume that Λ′ is also contained in Tτ∗>0 (M × R). Hence, for a given point y0 = (x0 , t0 ) in π˙ M ×R (Λ′ ) the strata Σj such that y0 ∈ Σj ⊂ π˙ M ×R (Λ′ ) do not meet the truncated cone given in local coordinates by {η > |t − t0 | > C∥x − x0 ∥}, for η > 0 small enough and C big enough. Then, for U = {∥x − x0 ∥ < η/2C} ⊂ M and I = {|t − t0 | < η} ⊂ R, we have y0 ∈ U × I and π˙ M ×R (Λ′ ) ∩ (U × ∂I) = ∅. We can then cover π˙ M ×R (Λ′ ) by open subsets of this kind and, taking a finite subcover, we obtain S a finite family of open subsets of M × R, Wa , a ∈ A, such that π˙ M ×R (Λ′ ) ⊂ a∈A Wa , Wa = Ua × Ia as in (i) of Definition XI.4.1 and Ua = {fa < 0} for a C ∞ function fa : M → − R. Moreover, for each a ∈ A there exists a compact inter−1 val Ka ⊂ Ia such that Λ′ ∩ T ∗ Wa is contained in πM ×R (Ua × Ka ). We can also assume that each Wa is small enough so that each component of Λ′ ∩ T ∗ Wa is contained in Φ1 (Λi ), for some i ∈ I. Let ε = {εa ; a ∈ A} be a family of negative numbers. We define Wa,εa = {fa < εa } × Ia . S We choose δ > 0 such that π˙ M ×R (Λ′ ) ⊂ a∈A Wa,−δ . Let E ⊂ [−δ, 0[A be the subset formed by the ε such that: (a) the hypersurfaces Xa,εa = {fa = εa }, a ∈ A, are smooth and intersect transversely, in the sense that their union is locally diffeomorphic to the embedding of coordinates hyperplanes in Rn ,

ASTÉRISQUE 440

CHAPTER XI.4. DOUBLED SHEAVES

(b) for any B ⊂ A, the manifold XB,ε = Σj of Σ transversely.

T

a∈B (Xa,εa

227

× Ia ) intersects each stratum

By the transversality theorem E is dense in [−δ, 0[A . Hence we only have to prove that, for a given ε ∈ E, the family Va = Wa,εa , a ∈ A, satisfies the conclusion of the lemma. The condition D′ (kV B ) ≃ kV B is local on M × R and follows from the above condition (a), up to modifying slightly the intervals Ia ’s so that they have distinct ends. S We set Y = a∈A (Xa,εa × ∂Ia ) and Ω = (M × R) \ Y . We remark that π˙ M ×R (Λ′ ) ∩ Y = ∅. Hence, to prove (XI.4.2) it is enough to see b SS(kV B′ )a ) ∩ ΛΣ ∩ T ∗ Ω ⊂ TΩ∗ Ω. (SS(kV B ) + In Ω we have the family of hypersurfaces (Xa,εa × Ia ) ∩ Ω, a ∈ A, which are closed (not only locally closed), smooth and intersect transversely. This family generates a stratification Σ′ (ε) of Ω such that the closures of the strata are the XB,ε ∩Ω, B ⊂ A. By b ΛΣ′ (ε) = ΛΣ′ (ε) and ΛΣ′ (ε) ∩ ΛΣ ∩ T ∗ Ω ⊂ TΩ∗ Ω. the transversality assumptions ΛΣ′ (ε) + B For B ⊂ A the open set V ∩ Ω is constructible with respect to Σ′ (ε) and we deduce SS(kV B ) ∩ T ∗ Ω ⊂ ΛΣ′ (ε) . This gives (XI.4.2) and the family {Va }a∈A is adapted to Λ′ . T (iii) Let A′ be the set of subsets of A and, for a′ ∈ A′ , set V˜a′ = a∈a′ Va . An open ′ subset V˜ B , for B ′ ⊂ A′ , is still bounded by the hypersurfaces Xa,εa introduced in the condition (a) of (ii). So we also have D′ (kV˜ B′ ) ≃ kV˜ B′ and SS(kV˜ B′ ) ⊂ ΛΣ′ (ε) as in (ii). Replacing A by A′ we have an adapted family which is stable by intersection. To ensure that Λ ∩ T ∗ V˜a′ has finitely many connected components, for each a′ ∈ A′ , − R, for each a ∈ A, we choose ε so that εa is a regular value of fa ◦ π : Λ′ ∩ T ∗ Wa → where π is the projection Λ′ → − M. Definition XI.4.4. — Let Λ and V = {Va ; a ∈ A} be as in Definition XI.4.1. Let b V ⊂ M × R be an open subset. We denote by Ddbl Λ,V (kV ) the subcategory of D (kVγ ) formed by the G such that any point of V has an open neighborhood W ⊂ V such that Λ∩T ∗ W has finitely many connected components, say {Λi }i∈I , and for each i ∈ I there exist Ai ⊂ A and Fi ∈ Db[Λi ] (kW ) satisfying M ≃ (XI.4.3) G| RΓV Ai ×R>0 (ΨW (Fi )), Wγ i∈I

where V

Ai

=

S

a∈Ai

Va and we use the abusive Notation XI.4.2.

For G and W as in (XI.4.3) we have supp(G) ⊂ Wγ ∩(γ⋆π˙ M ×R (Λ)) by Lemma XI.1.5 (see (XI.1.13) for ⋆). We can cover V Sby open subsets Wj , j ∈ J, for which a decomposition (XI.4.3) holds; setting V ′ = j∈J Wj,γ we then have (XI.4.4)

supp(G) ∩ V ′ ⊂ (γ ⋆ π˙ M ×R (Λ)) ∩ V ′

and (V × {0}) ⊔ V ′ is open in M × R × R≥0 (as in (XI.3.1)).

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For an open subset W ⊂ M × R we recall the maps qW , rW : Wγ → − M ×R of (XI.1.6). For subsets U of M × R or Ξ of T ∗ (M × R) we introduce the notations −1 U q = U qW = q W (U ∩ W ), −1 (Ξ ∩ T ∗ W ) = (Ξ × TR∗>0 R>0 ) ∩ T ∗ Wγ , Ξq = ΞqW = qW,d qW,π

(XI.4.5)

−1 Ξr = ΞrW = rW,d rW,π (Ξ ∩ T ∗ W ).

(W is in general suppressed from notation if there is no ambiguity.) The next lemma gives results about the sheaves appearing in (XI.4.3). Lemma XI.4.5. — Let Λ and V = {Va ; a ∈ A} be as in Definition XI.4.1. Let W be an open subset of M × R and let Λ1 be a connected component of Λ ∩ T ∗ W . Let F ∈ Db[Λ1 ] (kW ) and B ⊂ A be given. Then, using Notation XI.3.2 and (XI.4.5), we have (i) RΓV B,q (ΨW (F )) ≃ (ΨW (F ))V B,q , (ii) SS(RΓV B,q (ΨW (F ))) ∩ Λq ⊂ (Λ1 ∩ T ∗ V B )q , (iii) for any G ∈ Ddbl Λ,V (kW ) we have (XI.4.6)

−1 i−1 W Rj∗ RHom (G, RΓV B,q (ΨW (F ))) ≃ RΓV B (iW Rj∗ RHom (G, ΨW (F ))),

(iv) for any x ∈ W \ π˙ M ×R (Λ1 ) there exists a neighborhood W ′ of x such that (RΓV B,q (ΨW (F )))| ′ ≃ 0, Wγ

(v) for any x ∈ W ∩ π˙ M ×R (Λ1 ) there exists a neighborhood W ′ of x such that (XI.4.7)

(RΓV B,q (ΨW (F )))|

Wγ′

and SS(RΓW ′ ∩V B (F |

W′

≃ ΨW ′ (RΓW ′ ∩V B (F |

W′

))

)) ⊂ Tτ∗≥0 W ′ .

Proof. — (i)–(ii) We will show that the first two claims follow from Theorem I.2.13, q r ˙ the bound SS(Ψ W (F )) ⊂ Λ1 ∪ Λ1 of Lemma XI.1.7 and the hypotheses of Definition XI.4.1. ˙ More precisely, to deduce (i) from Theorem I.2.13 we need to check that SS(Ψ W (F )) q and SS(kV B,q ) = (SS(kV B )) do not intersect (here F is a sheaf on W and kV B means kV B | = kV B ∩W with the same abuse as in Notation XI.3.2). We take coorW dinates (ξ, τ, υ) in a fiber of T ∗ (M × R × R>0 ). The points of Λq1 , Λr1 and SS(kV B,q ) are respectively of the form (ξ1 , τ1 , 0), (ξ2 , τ2 , −τ2 ), (ξ3 , τ3 , 0) with τ1 , τ2 > 0 (since Λ1 ⊂ Λ). Hence the subsets Λr1 and SS(kV B,q ) cannot intersect. So it remains to ˙ ˙ V B ) is concheck that SS(kV B ) and SS(F ) do not intersect. We recall that SS(k B B tained in the fibers over ∂V and that V is a finite union of products Ua × Ia . We S remark that ∂V B ⊂ a∈B ∂(Ua × Ia ). Let us call the “vertical part” of ∂V B its subS set ∂V B \(∂V B ∩ a∈B (Ua ×∂Ia )). Near a point in the vertical part ∂V B is then of the form ∂V × I for some open subsets V ⊂ M , I ⊂ R and a point of SS(kV B,q ) is of the form (ξ3 , 0, 0). The hypothesis (i) in Definition XI.4.1 implies that πM ×R (Λ) and ∂V B can only meet at points in the vertical part of ∂V B . Near such a point SS(kV B ) is contained in {τ = 0} and cannot meet Λ. This proves (i).

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˙ Since SS(Ψ W (F )) and SS(kV B,q ) do not intersect, we know by Theorem I.2.13 and Example I.2.12 that SS(RΓV B,q (ΨW (F ))) is bounded by the pointwise sum ∗ Ξ = Λ′1 + SS(kV B,q ), where Λ′1 = Λq1 ∪ Λr1 ∪ TW Wγ . We need to understand the γ q ′ intersection of Ξ and Λ . It is clear that Λ1 + (TV∗ B V B )q = Λ′1 ∩ T ∗ (q −1 (V B )) and this satisfies the required bound. Hence we can concentrate on the part of (SS(kV B ))q outside the zero section, hence above q −1 (∂V B ). Since we are interested in Λq ∩ Ξ, by the preceding discussion we only need to consider points above the vertical part of q −1 (∂V B ) and a point of (SS(kV B ))q is then of the form (ξ0 , 0, 0). It follows that a point of Λr1 + (SS(kV B ))q is of the form (ξ, τ, −τ ), τ > 0, and cannot belong to Λq . ∗ We also see that TW Wγ + (SS(kV B ))q does not meet Λq . So it only remains to γ q understand (Λ1 + (SS(kV B ))q ) ∩ Λq and the statement follows from the hypothesis (ii) of Definition XI.4.1.

(iii-a) The isomorphism (iii) is local on W and we can shrink W if necessary. Since G ∈ Ddbl Λ,V (kW ) we can thus assume that G satisfies (XI.4.3). Taking one summand we assume in fact G = RΓV B′ ,q (ΨW (F ′ )), for some B ′ ⊂ A and F ′ ∈ Db[Λ] (kW ). We will prove (XI.4.8)

′ ′ i−1 W Rj∗ RHom (RΓV B ,q (ΨW (F )), RΓV B,q (ΨW (F ))) ′ ≃ RΓV B′ ∩V B (i−1 W Rj∗ RHom (ΨW (F ), ΨW (F ))).

This implies (XI.4.6): use (XI.4.8) as it is stated and also in the case where V B ⊃ W , together with RΓV B′ ∩V B (−) ≃ RΓV B (RΓV B′ (−)). We set H = RHom (ΨW (F ′ ), ΨW (F )). Using part (i) of the lemma and the isomorphism RHom ((−)Z ′ , RΓZ (−)) ≃ RΓZ ′ ∩Z RHom (−, −), we deduce that the left hand side of (XI.4.8) is −1 i−1 W Rj∗ RΓ(V B ′ ∩V B )q (H) ≃ iW RΓ(V B ′ ∩V B )×R (Rj∗ H)

and we are reduced to prove −1 i−1 W RΓ(V B ′ ∩V B )×R (Rj∗ H) ≃ RΓV B ′ ∩V B (iW Rj∗ H).

For this it is enough to check, for some ε > 0, setting Jε = ]−∞, ε[, ˙ (SS(kV B′ ∩V B ) × TJ∗ε Jε ) ∩ SS(Rj ∗ (H)) = ∅, (XI.4.9) −1 ˙ SS(kV B′ ∩V B ) ∩ SS(i W Rj∗ (H)) = ∅. Indeed, (XI.4.9), (XI.4.1) and Theorem I.2.13 imply the isomorphisms L

RΓ(V B′ ∩V B )×J (Rj∗ H) ≃ D′ (k(V B′ ∩V B )×J ) ⊗ Rj∗ H ε

ε

L

≃ (D′ (kV B′ ∩V B ) ⊠ kJ ε ) ⊗ Rj∗ H, L

−1 ′ RΓV B′ ∩V B (i−1 W Rj∗ H) ≃ D (kV B ′ ∩V B ) ⊗ iW Rj∗ H.

We then use the commutativity of i−1 W with the tensor product.

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(iii-b) Let us prove (XI.4.9). Proposition I.1.1-(i) gives Rj∗ H ≃ RHom (Rj! ΨW (F ′ ), Rj∗ ΨW (F )). For F ′′ ∈ Dτ ≥0 (kW ) Lemma XI.1.7 says Rj! ΨW (F ′′ ) ≃ Rj∗ ΨW (F ′′ ) and gives a bound for SS(Rj! ΨW (F ′′ )). We deduce the following lessSprecise bound which is easier to handle. For S ⊂ T ∗ W and ε > 0 we define Nε (S) = c∈[0,ε] Tc (S) where Tc is the vertical translation Tc (x, t; ξ, τ ) = (x, t + c; ξ, τ ). Lemma XI.1.7 implies SS((Rj! ΨW (F ′′ ))|

W ×Jε

) ⊂ Nε (SS(F ′′ )) × T ∗ Jε .

By Theorem I.2.13 we deduce (XI.4.10)

b Λ+ ) × T ∗ Jε . SS(Rj∗ (H)) ⊂ Nε (Λa+ +

When we take the inverse image by iW in (XI.4.10) we can assume ε as small as we want and Theorem I.2.8 gives (XI.4.11)

a b SS(i−1 W Rj∗ (H)) ⊂ Λ+ + Λ+ .

b SS(kV B′ )a . Now we deduce the relaFinally we remark SS(kV B′ ∩V B ) ⊂ SS(kV B ) + tions (XI.4.9) from (XI.4.10) (taking ε as small as required), (XI.4.11) and (XI.4.1). (iv) follows from the bound (XI.4.4) and the remark that, for any subset S of M × R and any ε > 0, [
(γ ⋆ S) ∩ (M × R × ]0, ε]) ⊂ tc (S) × ]0, ε], c∈[0,ε]

where tc is the vertical translation tc (x, t) = (x, t + c). (v) If x ∈ V B , then we choose W ′ = W ∩ V B and the result is obvious. If x ̸∈ V B , we choose W ′ such that W ′ ∩ V B = ∅ and both sides are zero. So we can assume x ∈ ∂V B . Since x ∈ π˙ M ×R (Λ1 ) we have seen in the proof of (i) that x belongs to the “vertical part” of ∂V B . Hence we can find a neighborhood W ′ of x such that W ′ ∩ V B = W ′ ∩ (U × R),Sfor some open subset U of M : recalling that S V B = a∈B (Ua × Ia ), we have U = a∈B,x∈Ua ×Ia Ua . It follows from the projection formula (see Proposition I.1.1-(g)) that ΨW ′ ((H)W ′ ∩(Z×R) ) ≃ (ΨW ′ (H))Wγ′ ∩(Z×R×R>0 ) , for any sheaf H on W ′ and any locally closed subset Z of M . Using part (i) of the lemma (and the similar isomorphism RΓV B (F ) ≃ (F )V B ) we deduce the result. We will state the analog of Proposition XI.3.7 for Ddbl Λ,V (kV ). We recall that the restriction of µhom(ΨU (F ), ΨU (G)) to {τ > 0} is decomposed, by Proposition XI.1.9, −1 −1 −1 −1 as the sum of µhom(qU (F ), qU (G)) and µhom(rU (F ), rU (G)). Hence in an analog of Proposition XI.3.7 we can expect that µhom(F, G) (for F, G ∈ D(kV )) should be replaced by µhom(F ′ , G′ )| q (for F ′ , G′ ∈ Ddbl Λ,V (kV )). We will make this more precise Λ soon and we will use the following lemma. Lemma XI.4.6. — Let G, G′ ∈ Ddbl Λ,V (kV ). Then there exist a uniquely defined sheaf µhomdbl (G, G′ ) in Db (kΛ∩T ∗ V ) and an open subset V ′ of Vγ such that (V ×{0})⊔V ′ is

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open in M × R × R≥0 and µhom(G, G′ )| q

∗

Λq ∩T ∗ V ′

dbl ≃ p−1 (G, G′ ))| Λ (µhom

Λq ∩T ∗ V ′

,

′

where pΛ : Λ ∩ T V → − Λ is the projection. Moreover, for W ⊂ V open, if G| ≃ ΨW (F ), G′ | ≃ ΨW (F ′ ) for some F, F ′ ∈ Dbτ ≥0 (kW ), then Wγ Wγ dbl ′ ′ µhom (G, G )| ≃ µhom(F, F )| . ∗ ∗ Λ∩T W

Λ∩T W

Proof. — By Lemma XI.4.5-(iv-v), any point x ∈ V has a neighborhood W such that G| , G′ | are of the form ΨW (F ), ΨW (F ′ ) for some F, F ′ ∈ Dbτ ≥0 (kW ). Then Wγ Wγ Proposition XI.1.9 implies that the restriction of µhom(G, G′ ) to Λq ∩ T ∗ Wγ is of the −1 −1 −1 form µhom(qW (F ), qW (F ′ )) ≃ qW,d ! qW,π µhom(F, F ′ ). In particular µhom(G, G′ )|

Λq ∩T ∗ Wγ

≃ (pΛ |

Λq ∩T ∗ Wγ

)−1 µhom(F, F ′ ),

which proves the last assertion and shows that µhom(G, G′ )| q

∗

Λq ∩T ∗ Wγ

is constant on

the fibers of the projection Λ ∩ T Wγ → − Λ. Now we choose a family Wi , i ∈ I, of such open subsets W which covers V and is locally S finite (each compact subset ′of V meets finitely many Wi ’s). We set V ′ = is open in M × R × R≥0 and, for i∈I Wi,γ . Then (V × {0}) ⊔ V any x ∈ V , there exists i ∈ I such that ({x} × R>0 ) ∩ V ′ = ({x} × R>0 ) ∩ Wi,γ . It follows that µhom(G, G′ )| q ∗ ′ is constant on the fibers of the projecΛ ∩T V tion pΛ : Λq ∩ T ∗ V ′ → − Λ, which are open intervals. Hence it is the inverse image of some uniquely defined sheaf µhomdbl (G, G′ ) by pΛ (and we have µhomdbl (G, G′ ) = RpΛ ∗ (µhom(G, G′ )| q ∗ ′ )). Λ ∩T V The following result follows easily from the definition of Ddbl Λ,V (kV ) and Lemma XI.4.5-(ii). ∗ Lemma XI.4.7. — For G ∈ Ddbl Λ,V (kV ) there exists a unique open subset ΛG ⊂ Λ∩T V , dbl that we denote SS (G), such that

(i) for any open subset W ⊂ V where a decomposition (XI.4.3) holds we have S ˙ i ) ∩ T ∗ V Ai (with the notations of Definition XI.4.4— ΛG ∩ T ∗ W = i∈I SS(F ˙ remark that SS(Fi ) is ∅ or Λi ). (ii) there exist an open subset V ′ of Vγ and a neighborhood Ω of ΛG in T ∗ V such that (V × {0}) ⊔ V ′ is open in M × R × R≥0 and q

SS(G) ∩ Λq ∩ T ∗ V ′ = ΛG ∩ T ∗ V ′ , SS(G) ∩ Ωq ∩ T ∗ V ′ = ΛqG ∩ T ∗ V ′ . Remark XI.4.8. — The microsupport of G ∈ Ddbl Λ,V (kV ) is in general bigger dbl dbl q r ˙ than (SS (G)) ∪ (SS (G)) ; the last formula in Lemma XI.4.7 says SS(G) dbl dbl q q ˙ coincides with (SS (G)) in some neighborhood of (SS (G)) , but SS(G) is a priori not contained in Λq ∪ Λr . Namely, the microsupport of RΓV Ai ×R>0 (ΨW (Fi )) could be as big as the bound given in the proof of Lemma XI.4.5 (ii).

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We can define the notions of pure or simple doubled sheaf and also mΛ0 (G) for a doubled sheaf G; this will be used in Part XII. Let G ∈ Ddbl Λ,V (kV ). Lemma XI.4.7 dbl defines an open subset SS (G) of Λ. Let Λ0 be an open subset of SSdbl (G). With V ′ as in the lemma we have G| ′ ∈ Db(Λq ∩T ∗ V ′ ) (kV ′ ) (we recall that this means V 0 that SS(G) coincides with Λq0 on some neighborhood of Λq0 ). We say that G is pure (or simple) along Λ0 if G| ′ is pure (or simple) in the usual sense along Λq0 ∩ T ∗ V ′ . V Since G| ′ ∈ Db(Λq ∩T ∗ V ′ ) (kV ′ ), we can also consider mΛq0 ∩T ∗ V ′ (G| ′ ) which is an V V 0 object of µSh(kΛq0 ∩T ∗ V ′ ). Up to shrinking V ′ we can assume that the fibers of the − Λ0 are open intervals. Then the inverse image by pΛ0 projection pΛ0 : Λq0 ∩ T ∗ V ′ → induces an equivalence ∼ (XI.4.12) µSh(kΛ ) −− → µSh(k q ∗ ′ ). Λ0 ∩T V

0

In this way we can identify mΛq0 ∩T ∗ V ′ (G| ′ ) with an object of µSh(kΛ0 ) that we denote V ∗ by mdbl Λ0 (G). We obtain a functor, for any open subset Λ0 of Λ ∩ T V , dbl dbl mdbl (G)} → − µSh(kΛ0 ) Λ0 : {G ∈ DΛ,V (kV ); Λ0 ⊂ SS

(XI.4.13)

and (X.1.6) together with (XI.4.12) give, for G, G′ Λ0 ⊂ SSdbl (G), Λ0 ⊂ SSdbl (G′ ),

∈ Ddbl Λ,V (kV ) such that

dbl ′ 0 0 dbl Hom(mdbl (G, G′ ))). Λ0 (G), mΛ0 (G )) ≃ H (Λ0 ; H (µhom

(XI.4.14)

(Here the condition Λ0 ⊂ SSdbl (G) ensures that G ∈ Db(Λq ) (kV ′ ).) Let us set 0 Vu = Vγ ∩ (M × R × {u}) for u > 0. We have a natural inclusion T ∗ Vu ⊂ T ∗ Vγ . We remark that, if V = V B for some B ⊂ A and u is small enough, we have Λq0 ∩ T ∗ Vu = Λ0 . Then G| belongs to Db(Λ0 ) (kVu ) and we have, by the construction Vu of mdbl Λ0 , mΛ0 (G| ) ≃ mdbl Λ0 (G).

(XI.4.15)

Vu

Now we define a version of the morphism b(F, G) of Definition XI.3.4. Let dbl G, G′ ∈ Ddbl (G), Λ′0 = SSdbl (G′ ) (they are open subsets Λ,V (kV ) and set Λ0 = SS ′ of Λ). We choose an open subset V of Wγ satisfying the conclusions of Lemmas XI.4.6 and XI.4.7 for G, G′ . We denote by πΛq , πΛ the projections from Λq ∩ T ∗ V ′ , Λ ∩ T ∗ V respectively to V ′ , V . The support of µhom(G, G′ )| q ∗ ′ is contained q

Λ ∩T V

q

in Λq ∩ T ∗ V ′ ∩ Λ0 ∩ Λ′0 , hence the support of µhomdbl (G, G′ ) is contained in Λ0 ∩ Λ′0 . Since Λ′0 is open in Λ we have a natural morphism (−) → − RΓΛ′0 (−) and we deduce the following sequence of morphisms RHom (G, G′ )|

V′

≃ RπV ′ ∗ (µhom(G, G′ )| ′

T ∗V ′

→ − RπΛq ∗ (µhom(G, G )|

) )

Λq ∩T ∗ V ′ −1 dbl ≃ (qV RπΛ∗ µhom (G, G′ ))| ′ V → − (qV−1 RπΛ∗ RΓΛ′0 µhomdbl (G, G′ ))| ′ . V

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Let iV : V × {0} − → M × R2 and j : V ′ → − M × R2 be the inclusions. We apply the −1 functor iV Rj∗ to the above sequence and we obtain a version of the morphism b(F, G) of Definition XI.3.4 for Ddbl Λ,V (kV ): (XI.4.16)

′ b′ (G, G′ ) : i−1 − RπΛ∗ RΓΛ′0 µhomdbl (G, G′ ), V Rj∗ RHom (G, G ) →

where Λ′0 = SSdbl (G′ ). The Proposition XI.3.7 generalizes to this setting as follows. ′ ′ Proposition XI.4.9. — Let G, G′ ∈ Ddbl Λ,V (kV ). Then the morphism b (G, G ) in (XI.4.16) is an isomorphism. In particular, taking global sections gives ∼ → H 0 (Λ′0 ; µhomdbl (G, G′ )), lim Hom(G| ′ , G′ | ′ ) −− V V −→ ′ V

′

where V runs over the open subsets of M × R × R>0 such that (V × {0}) ⊔ V ′ is open in M × R × R≥0 and Λ′0 = SSdbl (G′ ) is the open subset of Λ ∩ T ∗ V defined in Lemma XI.4.7. Proof. — Since the statement is local on V we may as well assume that G and G′ are decomposed as in (XI.4.3) and it is enough to consider one summand in their decompositions. By Lemma XI.4.5-(iv-v) we can assume that G = ΨW (RΓV B (F )) and G′ = ΨW (RΓV B′ (F ′ )) for some open subset W ⊂ V , F, F ′ ∈ Db[Λ] (kW ) and B, B ′ ⊂ A. By Lemma XI.4.5-(iii) the restriction to W of the left hand side of (XI.4.16) becomes (XI.4.17)

−1 ′ ′ ′ i−1 W Rj∗ RHom (G, G ) ≃ RΓV B (iW Rj∗ RHom (G, ΨW (F ))).

We have µhomdbl (G, G′ ) ≃ µhom(RΓV B (F ), RΓV B′ (F ′ ))| by the construction Λ ′ ′ ˙ of µhomdbl in Lemma XI.4.6. Let us set Λ′ = SS(F ). We have SSdbl (G′ ) = Λ′ ∩T ∗ V B and the restriction to W of the right hand side of (XI.4.16) becomes RπΛ∗ RΓΛ′ ∩T ∗ V B′ (µhom(RΓV B (F ), RΓV B′ (F ′ ))| ) Λ

≃ RπΛ∗ RΓΛ′ ∩T ∗ V B′ (µhom(RΓV B (F ), F ′ )| ) Λ

≃ RπΛ∗ RΓT ∗ V B′ (µhom(RΓV B (F ), F ′ )| ) Λ

≃ RΓV B′ RπΛ∗ (µhom(RΓV B (F ), F ′ )| ), Λ

where the first isomorphism follows from RΓΛ′ ∩T ∗ V B′ ≃ RΓΛ′ RΓT ∗ V B′ and the fact ′ that RΓU (H) only depends on H | when U is open (here U = T ∗ V B ) and the U ′ ˙ second isomorphism follows from the inclusion supp(µhom(H, F ′ )| ) ⊂ SS(F ) = Λ′ , Λ

′ ˙ whatever H. Since SS(F ) ⊂ {τ > 0} we can apply Proposition XI.3.7 and we obtain that the right hand side of (XI.4.16) is isomorphic to ′ RΓV B′ (i−1 W Rj∗ RHom (ΨW (RΓV B (F )), ΨW (F ))).

Comparing with (XI.4.17) we obtain the result. The last assertion follows by applying the functor H 0 (V ; −) to both sides of (XI.4.16) and using H 0 (V ′ ; RHom (G, G′ )) ≃ Hom(G| ′ , G′ | ′ ). V

V

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We have seen a notion of pure doubled sheaf and we have defined mdbl Λ0 in the paragraph before (XI.4.13). In the same way that we deduced Corollary XI.3.8 from Proposition XI.3.7, we obtain the following result, using Proposition XI.4.9 and (XI.4.14): Corollary XI.4.10. — Let Λ0 ⊂ Λ ∩ T ∗ V be an open subset. Let G, G′ ∈ Ddbl Λ,V (kV ) be dbl dbl ′ such that Λ0 ⊂ SS (G) and SS (G ) = Λ0 . We assume that G and G′ are pure with the same shift. Then (XI.4.18)

dbl ′ lim Hom(G| , G′ | ) ≃ Hom(mdbl Λ0 (G), mΛ0 (G )), V V −→ V

where V runs over the open subsets of Uγ such that (U × {0}) ⊔ V is open in dbl ′ M × R × R≥0 . In particular, if SSdbl (G) = SSdbl (G′ ) = Λ0 and mdbl Λ0 (G) ≃ mΛ0 (G ), ′ then there exists such an open subset V such that G| ≃ G | . V

dbl

V

′

We remark that the hypothesis SS (G ) = Λ0 implies that Λ0 is not any open subset of Λ but of the form described in (i) of Lemma XI.4.7. In Lemma XI.4.13 below we describe the isomorphism (XI.4.18) locally when we have a decomposition of G and G′ as in (XI.4.3). Before that we show that the subsets Λ ∩ T ∗ V B , B ⊂ A, have a local connectedness property near the boundary. For example the following situation is excluded. Assume M = R and Λ is half of the conormal bundle of the cusp {(x, t); x3 = t2 }, say Λ = {(z 2 , z 3 ; − 23 zτ, τ ); z ∈ R, τ > 0} (this example is explained in Lemma VIII.1.1) and set V = ]0, 1[ × ]−1, 1[. Then Λ ∩ T ∗ V has two −1 connected components but Λ ∩ πM ×R (V ) has only one component. In fact V cannot belong to an adapted family for Λ. The next lemma says a bit more. Lemma XI.4.11. — Let Λ and V = {Va ; a ∈ A} be as in Definition XI.4.1. Then, for any B ⊂ A, Λ ∩ T ∗ V B has the following local connectedness property. For any (x, t) ∈ M × R and any smallFenough neighborhood W of (x, t), denoting by Λ∩T ∗ W = F ∗ B = k∈Kj Λkj the decompositions into connected components j∈J Λj and Λj ∩ T V ∗ we have: for any j ∈ J there exists at most one k ∈ Kj such that Λkj ∩ T(x,t) (M × R) is ′ non empty. In other words, there exists a smaller neighborhood W of (x, t) such that the inclusion of Λj ∩ T ∗ V B ∩ T ∗ W ′ in Λj ∩ T ∗ V B factorizes through a connected set.

Remark XI.4.12. — A stronger statement would be that the subsets Λj ∩ T ∗ V B in the lemma are connected (that is, Kj is a singleton), but this would require a good choice of W . The lemma only says that, when restricting to a smaller neighborhood ∗ of T(x,t) (M × R), at most one component of Λj ∩ T ∗ V B survives. Proof. — (i) We set FΛ = R(π˙ M ×R )∗ (kΛ ). Let us prove that the natural mor− RΓV B (FΛ ) is an isomorphism, for any B ⊂ A. By Theorem I.2.13 phism u : (FΛ )V B → and the condition D′ (kV B ) ≃ kV B , it is enough to see that SS(kV B ) and SS(FΛ ) do not meet outside the zero section. This follows from (XI.4.1) and the inclusion ∗ b Λ+ , where Λ+ = Λ ∪ TM SS(FΛ ) ⊂ Λa+ + ×R (M × R), that we prove now. Since this is a local problem, we can assume by Lemma X.2.5 that there exists F such that SS(F ) ⊂ Λ+ and µhom(F, F )| ˙ ∗ ≃ kΛ (see (I.4.5)). Then the inclusion T (M ×R)

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follows from the triangle (XI.3.11), the triangular inequality for the microsupport and the bounds in Lemma XI.3.6 and Theorem I.2.13. ∗ (ii) Let J be as in the lemma and let J ′ be the set of half lines in Λ ∩ T(x,t) (M × R). ′ We choose W small enough so that the obvious map J → − J is injective. Applying H 0 (W ; −) to the isomorphism u of (i) we obtain ∼ H 0 (T˙ ∗ W ; k − → H 0 (T˙ ∗ W ∩ π −1 (V B ); kΛ ). −1 B ) − Λ∩πM ×R (V

)

M ×R

−1 B ˙∗ This says that the connected components of Λ ∩ πM ×R (V ) ∩ T W and −1 B ˙∗ Λ ∩ πM ×R (V ) ∩ T W are in bijection. Now we assume that there exist j ∈ J ′

′

∗ (M × R). Then Λkj and Λkj both and k, k ′ ∈ Kj such that Λkj and Λkj both meet T(x,t) ′ contain the same half line of J . In particular they have a non empty intersection and −1 ′ B ˙∗ must be the same connected component of Λ ∩ πM ×R (V ) ∩ T W , that is, k = k , as required.

The next result says that the local decomposition (XI.4.3) can be written in a canonical way and describe the morphism (XI.4.18) locally when we have such a decomposition. Lemma XI.4.13. — Let V be an open subset of M × R and let (x, t) ∈ V be given. Let ∗ {λi }i∈I be the set of half lines in Λ ∩ T(x,t) (M × R). Let W0 be a neighborhood of (x, t) small enough so that the map I → − π0 (Λ ∩ T ∗ W0 ) is injective; let Λi be the connected ∗ component of Λ ∩ T W0 containing λi . We assume that, for each i ∈ I, there exists a simple sheaf Fi ∈ D[Λi ] (kW0 ). Now let G, G′ ∈ Ddbl Λ,V (kV ) be pure objects with the dbl dbl ′ ′ same shift. We set Λ0 = SS (G), Λ0 = SS (G ). Then we have: (i) There exists an isomorphism, for some smaller neighborhood W of (x, t), M L G| ≃ RΓWi ×R>0 (ΨW (Fi ⊗ (Ei )W )), Wγ i∈I

where Wi = W ∩ π˙ M ×R (Λi ∩ Λ0 ) and Ei ∈ D(k) is given by Ei = (µhomdbl (ΨW (Fi ), G))pi for any pi ∈ Λi ∩ Λ0 (and Ei = 0 if Λi ∩ Λ0 is empty). (ii) We assume that Λ′0 ⊂ Λ0 and we define Wi′ , Ei′ like Wi , Ei in (i), choosing the same pi for G and G′ when Λi ∩ Λ′0 ̸= ∅. For a given u in dbl ′ 0 ′ dbl Hom(mdbl (G, G′ )) Λ′0 (G), mΛ′0 (G )) ≃ H (Λ0 ; µhom µ

we let ui : Ei → − Ei′ , e 7→ upi ◦ e, be the morphism induced by the composition (X.1.2) (we use the notation X.1.4). Then, up to shrinking W , the inverse image of dbl ′ u| ∗ ∈ Hom(mdbl ′ (G )) through (XI.4.18) (replacing V in the T ∗ W ∩Λ′0 (G), mT ∗ W T W L∩Λ0 corollary by W ) is represented by i∈I vi , where L

L

vi : RΓWi ×R>0 (ΨW (Fi ⊗ (Ei )W )) → − RΓWi′ ×R>0 (ΨW (Fi ⊗ (Ei′ )W ))

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CHAPTER XI.4. DOUBLED SHEAVES

is the composition of the morphism RΓWi ×R>0 (−) → − RΓWi′ ×R>0 (−) induced by the L

L

open inclusion Wi′ ⊂ Wi and the morphism ΨW (Fi ⊗ (Ei )W ) → − ΨW (Fi ⊗ (Ei′ )W ) induced by ui . Proof. — (i) We first take W so that we have a decomposition (XI.4.3) L ′ ′ ∗ ′ b G| ≃ i∈I ′ RΓV Ai ×R>0 (ΨW (Fi )) where I = π0 (Λ ∩ T W ) and Fi ∈ D[Λ] (kW ) Wγ ′ (I contains I but could be bigger). Shrinking W we can forget the components in I ′ \ I (maybe Λi ∩ T ∗ W will be no longer connected but we don’t care). Hence we can assume I ′ = I. Up to shrinking W several times, Lemma X.2.2 implies that, for each i ∈ I, L

mΛi (Fi′ ) is isomorphic to mΛi (Fi ⊗ (Ei0 )W ), where Ei0 = (µhom(Fi , Fi′ ))pi , and CorolL

lary XI.3.8 implies ΨW (Fi′ ) ≃ ΨW (Fi ⊗ (Ei0 )W ). We remark that µhom(Fi , Fi′ )|

Ξi

≃ µhomdbl (ΨW (Fi ), ΨW (Fi′ ))|

Ξi

dbl

≃ µhom dbl

∗

(ΨW (Fi ), G)| , Ξi

Ei0

where Ξi = Λi ∩ SS (G) ∩ T W . Hence ≃ Ei . Finally W ∩ V Ai = W ∩ π˙ M ×R (Λi ∩ Λ0 ) by Lemma XI.4.7 (i), proving the formula in (i). (ii) Using the decomposition in (i) for G and G′ , we can assume that L

L

G = RΓWi ×R>0 (ΨW (Fi ⊗ (Ei )W )), G′ = RΓWi′ ×R>0 (ΨW (Fi ⊗ (Ei′ )W )). The hypothesis that G, G′ are pure with the same shifts says that the complexes Ei , Ei′ is a constant are concentrated in the same degree. Hence µhomdbl (G, G′ )| ′ Λ0 ∩Λi

sheaf concentrated in degree 0 with stalks Hom(Ei , Ei′ ). By Lemma XI.4.11 we may consider that Λ′0 ∩ Λi is connected, up to shrinking W . Then u is determined by its germ at pi . By the construction of vi in the lemma, the morphism viµ has the same germ as u at pi . Hence vi represents u. dbl Corollary XI.4.14. — Let G ∈ Ddbl (G) ⊂ T ∗ V (see Λ,V (kV ) and Λ0 = SS 1 2 Lemma XI.4.7). Let Λ0 = Λ0 ⊔ Λ0 be a decomposition of Λ0 into two open and closed subsets. Then there exists an open subset V ′ of Vγ such that (V × {0}) ⊔ V ′ is open dbl ′ in M × R × R≥0 and there exist G1 , G2 ∈ Ddbl (Gi ) = Λi0 ∩ T ∗ V ′ , Λ,V (kV ) such that SS i = 1, 2, and G| ′ ≃ G1 | ′ ⊕ G2 | ′ . V

V

V

Proof. — (i) By Proposition XI.4.9 we have ∼ (XI.4.19) lim Hom(G| ′ , G| ′ ) −− → H 0 (Λ0 ; µhomdbl (G, G)), V V −→ ′ V

′

where V runs over the open subsets of M × R × R>0 such that (V × {0}) ⊔ V ′ is open in M × R × R≥0 . The identity morphism of G induces an element 1G of the left hand side of (XI.4.19). We let 1µG be the corresponding section of µhomdbl (G, G). Since

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CHAPTER XI.4. DOUBLED SHEAVES

Λ0 is split into two open and closed subsets we have M H 0 (Λ0 ; µhomdbl (G, G)) ≃ H 0 (Λi0 ; µhomdbl (G, G)) i=1,2

1µG

and we write = e1 + e2 according to this decomposition. Since Λ10 and Λ20 are disjoint we have e1 e2 = e2 e1 = 0. Hence e1 and e2 are orthogonal idempotents. By (XI.4.19) again we deduce a decomposition 1G = f1 + f2 where f1 and f2 are also orthogonal idempotents. (ii) We can find V ′ as in (XI.4.19) and f1′ , f2′ in Hom(G| ′ , G| ′ ) which represent f1 , V V f2 . Up to shrinking V ′ we can also assume that f1′ , f2′ are orthogonal idempotents (using (XI.4.19) again). By [8, Prop. 3.2] we deduce a corresponding decomposition G| ′ ≃ G1 ⊕ G2 in Db (kV ′ ). V

(iii) We extend G1 , G2 arbitrarily to Vγ . It remains to check that Gi ∈ Ddbl Λ,V (kV ) and SSdbl (Gi ) = Λi0 ∩ T ∗ V ′ . We will prove that we have a local decomposition (XI.4.3) of Gi in a neighborhood W of any given point (x, t) ∈ V . We first choose W (and we use the so that Wγ ⊂ V ′ and a decomposition (XI.4.3) holds for G| Wγ

corresponding notations I, Ai ). For i ∈ I the set Λi ∩ T ∗ V Ai may be non connected, but, by Lemma XI.4.11, if we start with W small enough, at most one component of Λi ∩ T ∗ V Ai can meet T ∗ W ′ for some smaller neighborhood W ′ . Hence Λi ∩ T ∗ V Ai ∩ T ∗ W ′ is contained in either Λ10 or Λ20 . We choose W ′ so that this holds for all i ∈ I. We define G′1 (resp. G′2 ) to be the sum of the summands of G| ′ in (XI.4.3) inWγ

dexed by the j ∈ I so that Λj ∩ T ∗ V Aj ∩ T ∗ W ′ is contained in Λ10 (resp. Λ20 ). Then G′i ∈ Ddbl Λ,V (kV ). This gives another decomposition of G| ′ and corresponding idemWγ

potents f1′′ , f2′′ . Since SS(G′i ) ∩ Λ = T ∗ Wγ′ ∩ Λi0 , the section of µhomdbl (G, G)| ∗ ′ T W ∩Λ associated with fi′′ by (XI.4.19) must be ei | ∗ ′ . Up to restricting W ′ once more, T W ∩Λ we deduce fi′′ = fi′ | ′ and then G′i ≃ Gi | ′ , which proves the result. Wγ

W ×]0,ε[

SOCIÉTÉ MATHÉMATIQUE DE FRANCE 2023

PART XII QUANTIZATION

The main result of this part is that, for any global object F of the Kashiwara˙ Schapira stack µSh(kΛ ), there exists F ∈ Db (kM ×R ) with SS(F ) = Λ which represents F , when Λ is the conification of a compact exact Lagrangian submanifold of T ∗ M (see (XII.0.1) below). This recovers a result of Viterbo in [52] who proves the existence of such a sheaf using Floer theory (the proof of [52] was sketched in [51] in 2011). We first consider a compact Legendrian submanifold of J 1 M , or equivalently, a closed conic Lagrangian submanifold Λ of Tτ∗>0 (M × R) such that Λ/R>0 is compact. We apply the procedure sketched in the introduction of Part XI and we prove that any object F ∈ µSh(kΛ ), or F ∈ µSh/[1] (kΛ ), is represented by some F ∈ Db (kM ×R ), ˙ or F ∈ D/[1] (kM ×R ), such that SS(F ) = Λ ⊔ Tε (Λ) for ε > 0 small, where Tε is the translation along the factor R. Then, assuming that Λ/R>0 has no Reeb chord we ′ ˙ prove that there exists another representative F ′ such that SS(F ) = Λ. We recall that ∗ ∗ “having no Reeb chord” means that ρM : Tτ >0 (M × R) → − T M , (x, t; ξ, τ ) 7→ (x; ξ/τ ), e = ρM (Λ) is then a compact exact induces an injection Λ/R>0 ,→ T ∗ M . The image Λ ∗ Lagrangian submanifold of T M in the sense that αM | e is exact. The link between Λ e is given by Λ and Λ (XII.0.1)

e t = −f (x; ξ/τ )}, Λ = {(x, t; ξ, τ ); τ > 0, (x; ξ/τ ) ∈ Λ,

where f is a primitive of αM | e . Λ

Now we introduce some notations. We first remark that, by Theorem II.1.1, we can as well move Λ by any contact isotopy of J 1 M and assume from the beginning that it satisfies some genericity hypotheses. Hence we can assume that the map Λ/R>0 → − M has finite fibers and, by Lemma XI.4.3, we can assume that there exists an adapted family V = {Va ; a ∈ A} in the sense of Definition F XI.4.1. We let Λb , b ∈ Ba , be the family of components of Λ ∩ T ∗ Va . We set B = a∈A Ba . We have introduced the subcategory Ddbl Λ,V (kM ×R ) of D(kM ×R×R>0 ) in Definidbl tion XI.4.4. For G ∈ DΛ,V (kM ×R ) we have defined SSdbl (G) in Lemma XI.4.7; it is

240

S an open subset of Λ of the form b∈B ′ Λb for some B ′ ⊂ B. For an open subset Λ0 of Λ we also have a functor (see (XI.4.13)) dbl dbl mdbl (G)} → − µSh(kΛ0 ). Λ0 : {G ∈ DΛ,V (kV ); Λ0 ⊂ SS

The same definitions make sense for the orbit category. We define the subcategory Ddbl /[1],Λ,V (kM ×R ) of D/[1] (kM ×R×R>0 ) as in Definition XI.4.4, replacing everywhere D(kX ) by D/[1] (kX ). We can also define SSdbl (G) for G ∈ Ddbl /[1],Λ,V (kM ×R ) and a functor (XII.0.2)

dbl dbl mdbl (G)} → − µSh/[1] (kΛ0 ). /[1],Λ0 : {G ∈ D/[1],Λ,V (kV ); Λ0 ⊂ SS

dbl In Theorems XII.1.1 and XII.2.2 we see that the functors mdbl Λ0 and m/[1],Λ0 are dbl essentially surjective. For a given G ∈ DΛ,V (kM ×R ) the microsupport of G| , M ×R×{ε} for ε > 0 small enough, is made of two copies of Λ. In Corollary XII.3.2 we see that, if we have no Reeb chords, we can translate one copy of Λ vertically using a Hamiltonian isotopy and obtain an object of Db[Λ] (kM ×R ) from a given one in Ddbl Λ,V (kM ×R ). In §XII.4 b we see a relation between objects of D[Λ] (kM ×R ), their restrictions to M ×{t0 }, t0 ≫ 0, and their microlocalizations; in particular we see that F 7→ F | induces a fully M ×{t0 }

faithful functor from sheaves on M ×R with microsupport Λ and vanishing on M ×{t}, t ≪ 0, and locally constant sheaves on M .

ASTÉRISQUE 440

CHAPTER XII.1 QUANTIZATION FOR THE DOUBLED LEGENDRIAN

Let Λ be a closed conic Lagrangian submanifold of Tτ∗>0 (M ×R) such that Λ/R>0 is compact. By Lemma XI.4.3, we can assume that there exists an adapted family V = {Va ; a ∈ A} for Λ which is stable by intersection. We can also assume that Λ ∩ T ∗ Va has finitely many connected components, for each a ∈ A. We let Λb , F b ∈ Ba , be the family of components of Λ ∩ T ∗ Va . We set B = a∈A Ba and we let σ : B → − A be the obvious map. Hence Λb is a component of Λ ∩ T ∗ Vσ(b) . When we use Lemma XI.4.3 we can also assume that the family {Λb }b∈B refines any given family. Since we know that simple sheaves along Λ locally exist (Lemma X.2.5), we can assume that, for each b ∈ B, there exist a neighborhood Vb′ of Vσ(b) , a contractible component Λ′b of Λ ∩ T ∗ Vb′ and Fb ∈ Db (kVb′ ) such that Λb is a component ˙ b ) = Λ′ and Fb is simple. of Λ′b ∩ T ∗ Vσ(b) , SS(F b S Theorem XII.1.1. — In the above setting let B ′ be a subset of B and set Λ0 = b∈B ′ Λb , S Λ′0 = b∈B ′ Λ′b . Then, for any pure object F ∈ µSh(kΛ′0 ) there exists F ∈ Ddbl Λ,V (kM ×R ) dbl such that SSdbl (F ) = Λ0 and mdbl (F ) ≃ F , where m is defined in (XI.4.13). | Λ0 Λ0 Λ0

Proof. — (i) We proceed by induction on |B ′ |. Let b ∈ B. Since Λ′b is contractible, L

the objects of µSh(kΛ′b ) are of the form mΛ′b (EVb′ ⊗ Fb ) for some E ∈ Db (k). So we write F |

Λ′b

L

= mΛ′b (EVb′ ⊗ Fb ) and we set L

G = RΓ(Vσ(b) ×R>0 )∩(Vb′ )γ (ΨVb′ (EVb′ ⊗Fb )), extended by zero outside (Vb′ )γ (the formula defines G on (Vb′ )γ with a support contained in (Vb′ )γ ∩ (Vσ(b) × R>0 )). Let us prove that G belongs to Ddbl Λ,V (kM ×R ). We check Definition XI.4.4 around a point (x, t) ∈ M × R: If (x, t) ̸∈ Vσ(b) , we choose W such that W ∩ Vσ(b) = ∅ and we have G| = 0 Wγ

so (XI.4.3) is trivial. If (x, t) ∈ Vb′ , we choose W = Vb′ and the defining formula for G satisfies (XI.4.3). We remark that the family I in (XI.4.3) consists of one element, say I = {i0 }, and Ai0 = {σ(b)}.

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CHAPTER XII.1. QUANTIZATION FOR THE DOUBLED LEGENDRIAN

We have SSdbl (G) = Λ′b ∩ T ∗ Vσ(b) and Λb is a connected component of SSdbl (G). dbl By Corollary XI.4.14 there exists Gb , G′ ∈ Ddbl (Gb ) = Λb , Λ,V (kM ×R ) such that SS dbl dbl ′ ′ SS (G ) = SS (G) \ Λb and G| ≃ Gb | ⊕ G | for some open subset O of O O O M × R × R>0 satisfying (XII.1.1)

(M × R × {0}) ∪ O is open in M × R × R≥0 .

By construction we have mdbl Λb (Gb ) ≃ F |

Λσ(b)

, which proves the case B ′ = {b}.

(ii) Now we write B ′ = B ′′ ∪ {b} and assume that the result holds for B ′′ . We S set Λ1 = c∈B ′′ Λc and Λ1b = Λ1 ∩ Λb . By the induction hypothesis and by (i) there dbl exist G1 , Gb ∈ Ddbl (G1 ) = Λ1 , SSdbl (Gb ) = Λb and isomorphisms Λ,V (kM ×R ) with SS ∼ ∼ φ1 : mdbl − → F | , φb : mdbl − → F | . We have assumed that the adapted Λ1 (G1 ) − Λb (Gb ) − Λ1 Λb family {Va }a∈A is stable by intersection. Hence G′1 = RΓ(Vσ(b) ×R>0 ) (G1 ) belongs to Ddbl Λ,V (kM ×R ) (this can be checked directly on the local decomposition(XI.4.3)). We have SSdbl (G′1 ) = Λ1 ∩ T ∗ Vσ(b) and Λ1b is open and closed in SSdbl (G′1 ). As in (i) we use Corollary XI.4.14 to find G1b ∈ Ddbl Λ,V (kM ×R ) such dbl ′ that SS (G1b ) = Λ1b and G1b | ′ is the summand of G1 | ′ corresponding to Λ1b , for O O some open subset O′ satisfying (XII.1.1). − G1b | ′ given by the composition of the We have a natural morphism g1 : G1 | ′ → O O ′ natural morphism G1 → − G1 and the projection to the summand G′1 | ′ → − G1b | ′ . By O O ∼ construction we have an isomorphism φ1b : mdbl (G ) − − → F compatible with φ1 1b | Λ1b Λ1b

dbl dbl in the sense that φ1 | = φ1b ◦ mdbl Λ1b (g1 ) (we use mΛ1b (G1 ) = mΛ1 (G1 )|Λ1b ). Λ1b dbl ∼ We define g˜b = φ−1 − → mdbl Λ1b (G1b ). Since F is pure, we can 1b ◦ (φb |Λ1b ) : mΛ1b (Gb ) − ′ − G1b | ′ which apply Corollary XI.4.10 and, up to shrinking O , we obtain gb : Gb | ′ → O O b represents g˜b . Finally we define G ∈ D (kM ×R×R>0 ) by the distinguished triangle on O′

(XII.1.2)

G→ − G1 |

O′

⊕ Gb |

O′ ′

(g1 ,−gb )

−−−−−→ G1b |

+1

O′

−−→

and by extending G by zero outside O . We prove in (iii) that G represents F over Λ0 = Λ1 ∪ Λb . (iii) We first have to check that G belongs to Ddbl Λ,V (kM ×R ), which means that it can be decomposed as in (XI.4.3), around any given point (x, t) ∈ M × R. We use Lemma XI.4.13 to describe G∗ , for ∗ = 1, ∗ = b or ∗ = 1b, F over a small enough neighborhood W of (x, t). We have a partition Λ ∩ T ∗ W = i∈I Ξi into connected components and a simple sheaf Fi ∈ D[Ξi ] (kW ) for each i ∈ I. We choose pi ∈ Ξi and set Ei = (Hom (mΞi (Fi ), F ))pi (this does not depend on pi ). Over SSdbl (G∗ ) we have dbl µhomdbl (ΨW (Fi ), G∗ ) ≃ Hom (mdbl Ξi (ΨW (Fi )), mΞi (G∗ ))

≃ Hom (mdbl Ξi (ΨW (Fi )), F )

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CHAPTER XII.1. QUANTIZATION FOR THE DOUBLED LEGENDRIAN

≃ Hom (mΞi (Fi ), F ). L

Hence, setting for short Hi = ΨW (Fi ⊗ (Ei )W ) and maybe shrinking W , we have by Lemma XI.4.13 (i) M ∗ = 1, b, 1b, G∗ | ≃ RΓWi∗ ×R>0 (Hi ), Wγ i∈I

where Wi∗ = W ∩ π˙ M ×R (Ξi ∩ SSdbl (G∗ )) = W ∩ π˙ M ×R (Ξi ∩ Λ∗ ). By Lemma XI.4.13 (ii) the morphisms g1 , gb in (XII.1.2) are obtained by compos− RΓWi1b ×R>0 (−), ∗ = 1, b, induced by the open ing morphisms αi∗ : RΓWi∗ ×R>0 (−) → inclusions Wi1b ⊂ Wi∗ , and morphisms βi∗ : Hi → − Hi , induced by mdbl Ξi (g∗ ). For the i’s ∗ such that Ξi ∩ Λ1b ̸= ∅, mdbl (g ) is an isomorphism, and so is β (for the other i’s we ∗ i Ξi 1b ∗ 1b have Wi = ∅). Setting βi = id when Wi = ∅ we obtain the commutative square L (g1 ,−gb ) L i∈I (RΓWi1 ×R>0 (Hi ) ⊕ RΓWib ×R>0 (Hi )) i∈I RΓWi1b ×R>0 (Hi ) ≀ β

L ⊕ RΓWib ×R>0 (Hi )) α i∈I RΓWi1b ×R>0 (Hi ),   L L βi1 0 1 b where α = . Since β is an isomorphism, we i∈I (αi , −αi ) and β = i∈I 0 βb L

i∈I (RΓWi1 ×R>0 (Hi )

i

obtain that (XII.1.2) is isomorphic to the sum of the Mayer-Vietoris distinguished triangles +1

− RΓWi1b ×R>0 (Hi ) −−→ . RΓ(Wi1 ∪Wib )×R>0 (Hi ) → − RΓWi1 ×R>0 (Hi ) ⊕ RΓWib ×R>0 (Hi ) → L It follows that G| ≃ i∈I RΓ(Wi1 ∪Wib )×R>0 (Hi ) which is a decomposition Wγ

dbl like (XI.4.3). This proves that G ∈ Ddbl (G) = Λ0 . Λ,V (kM ×R ) and also that SS dbl By construction we also have mΛ0 (G) ≃ F | since g1 , gb are defined to represent Λ0 the morphisms gluing F | and F | into F | . Λ1

Λb

Λ0

SOCIÉTÉ MATHÉMATIQUE DE FRANCE 2023

CHAPTER XII.2 THE TRIANGULATED ORBIT CATEGORY CASE

We checked in Sections IX.1 and IX.2 that the results we used in the category Db (kM ) have analogs in the category D/[1] (kM ). In particular we have already defined a Kashiwara-Schapira stack µSh/[1] (kΛ ) in this situation. However the proof of Theorem XII.1.1 does not work because we used the fact that some µhom sheaf was concentrated in degree 0 (we used Corollary XI.4.10), which makes no sense in the orbit category. We prove the result for the triangulated orbit category by gluing two sheaves obtained on two open subsets by Theorem XII.1.1. For this we first remark in Lemma XII.2.1 below that we can decompose Λ in two open subsets with vanishing Maslov classes. Lemma XII.2.1. — Let X be a manifold and let c ∈ H 1 (X; ZX ). Then there exists a map f : X → − S 1 of class C ∞ such that c = f ∗ (δ), where δ ∈ H 1 (S 1 ; ZS 1 ) is the canonical class. In particular, for any covering S1 = I1 ∪ I2 by two open intervals, the restrictions c| −1 ∈ H 1 (f −1 (Ii ); Zf −1 (Ii ) ) vanish, for i = 1, 2. f (Ii ) Proof. — Since H 1 (X; ZX ) → − H 1 (X; RX ) is injective, it is enough to prove the result for the image of c in H 1 (X; RX ), which we represent by a 1-form α. Let r : X ′ → − X be ′ ∗ the universal covering of X and let g : X → − R be a primitive of r (α). Then, for any x1 , x2 ∈ X ′ such that r(x1 ) = r(x2 ) we have g(x1 ) − g(x2 ) = ⟨c, γ⟩, where γ is the loop at g(x1 ) determined by x1 , x2 . Hence g(x1 ) − g(x2 ) is an integer and g descends to a map f : X → − S 1 which satisfies the conclusion of the lemma. For the next result the ring is k = Z/2Z. Theorem XII.2.2. — For any global object F of µSh/[1] (kΛ ) there exists an object F ∈ Ddbl /[1],Λ,V (kM ×R ), for some finite family V of open subsets of M × R which is adapted to Λ, such that mdbl /[1],Λ (F ) ≃ F . Proof. — (i) By Proposition X.4.4 µSh/[1] (kΛ ) has a unique simple object, F0 , and is of the type F ≃ F0 ⊗εkΛ L for some L ∈ Loc(kΛ ). We let µ1 (Λ) ∈ H 1 (Λ; ZΛ ) be the sheaf obstruction class of Λ (which coincides with the Maslov class). We apply Lemma XII.2.1 to obtain f : Λ → − S 1 such that µ1 (Λ) is F

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CHAPTER XII.2. THE TRIANGULATED ORBIT CATEGORY CASE

the pull-back of the fundamental class of S 1 . We choose a covering S1 = I+ ∪ I− by two open intervals and set Λ± = f −1 (I± ). Then µ1 (Λ)| = 0 and the categories Λ±

µSh(kΛ± ) have simple objects, say F0,± . Their images in µSh/[1] (kΛ± ) are F0 | Λ± because µSh/[1] (kΛ± ) has a unique simple object. The intersection I+ ∩I− is the union of two intervals, say Ia , Ib . We set Λa = f −1 (Ia ), Λb = f −1 (Ib ) and F0,a = F0,+ | , Λa F0,b = F0,+ | . We remark that simple objects in µSh(kΞ ), for Ξ open in Λ, with Λb coefficients k = Z/2Z, are unique up to shift and up to a unique isomorphism. Hence we have the following canonical isomorphisms ∼ ∼ φa+ = id : F0,a −− → F0,+ | , φb+ = id : F0,b −− → F0,+ | , Λa Λb a b ∼ ∼ φ− : F0,a −−→ F0,− | [da ], φ− : F0,b −−→ F0,− | [db ], Λa Λb where da , db ∈ Z are locally constant functions on Λa , Λb . We set F± = F0,± ⊗ L, ∗ Fa = F0,a ⊗ L, Fb = F0,b ⊗ L and Φ∗ ± = φ± ⊗ idL , for ∗ = a, b. (ii) Up to shrinking Λ0 and Λ1 we can find a finite family V = {Va ; a ∈ A} of open subsets of M × R which is adapted to Λ such that Λ+ and Λ− are of the form ΛB for some B ⊂ A (see Lemma XI.4.3). We can also assume that the adapted family is stable by intersection. Hence Λa and Λb are also of the form ΛB . By Theorem XII.1.1 there exist F• ∈ Ddbl Λ,V (kM ×R ), for • = +, −, a, b, such dbl dbl that SS (F• ) = Λ• and m/[1],Λ• (F• ) ≃ F• . Moreover, by Proposition XI.4.9 we have morphisms, for ∗ = a, b, Ψ∗+ : F∗ | → − F+ | and Ψ∗− : F∗ | → − F− | [d∗ ] representing V V V V ∗ Φ± , where V is some open subset of M × R × R>0 satisfying (XII.1.1). In D/[1] (kV ) we have F− ≃ F− [d∗ ] and we can define F ∈ D/[1] (kV ) by the distinguished triangle b Ψa + Ψ+ b Ψa − Ψ−

! +1

− F −−→ . Fa ⊕ Fb −−−−−−−→ F+ ⊕ F− → We extend F by zero outside V . We can then check as in part (iii) of the proof of dbl Theorem XII.1.1 that F ∈ Ddbl /[1],Λ,V (kM ×R ) and that m/[1],Λ (F ) ≃ F . Remark XII.2.3. — By Proposition X.4.4 µSh/[1] (kΛ ) has a simple object. Hence Theorem XII.2.2 gives the existence of a simple object in Ddbl /[1],Λ,V (kM ×R ), say F0 . In the case where Λ has no Reeb chord, we will see that Loc(kΛ ) is equivalent to Loc(kM ) and ε all objects of Ddbl /[1],Λ,V (kM ×R ) are of the form F0 ⊗kM ×R L for some L ∈ Loc(kM ×R ). (However in the course of the proof of the previous theorem this is unknown, so we cannot deduce a representative for F from one for F0 .)

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CHAPTER XII.3 TRANSLATION OF THE MICROSUPPORT

Now we assume that the Legendrian submanifold Λ/R>0 of J 1 M has no Reeb chord, that is, the map Tτ∗>0 (M × R) → − T ∗ M , (x, t; ξ, τ ) 7→ (x; ξ/τ ), induces an ∗ embedding Λ/R>0 ,→ T M (see (XII.0.1)). For u ∈ R we define the translation Tu : M × R → − M × R, (x, t) 7→ (x, t + u). We denote by Tu′ : T ∗ (M × R) → − T ∗ (M × R), (x, t; ξ, τ ) 7→ (x, t + u; ξ, τ ), the induced map on the cotangent bundle. We also introduce some notations, for Λ ⊂ Tτ∗>0 (M × R): Λu = Λ ∪ Tu′ (Λ)

(XII.3.1)

⊂

Tτ∗>0 (M × R),

Λ+ = qd qπ−1 (Λ) ⊔ rd rπ−1 (Λ)

⊂

for u > 0,

Tτ∗>0 (M × R × R>0 ),

where q, r : M ×R×R>0 → − M ×R are given by q(x, t, u) = (x, t), r(x, t, u) = (x, t−u). We remark that Λ+ is non-characteristic for the inclusions iu : M × R × {u} → − M × R × R>0 , + u > 0, and that Λu = (iu )d ((iu )−1 π (Λ )).

Lemma XII.3.1. — There exists ϕ : T˙ ∗ (M × R) × R>0 → − T˙ ∗ (M × R), a homogeneous Hamiltonian isotopy, such that ϕ1 = id and, using the notations (I.5.5) and (XII.3.1), we have Γϕ ◦a Λ1 = Λ+ . In particular ϕu (Λ1 ) = Λu , for all u > 0. Proof. — (i) We set I = R>0 . Since the map (x, t; ξ, τ ) 7→ (x; ξ/τ ) induces an injection Λ/R>0 ,→ T ∗ M , the sets Λ and Tu′ (Λ) are disjoint for all u > 0. Considering all u > 0 at once we define the following closed subsets of T˙ ∗ (M × R) × I: G Λ0 = Λ × R>0 , Λ1 = (Tu′ (Λ) × {u}). u>0 0

1

Then Λ and Λ are disjoint and the projections Λi /R>0 → − I are proper for i = 0, 1. Hence we can find a conic neighborhood Ω of Λ1 in T˙ ∗ (M ×R)×I such that Ω∩Λ0 = ∅ and the projection Ω/R>0 → − I is proper, that is, Ω ∩ (T˙ ∗ (M × R) × {u}) is compact for all u > 0. (ii) We choose a C ∞ -function h : T˙ ∗ (M × R) × I → − R such that, (a) hu := h| ˙ ∗

T (M ×R)×{u}

is homogeneous of degree 1, for all u ∈ I,

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(b) h vanishes outside Ω, (c) there exists a neighborhood Ω′ of Λ1 such that h(x, t; ξ, τ ) = −τ , for all (x, t; ξ, τ ) ∈ Ω′ . By (a), (b) and the compactness of (Ω ∩ (T˙ ∗ (M × R) × {u}))/R>0 , the Hamiltonian flow of h, say ϕ, is defined on I (the initial time here is t0 = 1). Then ϕu is the identity map outside Ω for all u ∈ I and ϕu (x, t; ξ, τ ) = (x, t + u − 1; ξ, τ ), for all ((x, t; ξ, τ ), u) ∈ Ω′ . Since Λ0 ⊂ (T˙ ∗ (M × R) \ Ω) and Λ1 ⊂ Ω′ , the lemma follows. Corollary XII.3.2. — Let Λ be a closed conic Lagrangian submanifold of Tτ∗>0 (M × R) coming from a compact exact submanifold of T ∗ M as in (XII.0.1). Then, for any pure object F ∈ µSh(kΛ ) there exists F ∈ Db[Λ] (kM ×R ) such that mΛ (F ) ≃ F and F| ≃ 0 for t ≪ 0. In the same way, for any F ∈ µSh/[1] (kΛ ) there exists M ×{t}

F ∈ D/[1],[Λ] (kM ×R ) such that m/[1],Λ (F ) ≃ F and F |

M ×{t}

≃ 0 for t ≪ 0.

Proof. — (i) The proofs are the same for F ∈ µSh(kΛ ) or F ∈ µSh/[1] (kΛ ). We first consider F ∈ µSh(kΛ ) and then emphasize some points for the other case. By Theorem XII.1.1 there exists F0 ∈ Ddbl Λ,V (kM ×R ), for some finite family V of open subsets of M × R which is adapted to Λ, such that SSdbl (F0 ) = Λ and dbl mdbl Λ (F0 ) ≃ F . By the definition of DΛ,V (kM ×R ) any point x ∈ M × R has a neighborhood Wx such that a decomposition (XI.4.3) holds for F0 | . Since Wx,γ

SSdbl (F0 ) = Λ, we have in fact F0 | ≃ ΨWx (Fx ) for some Fx ∈ Db[Λ] (kWx ) (in Wx,γ S Lemma XI.4.13-(i) all Wi coincide with Wx ). Setting V = x∈M ×R Wx,γ we then have ˙ 0 ) ∩ T ∗ V ⊂ Λ+ ∩ T ∗ V by Lemmas XI.1.5 supp(F0 ) ∩ V ⊂ (γ ⋆ π˙ M ×R (Λ)) ∩ V and SS(F and XI.1.7. In particular we can find a compact neighborhood C of π˙ M ×R (Λ) such that, for u ≤ 1, the support of F0 | is contained in C × ]0, u[. For u > 0 V ∩(M ×R×]0,u[)

small enough we have (C × ]0, u[) ⊂ V and we obtain F1 ∈ Db[Λ+ ] (kM ×R×]0,u[ ) by ˙ 1| extending F0 | by zero. For any v ∈ ]0, u[ we have SS(F ) = Λv V ∩(M ×R×]0,u[)

M ×R×{v}

and it follows from (XI.4.15) that mΛ (F1 |

M ×R×{v}

(ii) We set F2 = F1 |

M ×R×{ 21 u}

)≃F.

˙ 2 ) = Λ 1 u . We consider the isotopy ϕ of . Then SS(F 2

Lemma XII.3.1 and let Kϕ ∈ Dlb (k(M ×R)2 ×R>0 ) be the sheaf associated with ϕ by Theorem II.1.1. For given 0 < u1 < u2 the restriction of ϕ to T˙ ∗ (M × R × ]u1 , u2 [) has compact support (after quotienting by the R>0 action in the fibers) and Kϕ | coincides with k∆M ×R ×]u1 ,u2 [ outside a compact set, hence is 2 (M ×R) ×]u1 ,u2 [

−1 ◦ F2 belongs bounded. In particular, for any v > 0, the sheaf F2,v = Kϕ,v ◦ Kϕ, 1 2u −1 b ˙ to D (kM ×R ) and satisfies SS(F2,v ) = (ϕv ◦ ϕ 1 u )(Λ 12 u ) = Λv . Moreover, for given 2 0 < u1 < u2 , ϕu is the identity map on some neighborhood Ω of Λ, for any u ∈ ]u1 , u2 [, and it follows that Kϕ,u ◦ − induces the identity functor on D(kM ×R ; Ω). Hence mΛ (F2,v ) ≃ mΛ (F2 ) ≃ F .

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CHAPTER XII.3. TRANSLATION OF THE MICROSUPPORT

249

(iii) Since Λ/R>0 is compact we can choose A > 0 such that Λ ⊂ Tτ∗>0 (M × ]−A, A[). For v ≥ 2A + 1 we have Tv′ (Λ) ⊂ Tτ∗>0 (M × ]A + 1, +∞[). ˙ 3 ) = Λ. We choose a diffeomorWe set F3 = F2,2A+1 | . Then SS(F M ×]−∞,A+1[ ∼ phism f : ]−∞, A + 1[ −− → R such that f is the identity map on ]−∞, A[. Then ˙ F = R(idM × f )∗ (F3 ) satisfies SS(F ) = Λ and mΛ (F ) ≃ F . ˙ 2,v | ) is empty for all v > 0, We set UA = M × ]−∞, −A[. We remark that SS(F UA

hence F2,v is locally constant on UA . If F2,v | does not vanish, then supp(F2,v ) must UA contain UA . We have seen that supp(F1 ) is contained in C × ]0, u[ for some compact set C. Hence F2 and all F2,v have compact support. It follows that F2,v | vanishes UA and so does F | . This concludes the proof for the case F ∈ µSh(kΛ ). UA

(iv) Now we assume F ∈ µSh/[1] (kΛ ). We use Theorem XII.2.2 instead of XII.1.1 in (i). We didn’t state Theorem II.1.1 for the orbit category but there is nothing really new to say: as soon as the sheaf Kϕ belongs to Db (k(M ×R)2 ×]u1 ,u2 [ ) (rather than the locally bounded category) we can take its image in D/[1] (k(M ×R)2 ×]u1 ,u2 [ ). The relation −1 Kϕ,u ◦ Kϕ,u ≃ k∆M ×R still holds in the orbit category and we deduce the same equiv∼ alence Kϕ,u ◦ − : D/[1],[A] (kM ×R ) −− → D/[1],[ϕu (A)] (kM ×R ), for any A ⊂ T˙ ∗ (M × R). Now the step (ii) works the same way. The step (iii) also, using Proposition IX.2.10 to see that F2,v is locally constant on UA . Remark XII.3.3. — In the proof of Theorem XII.1.1 the compactness of Λ/R>0 was used since we glued doubled sheaves defined on open subsets of a covering of π˙ M ×R (Λ) by a finite induction. We could probably use a countable covering (with some care to take the limit) and the compactness of Λ/R>0 was not essential. However in Corollary XII.3.2 this compactness is necessary to ensure that the sheaf given by Theorem XII.1.1 can be modified to have the required microsupport over M × R × ]0, u[ for some u > 0. Remark XII.3.4. — Using Remark XII.2.3 and Corollary XII.3.2 we see that, if Λ comes from a compact exact submanifold of T ∗ M , then there exists a simple object F ∈ D/[1],[Λ] (kM ×R ) such that F | ≃ 0 for t ≪ 0. M ×{t}

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CHAPTER XII.4 RESTRICTION AT INFINITY

As in §XII.3 we assume that Λ/R>0 has no Reeb chord. Since Λ/R>0 is compact, we can choose A > 0 such that Λ ⊂ Tτ∗>0 (M × ]−A, A[). Then, for any F ∈ Dlb [Λ] (kM ×R ), the restrictions F | and F | have locally constant cohomology M ×]−∞,−A[ M ×]A,+∞[ sheaves. lb Definition XII.4.1. — For F ∈ Dlb [Λ] (kM ×R ) we let F− , F+ ∈ D (kM ) be the restrictions at infinity F− = F | , F+ = F | , for any t ∈ [A, +∞[. Then F− , F+ M ×{−t}

M ×{t}

are indeed independent of t ∈ [A, +∞[ and have locally constant cohomology sheaves. lb We let Dlb [Λ],+ (kM ×R ) be the full subcategory of D[Λ] (kM ×R ) consisting of the F such that F− ≃ 0. Replacing Dlb [Λ] (−) by D/[1],[Λ] (−) we also define F− , F+ ∈ D/[1] (kM ) for F ∈ D/[1],[Λ] (kM ×R ) and a similar category D/[1],[Λ],+ (kM ×R ). For F ∈ Dlb [Λ],+ (kM ×R ) we have by definition (XII.4.1)

F|

M ×]A,+∞[

≃ F+ ⊠ k]A,+∞[ ,

F|

M ×]−∞,−A[

≃0

for A ≫ 0.

Lemma XII.4.2. — Let F ∈ Dlb (kM ×R ) (or F ∈ D/[1] (kM ×R )). We assume that there exists A > 0 such that supp(F ) ⊂ M × [−A, A]. We also assume either ˙ ˙ SS(F ) ⊂ Tτ∗>0 (M × R) or SS(F ) ⊂ Tτ∗ 0 so that (XII.4.1) holds. We set G = F ⊗ kM ×]−∞,A+1[ . Then supp(G) ⊂ [−A, A + 1] and SS(G) ⊂ Tτ∗>0 (M × R) by Theorem I.2.13. By Lemma XII.4.2 we obtain RpM ∗ G ≃ 0. +1 By (XII.4.1) we have the distinguished triangle G → − F → − F+ ⊠ k[A+1,+∞[ −−→ and we deduce RpM ∗ F ≃ RpM ∗ (F+ ⊠ k[A+1,+∞[ ) ≃ F+ , RpM ! F ≃ RpM ! (F+ ⊠ k[A+1,+∞[ ) ≃ 0. We recall the maps q, r, Tu (see around (XII.3.1)). By Lemma XI.1.4 we have a morphism q −1 F → − r−1 F , for any F ∈ Dτ ≥0 (kM ×R ). Restricting to M × R × {u} we obtain a morphism F → − Tu∗ F . Lemma XII.4.4. — For all F, F ′ in Dlb [Λ],+ (kM ×R ) (or in D/[1],[Λ],+ (kM ×R )) and ′ ′ any u ≥ 0, the morphism F → − Tu∗ F induces the isomorphism ∼ RHom(F, F ′ ) −− → RHom(F, Tu∗ F ′ ). Moreover, for any u > 0, we have RHom(Tu∗ F, F ′ ) ≃ 0. Proof. — (i) We extend q, r to M × R × R (with the same formulas) and we define p2 : M × R × R → − R, (x, t, u) 7→ u. We introduce G = Rp2 ∗ RHom (q −1 F, r! F ′ ). For u ∈ R, letting iu : M × R × {u} → − M × R × R be the inclusion we have q ◦ iu = id, r ◦ iu = T−u , and hence, by Proposition I.1.1-(h-j), RΓ{u} (G) ≃ RΓ(M × R; i!u RHom (q −1 F, r! F ′ )) ! ≃ RΓ(M × R; RHom (F, T−u F ′ ))

≃ RHom(F, Tu∗ F ′ ), Using the microsupport bounds of §I.2 or §IX.2 we obtain SS(G) ⊂ {(u; υ); ∃(x, t; ξ, τ ) ∈ SS(F ), ∃(x′ , t′ ; ξ ′ , τ ′ ) ∈ SS(F ′ ), x = x′ , t − u = t′ , (−ξ, −τ, 0) + (ξ ′ , τ ′ , −τ ′ ) = (0, 0, υ)}. ′ ˙ ˙ Using SS(F ) = SS(F ) = Λ and the fact that Λ has no Reeb chord (hence ˙ ⊂ {(0; υ); υ < 0}. Λ ∩ Tu′ (Λ) = ∅ for u ̸= 0), we deduce SS(G)

(ii) Using Corollary I.2.16 (or Corollary IX.2.11 for D/[1] (kM ×R )) we deduce RΓ[a,b[ (R; G) ≃ 0 for any a < b and RΓ]a′ ,b′ ] (R; G) ≃ 0 for any 0 ≤ a′ < b′ . In ∼ ∼ particular RΓ{u} (G) −− → RΓ[u,u′ ] (R; G) ← −− RΓ{u′ } (G) for 0 ≤ u ≤ u′ and the first isomorphism follows. By the same argument RHom(Tu∗ F, F ′ ) ≃ RHom(F, T−u ∗ F ′ ) is independent of u > 0. If we choose u > 2A, where A satisfies (XII.4.1) both for F and F ′ , then ′ T−u ∗ F ′ coincides with p−1 − M. M F+ over supp(F ), where pM is the projection M × R → Hence ′ ′ RHom(F, T−u ∗ F ′ ) ≃ RHom(F, p−1 M F+ ) ≃ RHom(RpM ! F, F+ )[−1]

vanishes by Lemma XII.4.3.

ASTÉRISQUE 440

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CHAPTER XII.4. RESTRICTION AT INFINITY

′ lb Theorem XII.4.5. — Let F, F ′ ∈ Dlb [Λ],+ (kM ×R ). We let F+ , F+ ∈ D (kM ) be their restrictions to M × {t}, t ≫ 0, as in Definition XII.4.1. Then ∼ (XII.4.2) RHom(F, F ′ ) −− → RHom(F+ , F ′ ). +

Dlb [Λ],+ (kM ×R )

lb

In particular the functor → − D (kM ) given by F 7→ F+ is fully faithful and we have: F ≃ F ′ if and only if F+ ≃ F+′ . lb The same statement holds with Dlb [Λ],+ (kM ×R ), D (kM ) replaced by D/[1],[Λ],+ (kM ×R ), ε D/[1] (kM ) (using RHom defined before IX.1.22). Proof. — Let pM : M × R → − M be the projection. Let us choose A > 0 so that (XII.4.1) holds for F and F ′ and let u > 2A. Then RHom(F, F ′ ) ≃ RHom(F, Tu∗ F ′ ) ′ ≃ RHom(p−1 M F+ , Tu∗ F )

≃ RHom(F+ , RpM ∗ Tu∗ F ′ ) ≃ RHom(F+ , F+′ ), where the first isomorphism follows from Lemma XII.4.4, the second one from supp(Tu∗ F ′ ) ⊂ M ×]A, +∞[ and the last one from Lemma XII.4.3. With Theorem XII.4.5 we can recover a classical result [35] of Lalonde and Sikorav: ˜ ⊂ T ∗ M be a compact exact Lagrangian submanifold. Then Corollary XII.4.6. — Let Λ ˜ the map Λ → − M is onto. In particular M is compact. ˜ as in (XII.0.1). By ReProof. — We let Λ ⊂ T˙ ∗ (M × R) be the conification of Λ mark XII.3.4 there exists a simple object F ∈ D/[1],[Λ],+ (kM ×R ). In particular F ̸= 0. By Theorem XII.4.5 (for the case of the orbit category) we have RHom(F+ , F+ ) ̸= 0, ˜→ hence F+ ̸= 0. Let us assume that Λ − M is not onto. Then there exists an open sub∗ ˙ set U of M such that Λ∩ T (U ×R) = ∅. Hence F | is locally constant, F | ≃0 U ×R U ×{t} for t ≪ 0 and F | ̸≃ 0 for t ≫ 0, which is a contradiction. U ×{t}

Theorem XII.4.7. — Let F, F ′ ∈ Dlb [Λ],+ (kM ×R ). Then we have an isomorphism (XII.4.3)

∼ RHom(F, F ′ ) −− → RΓ(Λ; µhom(F, F ′ )).

Its composition with (XII.4.2) gives a canonical isomorphism (XII.4.4)

RHom(F+ , F+′ ) ≃ RΓ(Λ; µhom(F, F ′ )).

As in Theorem XII.4.5 the same statement holds with Dlb [Λ],+ (kM ×R ) replaced by D/[1],[Λ],+ (kM ×R ). Proof. — (i) For an interval I of R>0 we set NI = M × R × I. Proposition XI.3.7 and the fact that N]0,ε[ is open give the isomorphisms, for any i ∈ Z, H i RΓ(Λ; µhom(F, F ′ )) ≃ lim H i RHom(ΨM ×R (F )| , ΨM ×R (F ′ )| )) ≃ lim H i Aε , N]0,ε[ N]0,ε[ −→ −→ ε>0

ε>0

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where we set Aε = RHom((ΨM ×R (F ))N]0,ε[ , ΨM ×R (F ′ )). It is enough to prove that Aε ≃ RHom(F, F ′ ) for all ε > 0. +1 The triangle (XI.1.10), ΨM ×R (F ′ ) → − q −1 (F ′ ) → − r−1 (F ′ ) −−→, yields another dis+1 tinguished triangle Aε → − Bε → − Cε −−→ with Bε = RHom((ΨM ×R (F ))N]0,ε[ , q −1 (F ′ )) ≃ RHom(Rq! ((ΨM ×R (F ))N]0,ε[ ), F ′ )[−1], Cε = RHom((ΨM ×R (F ))N]0,ε[ , r−1 (F ′ )) ≃ RHom(Rr! ((ΨM ×R (F ))N]0,ε[ ), F ′ )[−1]. We check in (ii) below that Bε ≃ RHom(F, F ′ ) and in (iii) that Cε ≃ 0, which proves the theorem. (ii) We compute Bε . The triangle (XI.1.10) again gives the distinguished triangle (XII.4.5)

+1

Rq! ((ΨM ×R (F ))N]0,ε[ ) → − Rq! ((q −1 F )N]0,ε[ ) → − Rq! ((r−1 F )N]0,ε[ ) −−→ .

In the following computations we consider q, r as maps defined on M × R2 (with the same formulas—see after (XII.3.1)). The projection formula (Proposition I.1.1˙ −1 F ) ⊂ {−υ = τ > 0} and (g)) gives Rq! ((q −1 F )N]0,ε[ ) ≃ F [−1]. We have SS(r −1 ˙ SS(kN]0,ε] ) ⊂ {υ ≤ 0}. Hence SS((r F )N]0,ε] ) ⊂ {υ < 0} and Lemma XII.4.2 (used 2 with q : M × R → − M × R instead of pM : M × R → − M ) gives Rq! ((r−1 F )N]0,ε] ) ≃ 0. +1

Using the triangle kN{ε} [−1] → − kN]0,ε[ → − kN]0,ε] −−→ we deduce Rq! ((r−1 F )N]0,ε[ ) ≃ Rq! ((r−1 F )N{ε} )[−1] ≃ Tε∗ F [−1]. Lemma XII.4.4 gives RHom(Tε∗ F, F ′ ) ≃ 0 for any ε > 0. Applying RHom(−, F ′ ) to the triangle (XII.4.5) we deduce Bε ≃ RHom(F, F ′ ), as claimed. (iii) Now we prove Rr! ((ΨM ×R (F ))N]0,ε[ ) ≃ 0, which implies Cε ≃ 0. As in (ii) we have the triangle (XII.4.6)

α

+1

Rr! ((ΨM ×R (F ))N]0,ε[ ) → − Rr! ((q −1 F )N]0,ε[ ) − → Rr! ((r−1 F )N]0,ε[ ) −−→

and an isomorphism Rr! ((r−1 F )N]0,ε[ ) ≃ F [−1]. The microsupport bound −1 ˙ SS((q F )N[0,ε[ ) ⊂ {τ > 0, υ ≥ 0} and Lemma XII.4.2 again give Rr! ((q −1 F )N[0,ε[ ) ≃ 0. We deduce Rr! ((q −1 F )N]0,ε[ ) ≃ Rq! ((r−1 F )N{0} )[−1] ≃ F [−1]. Moreover the morphism α in (XII.4.6) corresponds to idF through these isomorphisms and we deduce Rr! ((ΨM ×R (F ))N]0,ε[ ) ≃ 0. Remark XII.4.8. — We have recalled in (X.1.2) that µhom admits a composition morµ phism (denoted by ◦ in Notation X.1.4) compatible with the composition morphism for RHom . In particular the isomorphism (XII.4.3) is compatible with the compoµ sition morphisms ◦ and ◦. Since (XII.4.2) is clearly compatible with ◦, we deduce µ that (XII.4.4) also is compatible with ◦ and ◦.

ASTÉRISQUE 440

PART XIII EXACT LAGRANGIAN SUBMANIFOLDS IN COTANGENT BUNDLES In this part M is a connected manifold and Λ0 ⊂ T ∗ M is a compact exact Lagrangian submanifold. We use the sheaves constructed in Corollary XII.3.2 and Theorems XII.4.5, XII.4.7 to recover some results on the topology of Λ0 , namely that the projection Λ0 → − M is a homotopy equivalence and that the first and second Maslov classes of Λ0 vanish. The fact that Λ and M have the same homology was proved by Fukaya, Seidel, Smith in [15] and also by Nadler in [37] using the Fukaya category of the cotangent bundle and assuming that the first Maslov class vanishes. The fact that the projection Λ0 → − M is a homotopy equivalence was proved by Abouzaid in [2], also assuming the vanishing of the Maslov class. Then Kragh in [32] proved that this vanishing holds, also using [33] and [1]. Abouzaid and Kragh gave more precise results on the topology of Λ. In [4] they proved that the map Λ0 → − M is a simple homotopy equivalence and in [3] they proved a vanishing result for the higher Maslov classes (very roughly the images of the Maslov classes by the map BO → − BH vanish, where H is the group of homotopy equivalences of the sphere—this gives the vanishing of obstructions for the existence of a sheaf in spectra—see [24, 23]). and define as In the following we choose f : Λ0 → − R such that df = αM | Λ0 in (XII.0.1) Λ = {(x, t; ξ, τ ); τ > 0, (x; ξ/τ ) ∈ Λ0 , t = −f (x; ξ/τ )}. We have Λ/R>0 = Λ0 and we work with Λ instead of Λ0 .

CHAPTER XIII.1 FUNDAMENTAL GROUPS

We let πΛ : Λ → − M be the projection to the base and we denote by π1 (πΛ ) : π1 (Λ) → − π1 (M ) the induced morphism of fundamental groups. Proposition XIII.1.1. — The morphism π1 (πΛ ) : π1 (Λ) → − π1 (M ) is injective. Proof. — (i) We set k = Z/2Z and G = π1 (Λ). We let ρ : G → − GL(k[G]) be the regular representation of G. This means that k[G] is the vector space with basis {eg }g∈G and the action of G is given by g · eh = egh , for all g, h ∈ G. We let Lρ be the local system on Λ with stalks k[G] corresponding to this representation ρ. (ii) We recall some results of §X.4. The Hom sheaf in µSh/[1] (kΛ ) is induced by µhomε through the equivalence hΛ : OL(kΛ ) ≃ Loc(kΛ ) of Lemma X.4.3. By Proposition X.4.4 there exists a simple object F0 ∈ µSh/[1] (kΛ ) and Hom (F0 , ·) gives an equivalence ∼ µSh/[1] (kΛ ) −− → Loc(kΛ ). We let Fρ ∈ µSh/[1] (kΛ ) be the object associated with Lρ by this equivalence. By Corollary XII.3.2, there exist F0 , Fρ ∈ D/[1],[Λ],+ (kM ×R ) such that m/[1],Λ (F0 ) ≃ F0 and m/[1],Λ (Fρ ) ≃ Fρ . Moreover Lρ

≃ Hom (F0 , Fρ ) ≃ hΛ (µhomε (F0 , Fρ )| ). Λ

We define L0 , L1 ∈ D/[1] (kM ) by L0 = F0 | and L1 = Fρ | for t ≫ 0. M ×{t} M ×{t} ε We let p : M × R → − M be the projection and we set F = F0 ⊗kM ×R p−1 L1 and F ′ = Fρ ⊗εkM ×R p−1 L0 . Taking the tensor product with a locally constant sheaf (like L0 , L1 ) does not increase the microsupport and we still have F, F ′ ∈ D/[1],[Λ],+ (kM ×R ). Then F | ≃ L0 ⊗εkM L1 ≃ F ′ | for t ≫ 0 and Theorem XII.4.5 (for the orbit M ×{t} M ×{t} category) implies (XIII.1.1)

F0 ⊗εkM ×R p−1 L1 ≃ Fρ ⊗εkM ×R p−1 L0 .

−1 −1 (iii) Applying m/[1],Λ to (XIII.1.1) we find F0 ⊗εkΛ πΛ L1 ≃ Fρ ⊗εkΛ πΛ L0 in µSh/[1] (kΛ ). Using the equivalences Hom (F0 , ·) and hΛ recalled in (ii) we obtain −1 ′ −1 ′ πΛ L 1 ≃ L ρ ⊗ πΛ L0 , where L′i = hM (Li ), for i = 0, 1 (see Lemma X.4.3—L′i is the locally constant sheaf on M associated with the presheaf U 7→ HomD/[1] (kU ) (kU , Li )).

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(iv) We let ρ′0 and ρ′1 be the representations of π1 (M ) corresponding to the local systems L′0 and L′1 . They induce representations of G = π1 (Λ), say ρ′′0 and ρ′′1 , through the morphism π1 (πΛ ). Then the result of (iii) gives the isomorphism of representations of G, ρ′′1 ≃ ρ ⊗ ρ′′0 . We restrict these representations to the subgroup K = ker(π1 (πΛ )) of G. Then ρ′′0 | and ρ′′1 | are trivial representations and we deduce that ρ| also is K K K trivial. Since ρ is a faithful representation of G, this gives K = {1}, as required. We will see later (see Theorem XIII.5.1 and Proposition XIII.5.2) that the sheaf F0 introduced in part (ii) of the proof of Proposition XIII.1.1 satisfies F0 | ≃ kM M ×{t}

for t ≫ 0 (in general it is of rank one but here k = Z/2Z so it is constant). If we already knew this, (XIII.1.1) would give F0 ⊗εkM ×R p−1 L1 ≃ Fρ and we would have −1 ′ directly πΛ L1 ≃ Lρ , simplifying the end of the proof. ′ Let r : M → − M be a covering. The derivative of r induces a covering r′ : T ∗ M ′ → − T ∗ M . We let Λ′0 be a connected ′−1 ′ component of r (Λ). Then Λ0 → − Λ is a covering and π1 (Λ′0 ) is a subgroup of π1 (Λ). We have the commutative diagram π1 (Λ′0 )

π1 (Λ)

(XIII.1.2)

π1 (πΛ ) ′

π1 (M )

π1 (M ),

where π1 (πΛ ) is injective by Proposition XIII.1.1. This implies that the morphism π1 (Λ′0 ) → − π1 (M ′ ) is injective. In particular, if M ′ is the universal cover of M , ′ then π1 (Λ0 ) vanishes, that is, Λ′0 is the universal cover of Λ. We let mΛ : π1 (Λ) → − Z be the group morphism induced by the Maslov sh 1 class µ1 (Λ) ∈ H (Λ; ZΛ ) introduced in §X.3. We remark that mΛ determines µsh 1 (Λ). Corollary XIII.1.2. — There exists a covering map r : M ′ → − M and a closed conic connected Lagrangian submanifold Λ′ ⊂ T˙ ∗ (M ′ × R) such that the deriva∼ tive of r and the projection Λ′ → − M ′ induce isomorphisms Λ′ −− → Λ and ′ ∼ ′ ∼ π1 (Λ) ←−− π1 (Λ ) −−→ π1 (M ): Λ′

∼

M′

r

Λ

(XIII.1.3) M. ′ sh ∼ Moreover the isomorphism Λ −−→ Λ identifies µsh 1 (Λ ) and µ1 (Λ). ′

By Corollary XII.4.6 M and its covering M ′ are compact. ˜ → Proof. — As noticed around the diagram (XIII.1.2), if r˜ : M − M is the universal ′ cover of M and Λ0 a connected component of the pull-back of Λ by r˜, then Λ′0 is the ˜ (via the inclusion universal cover of Λ. We can see that the action of π1 (Λ) on M ∗ ˜ ′ π1 (Λ) ⊂ π1 (M )) induces an action on T M which preserves Λ0 . The result follows by ˜ /π1 (Λ) and Λ′ = Λ′ /π1 (Λ). setting M ′ = M 0

ASTÉRISQUE 440

CHAPTER XIII.2 VANISHING OF THE MASLOV CLASS

By Corollary X.6.3 the vanishing of µsh 1 (Λ) implies the vanishing of the usual Maslov class µ1 (Λ). We recall that mΛ : π1 (Λ) → − Z is the group morphism induced by µsh 1 (Λ) sh and that mΛ determines µsh (Λ). In this section we prove the vanishing of µ (Λ) as 1 1 ′ follows. Assuming µsh (Λ) = ̸ 0 we consider the cyclic cover M of M associated with 1 ′ ′ µsh 1 (Λ). We let ψn be the action of Z on M and we let Λ be the pull-back of Λ. We construct G ∈ D[Λ′ ] (kM ′ ×R ) which is quasi-periodic in the sense ψn−1 (G) ≃ G[−n], for each n ∈ Z. We then check that G| ′ is bounded for t ≫ 0, obtaining a M ×{t} ′ contradiction. Since Λ is not compact, we cannot apply Corollary XII.3.2 immediately and we will use the Z-action to construct G. We first give the statement at the level of stacks. In this section we take k = Z/2Z. By Corollary XIII.1.2, up to replacing M by some covering (and without changing Λ), ∗ ∼ ∼ : H 1 (M ; ZM ) −− → H 1 (Λ; ZΛ ). we can assume that π1 (Λ) −− → π1 (M ). Hence we have πΛ 1 sh ∗ ∗ f (δ), where By Lemma XII.2.1 there exists a map f : M → − S such that µ1 (Λ) = πΛ 1 1 ′ ′ δ ∈ H (S ; ZS 1 ) is the canonical class. We set M = M ×S 1 R and Λ = Λ ×S 1 R: Λ′ (XIII.2.1)

M′

r′

Λ

f′

R

r πΛ

M

f

S1

′ ′∗ sh ′ and we have µsh 1 (Λ ) = r (µ1 (Λ)) = 0. By construction M comes with a Z-action, ′ ′ denoted ψn : M → − M for n ∈ Z, which translates by n in the fibers of r. We abusively denote by ψn the diffeomorphisms induced by ψn on M ′ × R, M ′ × R × R>0 or Λ′ .

Lemma XIII.2.1. — There exists F ∈ µSh(kΛ′ ) which is simple and satisfies ψn−1 (F ) ≃ F [−n], for each n ∈ Z. If Λ′ is connected (that is, if µsh 1 (Λ) ̸= 0), Proposition X.2.4 then implies that all objects F ′ ∈ µSh(kΛ′ ) satisfy ψn−1 (F ′ ) ≃ F ′ [−n]. Proof. — In §X.3 we have defined µsh 1 (Λ) by a Čech cocycle as follows. First we choose a covering {Λi }i∈I of Λ by suitable small open subsets. We choose Fi ∈ Db(Λi ) (kM ×R ),

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for each i ∈ I, which is simple along Λi . We have seen that there exist isomor∼ phisms mΛi (Fi )| −− → mΛj (Fj )| [dij ], for some integers dij , and that the cochain Λij

Λij

{dij }i,j∈I is a cocyle whose cohomology class, µsh 1 (Λ), is independent of choices. ∗ ∗ sh The relation µsh 1 (Λ) = πΛ f (δ) gives a representative of µ1 (Λ), as follows. We 1 cover S by three arcs, say A0 , A1 , A2 , and represent δ by the cocycle {δij }i,j=0,1,2 , δ01 = δ12 = 0, δ20 = 1 (and δij = −δij ). Taking the Λi ’s small enough, we can assume that there exists σ : I → − {0, 1, 2} such that f (πΛ (Λi )) ⊂ Aσ(i) for each i ∈ I. Then ′ d′ij := δσ(i),σ(j) defines a cocycle representing µsh 1 (Λ). Hence {dij }i,j∈I and {dij }i,j∈I differ by a coboundary and, shifting the Fi ’s by this coboundary, we obtain ∼ (XIII.2.2) mΛi (Fi )| −− → mΛj (Fj )| [d′ij ]. Λij Λij F n We write the pull-back of Ai to R as n∈Z Ai in such a way that An+1 = 1 + Ani i 0 0 0 ∗ and A1 meets A0 and A2 . We numerate the pull-backs of the Λi ’s to T (M ′ × R) accordingly and obtain a covering {Λni }i∈I,n∈Z of Λ′ . The pull-back of Fi yields Fin ∈ Db(Λn ) (kM ′ ×R ), n ∈ Z (actually Fin = Fim but we see Fin and Fim in differi ent categories), and we set Fin = mΛni (Fin )[n]. Then the relation (XIII.2.2) gives Fin | n m ≃ Fjm | n m . Since k = Z/2Z and the Fin ’s are simple, the compatibility Λi ∩Λj

Λi ∩Λj

conditions on the triple intersections are trivial. Hence the Fin ’s glue together in an object F which satisfies ψn−1 (F ) ≃ F [−n], for each n ∈ Z. Theorem XIII.2.2. — We have µ1 (Λ) = 0. Proof. — Let F ∈ µSh(kΛ′ ) be given by Lemma XIII.2.1. We first build in (i) a doubled sheaf F on M ′ × R which represents F . Then in (ii) we deduce from F a usual sheaf, as in the proof of Corollary XII.3.2, and in (iii) we prove that µsh 1 (Λ) vanishes. (i) As in §XII.1 we choose an adapted family V = {Va ; a ∈ A} for Λ which is stable by intersection. We can assume fF (Va ) ̸= S 1 , for each a ∈ A, and hence (r ×idR )−1 (Va ) ′ ′ ∼ decomposes as a disjoint union n∈Z Va,n with Va,n −− → Va , for each n. In this way ′ the pull-back of V to M × R gives an adapted family V ′ = {Vb′ ; b ∈ B} for Λ′ which is stable by intersection and by the Z-action. S We set B0 = {b ∈ B; Vb′ ∩ f ′−1 ([0, 1]) ̸= ∅} and U0 = b∈B0 Vb′ , Λ′0 = Λ′ ∩ T ∗ U0 , ′ Un = ψn−1 (U0 ), Λ′n = ψn−1 (Λ′0 ). By Theorem XII.1.1 there exists F0 ∈ Ddbl Λ′ ,V ′ (kM ×R ) dbl −1 ′ dbl such that SS (F0 ) = Λ0 and mΛ′ (F0 ) ≃ F | ′ . By Z-invariance F1 := ψ1 F0 belongs Λ0

0

dbl −1 ′ to Ddbl (F1 ) = Λ′1 , and mdbl Λ′ (F1 ) ≃ (ψ1 F )| Λ′ ,V ′ (kM ×R ), SS

Λ′1

1

≃ F [−1]| ′ . Λ1

+ We set Un+ = Un × R>0 , U01 = U0+ ∩ U1+ and Λ′01 = Λ′0 ∩ Λ′1 . Since V ′ is ′ stable by intersection, RΓU + F0 and RΓU + F1 belong to Ddbl Λ′ ,V ′ (kM ×R ) and we have 01

01

dbl SSdbl (RΓU + F0 ) = SSdbl (RΓU + F0 ) = Λ′01 and mdbl Λ′ (RΓU + F0 ) ≃ mΛ′ (RΓU + F1 )[1] ≃ F | 01

01

01

01

01

01

Λ′01

.

By Corollary XI.4.10 there exists an open subset V of M ′ × R × R>0 such that (M ′ × R × {0}) ⊔ V is open in M ′ × R × R≥0 and an isomorphism ∼ u : (RΓU + F1 [1])| −−→ (RΓU + F0 )| . 01

ASTÉRISQUE 440

V

01

V

CHAPTER XIII.2. VANISHING OF THE MASLOV CLASS

261

Now we can glue F0 and F1 and define F 01 ∈ D(kM ′ ×R×R>0 ) by the triangle u′

+1

F 01 → − RΓV ∩U + F0 ⊕ RΓV ∩U + F1 [1]) −→ RΓV ∩U + F0 −−→, where u′ = (id, −u). 0 1 01 Then F 01 | ≃ Fi [i]| for i = 0, 1 and hence F 01 belongs (Ui ×R>0 )∩V

(Ui ×R>0 )∩V

′ ′ to Ddbl Λ′ ,V ′ (kM ×R ) and represents F |Λ′ ∪Λ′ . More generally, assuming that the Vb ’s 0 1 are small enough so that U0 ∩ U2 = ∅, we can glue all translates of F0 at once and we define F by the triangle M M v +1 F → − ψn−1 (RΓV ∩U + F0 )[n] − → ψn−1 (RΓV ∩U + F0 )[n] −−→, 0

n∈Z

01

n∈Z

−1 where the restriction of v to the summand ψn−1 (RΓV ∩U + F0 ) is ψn−1 (id) ⊕ ψn−1 (−u). 0 dbl Then F belongs to DΛ′ ,V′ (kM ′ ×R ) (but is only locally bounded in cohomological degrees) and represents F . Moreover, we have by construction ψn−1 F ≃ F [−n], for all n ∈ Z.

(ii) As in (i) of the proof of Corollary XII.3.2 we can see that there exists u > 0 such that, truncating F near (γ ⋆ π˙ M ′ ×R (Λ′ )), we obtain G ∈ Dlb [Λ′+ ] (kM ′ ×R×]0,u[ ), ′ ′ −1 ′+ −1 representing F , where Λ = qd qπ (Λ ) ⊔ rd rπ (Λ ) as in (XII.3.1) (here Λ′ is not compact but we can first consider the restriction of F to a neighborhood of U0 and then get the result by Z-equivariance). We still have ψn−1 G ≃ G[−n], for all n ∈ Z. ′ As in the proof of Corollary XII.3.2 we deduce G′ ∈ Dlb [Λ′ ] (kM ×R ), representing F , −1 ′ ′ and such that ψn G ≃ G [−n], for all n ∈ Z. (To define an isotopy ϕ which translates Λ′ as in Lemma XII.3.1, we first apply the lemma to Λ and pull back the isotopy to T˙ ∗ (M ′ × R).) for t ≫ 0 as in (iii) We have G′ | ′ ≃ 0 for t ≪ 0 and we define G′+ = G′ | ′ M ×{t} M ×{t} ′ ′ ∼ Definition XII.4.1. Theorem XII.4.5 gives RHom(G , G ) −−→ RHom(G′+ , G′+ ). Since G′ ̸≃ 0 it follows that G′+ ̸≃ 0. We also know that G′+ has locally constant cohomology sheaves. ′ ′ Let us assume that µsh 1 (Λ) ̸= 0. Then M is connected and the stalk (G+ )x is ′ ∗ ′ independent of x ∈ M . By Lemma I.4.7 dim(H (G+ )x ) is finite. In particular G′+ is bounded in degrees. On the other hand ψn−1 G′+ ≃ G′+ [−n], for all n ∈ Z, and we have a contradiction.

SOCIÉTÉ MATHÉMATIQUE DE FRANCE 2023

CHAPTER XIII.3 RESTRICTION AT INFINITY

We recall the notation Db[Λ],+ (kM ×R ) of Definition XII.4.1 and F+ = F |

M ×{t0 }

for t0 ≫ 0, for F ∈ Db[Λ],+ (kM ×R ). Proposition XIII.3.1. — We assume k = Z or k is a finite field. Let F ∈ Db[Λ],+ (kM ×R ). We assume that F is simple along Λ. Then F+ is concentrated in one degree, say i, and H i F+ is a local system with stalks isomorphic to k. Proof. — (i) We first assume that k is a finite field. Let us prove that F+ is concentrated in one degree. Let a ≤ b be respectively the minimal and maximal integers i such that H i F+ ̸≃ 0. By Lemma I.4.7 the local systems H i F+ are of finite rank. Since k is finite we can find a finite cover r : M ′ → − M such that r−1 (H i F+ ) is a ′ −1 constant sheaf, for i = a, b. We set F = (r × idR ) F and Λ′ = d(r × idR )−1 (Λ). Then r−1 (H i F+ ) ≃ H i F+′ , F ′ is simple along Λ′ and we have µhom(F ′ , F ′ ) ≃ kΛ′ . Since Λ′ /R>0 is compact, Theorem XII.4.7 gives (XIII.3.1)

RHom(F+′ , F+′ ) ≃ RΓ(Λ′ ; kΛ′ ).

On the other hand the complex G = RHom (F+′ , F+′ ) is concentrated in degrees greater than a − b and H a−b G ≃ Hom (H a F+′ , H b F+′ ) is a non zero constant sheaf. Hence H a−b RHom(F+′ , F+′ ) is non zero. By (XIII.3.1) we deduce that H a−b RΓ(Λ′ ; kΛ′ ) also is non zero, which implies a − b ≥ 0. Hence a = b and F+ is concentrated in a single degree. (ii) Now we prove that H a F+ is of rank one, that is, H a F+′ ≃ kM ′ . There exists d ≥ 1 such that H a F+′ ≃ kdM ′ . The isomorphism (XIII.3.1) gives in degree 0: (XIII.3.2)

Hom(kd , kd ) ≃ H 0 (Λ′ ; kΛ′ ).

By Remark XII.4.8 this isomorphism is compatible with the algebra structures of both terms. Let I be the set of connected L components of Λ′ . We obtain |I| = d2 . 0 ′ 0 The natural decomposition H (Λ ; kΛ′ ) ≃ (Λ′ ; k ′ ) gives an expression of i∈I H P i Λi the unit as a sum of orthogonal idempotents, 1 = i∈I ei , where ei is the projection ei : H 0 (Λ′ ; kΛ′ ) → − H 0 (Λ′i ; kΛ′i ),

i ∈ I.

d d We let mP The relai ∈ Hom(k , k ) = Matd×d (k) be the image of ei by (XIII.3.2). P tion 1 = i∈I ei gives a decomposition of the identity matrix Id = i∈I mi as a sum

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CHAPTER XIII.3. RESTRICTION AT INFINITY

of |I| non-zero orthogonal projections, that is, m2i = mi and mi mj = 0, for i ̸= j. We deduce that |I| ≤ d, that is, d2 ≤ d. Hence d = 1, as claimed. (iii) Now we assume that k = Z. By Lemma I.4.7 the stalks of F+ are of finite rank over Z. We recall that any object of Db (Z) is the sum of its cohomology. Hence, Lb for z = (x, t), t ≫ 0, we can write (F+ )z = i=a Mi [−i], where the Mi ’s are abelian groups of finite rank. We first prove that the Mi ’s are free. If this is not the case, there exist i ∈ Z and a L

prime p such that Mi has p-torsion. We set G = F ⊗Z Z/pZ. Then G is simple along L

Λ and (G+ )z ≃ (F+ )z ⊗Z Z/pZ. We have seen that (G+ )z is isomorphic to Z/pZ[j] for L

L

some shift j. On the other hand H −1 (Mi ⊗Z Z/pZ) and H 0 (Mi ⊗Z Z/pZ) are both non zero, since Mi has p-torsion. This gives a contradiction and proves that, for each i, we have Mi ≃ Zdi for some di . L We set G = F ⊗Z Z/2Z. We have again (G+ )z ≃ (F+ )z ⊗Z Z/2Z ≃ Z/2Z[j]. Hence di = 0 for all i ̸= j and dj = 1, as claimed. Corollary XIII.3.2. — We assume that k = Z or k is a finite field and that µSh(kΛ ) has at least one global simple object. Then the projection Λ → − M induces an isomorphism ∼ RΓ(M ; kM ) −− → RΓ(Λ; kΛ ). Proof. — We choose a simple object F ∈ µSh(kΛ ). By Corollary XII.3.2 there exists F ∈ Db[Λ],+ (kM ×R ) such that mΛ (F ) ≃ F . By Proposition XIII.3.1 F+ is concentrated in one degree, say i, and H i F+ is a local system with stalks isomorphic to k. Hence RHom (F+ , F+ ) ≃ kM and RHom(F+ , F+ ) ≃ RΓ(M ; kM ). Since F is simple we also have µhom(F, F )| ≃ kΛ . By Theorem XII.4.7 we deduce an isomorphism Λ RΓ(M ; kM ) ≃ RΓ(Λ; kΛ ).

(XIII.3.3)

By construction (XIII.3.3) is given by taking the global sections in the bottom morphism of the commutative diagram: a

kM ×R

≀ c

b

RHom (F, F )

Rπ˙ ∗ (kΛ )

∼

Rπ∗ µhom(F, F )

Rπ˙ ∗ (µhom(F, F )| ), Λ

where π = πM ×R , b and c map the sections 1 to the identity morphisms. When taking global sections, b and c induce isomorphisms and a induces the natural morphism RΓ(M ; kM ) → − RΓ(Λ; kΛ ) given by the projection of Λ to the base M . The bottom horizontal arrow induces (XIII.3.3). This shows that (XIII.3.3) is indeed induced by the projection to the base. Remark XIII.3.3. — By Theorem XIII.2.2 the first Maslov class of Λ vanishes. Hence, when k = Z/2Z the stack µSh(kΛ ) has a global simple object and Corollary XIII.3.2 applies: the projection Λ → − M induces an isomorphism ∼ RΓ(M ; Z/2ZM ) −− → RΓ(Λ; Z/2ZΛ ).

ASTÉRISQUE 440

CHAPTER XIII.4 VANISHING OF THE SECOND OBSTRUCTION CLASS

2 × We have seen the class µsh 2 (Λ) ∈ H (Λ; k ) in §X.3. By Corollary X.6.3, if k = Z, sh the vanishing of µ2 (Λ) implies the vanishing of the usual obstruction class µgf 2 (Λ). 2 Here we prove that µsh (Λ) ∈ H (Λ; Z/2Z ) vanishes. For this we will use CorolΛ 2 lary XII.3.2 in the framework of twisted sheaves. Let c ∈ H 2 (M ; Z/2Z) be given and let cˇ = {cijk }, i, j, k ∈, be a Čech cocycle representing c with respect to a finite covering {Ui }i∈I of M . We view Z/2Z as the multiplicative group {±1} and cijk = ±1, for all i, j, k.

Definition XIII.4.1. — A cˇ-twisted sheaf F on M is the data of sheaves Fi ∈ Mod(kUi ) ∼ satisfying the condition −− → Fi | and isomorphisms φij : Fj | Uij

Uij

φij ◦ φjk = cijk φik . ˇ ). We The cˇ-twisted sheaves form an abelian category that we denote by Mod(kcM b cˇ denote by D (kM ) its derived category. cˇ| The prestack U 7→ Mod(kU U ) is a stack which is locally equivalent to the stack of sheaves. The usual operations on sheaves extend to twisted sheaves. In particular ˇ ˇ if cˇ, dˇ are Čech cocycles on M and F ∈ Db (kcM ), F ′ ∈ Db (kdM ), we have a tensor L

ˇ

ˇ

ˇ+d d−ˇ c product F ⊗ F ′ ∈ D(kcM ) and a homomorphism sheaf RHom (F, F ′ ) ∈ D(kM ). If f: M → − N is a morphism of manifolds and dˇ is a Čech cocycle on N with values ∗ˇ ˇ in {±1}, we have inverse images f −1 , f ! : Db (kdN ) → − Db (kfM d ) and direct images ∗ˇ ˇ Rf∗ , Rf! : Db (kfM d ) → − Db (kdN ) with the usual adjunction properties. The notion of microsupport also generalizes to the twisted case (since this is a local notion and twisted sheaves are locally equivalent to sheaves) with the same behavior with respect to the sheaves operations. ˇ We can define a Kashiwara-Schapira stack µSh(kcΛ ) and formulate a version of cˇ ˇ Corollary XII.3.2 in this framework: for F ∈ µSh(kΛ ) there exists F ∈ Db (kcM ×R ) cˇ ˙ such that SS(F ) = Λ, F | ≃ 0 for t ≪ 0 and mΛ (F ) ≃ F . M ×{t}

2 Proposition XIII.4.2. — The class µgf 2 (Λ) ∈ H (Λ; Z/2ZΛ ) is zero.

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CHAPTER XIII.4. VANISHING OF THE SECOND OBSTRUCTION CLASS

Proof. — (i) By Corollary XIII.3.2 and Remark XIII.3.3 we have an isomorphism ∼ H 2 (M ; Z/2ZM ) −− → H 2 (Λ; Z/2ZΛ ). We let c ∈ H 2 (M ; Z/2ZM ) be the inverse image sh of µ2 (Λ) by this isomorphism and we choose a Čech cocycle cˇ representing c. Then ˇ the twisted Kashiwara-Schapira stack µSh(ZcΛ ) has a simple global object and the ˇ twisted version of Corollary XII.3.2 gives F ∈ Db[Λ],+ (ZcM ×R ) which is simple along Λ. ˇ By Proposition XIII.3.1 we have F+ ≃ L[d] where L ∈ Mod(ZcM ) is a twisted locally constant sheaf with stalks isomorphic to Z and d is some integer. ˇ (ii) Now we prove that the existence of L ∈ Mod(ZcM ) as in (i) implies that cˇ is a sh boundary, that is, µ2 (Λ) = 0. The cocycle cˇ is associated with a covering {Ui }i∈I ˇ of M . The object L ∈ Mod(ZcM ) is given by sheaves Li ∈ Mod(ZUi ) and isomorphisms ∼ φij : Lj | −−→ Li | , for any i, j ∈ I, such that φij ◦ φjk = cijk φik for all i, j, k ∈ I. Uij

Uij

We can assume that Ui is contractible and that Uij is connected for any i, j ∈ I. Since L is locally constant, we can choose an isomorphism φi : L| ≃ ZUi for each i ∈ I. Ui ∼ Then the composition bij = φi φij φ−1 is an isomorphism Z −− → Z, that is, bij = ±1. j We let ˇb be the 1-cochain defined by {bij }i,j∈I . Then the equality φij ◦ φjk = cijk φik says that cˇ is the boundary of ˇb, as required.

ASTÉRISQUE 440

CHAPTER XIII.5 HOMOTOPY EQUIVALENCE

Here we recover a result of [2] that the projection πΛ : Λ → − M is a homotopy equivalence, that is, it induces an isomorphism of all fundamental groups. It is well ∼ know that this is equivalent to the following: π1 (Λ) −− → π1 (M ) and, for any local −1 ∼ system L on M , RΓ(M ; L) −−→ RΓ(Λ; πΛ L). lb We recall the subcategory Dlb [Λ],+ (kM ×R ) of D[Λ] (kM ×R ) introduced in Definition XII.4.1. It consists of the F such that F− ≃ 0, where we have set F± = F | M ×{±t} for t ≫ 0. Theorem XIII.5.1. — Let k be a ring with finite global dimension. (i) There exists F ∈ Db[Λ],+ (kM ×R ) such that F+ ≃ kM ; it is a simple sheaf. (ii) The functor G 7→ G+ , Db[Λ],+ (kM ×R ) → − Db (kM ), induces an equivalence between b D[Λ],+ (kM ×R ) and the full subcategory of Db (kM ) formed by the locally constant sheaves. In particular the object F in (i) is unique up to a unique isomorphism. (iii) The projection πΛ : Λ → − M induces an isomorphism in cohomology ∼ RΓ(M ; kM ) −− → RΓ(Λ; kΛ ). −1 ∼ (iv) More generally we have RΓ(M ; L) −− → RΓ(Λ; πΛ L) for any local system L on M . Proof. — (i) We first assume that k = Z. By Theorem XIII.2.2 and Proposition XIII.4.2 we know that µ1 (Λ) = 0 and µsh 2 (Λ) = 0. By Corollary XII.3.2 there exists F 0 ∈ Db[Λ],+ (kM ×R ) which is simple along Λ. By Proposition XIII.3.1 we have F+0 ≃ L[d] where L ∈ Mod(kM ) is locally constant with stalks isomorphic to Z and d is some integer. Let p : M ×R → − M be the projection. Then F 1 = F 0 ⊗p−1 L⊗−1 [−d] L

satisfies the required properties. For a general ring k we set F = F 1 ⊗ZM ×R kM ×R . (ii) By Theorem XII.4.5 the functor G 7→ G+ is fully faithful. Let L ∈ Db (kM ) be L

a locally constant sheaf. Then F ⊗ p−1 L belongs to Db[Λ],+ (kM ×R ) and we have L

L

(F ⊗ p−1 L)+ ≃ F+ ⊗ L ≃ L, which proves that G 7→ G+ is essentially surjective. (iii) follows from Corollary XIII.3.2.

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L

(iv) We apply Theorem XII.4.7 with F given by (i) and F ′ = F ⊗ (L ⊠ kR ). L

We have in this case F+′ ≃ F+ ⊗ L ≃ L, hence RHom (F+ , F+′ ) ≃ L, and also L

−1 −1 µhom(F, F ′ ) ≃ µhom(F, F ) ⊗ πΛ L ≃ πΛ L. −1 The theorem then gives RΓ(M ; L) ≃ RΓ(Λ; πΛ L).

Now we prove that π1 (Λ) → − π1 (M ) is an isomorphism. It is equivalent to show that ∼ the inverse image by πΛ induces an equivalence of categories Loc(kM ) −− → Loc(kΛ ), for some field k. Proposition XIII.5.2. — Let k be a field. Let πΛ : Λ → − M be the projection. Then the −1 inverse image functor πΛ : Loc(kM ) → − Loc(kΛ ) is an equivalence of categories. −1 Proof. — (i) We first prove that πΛ is fully faithful. Let F ∈ Db[Λ],+ (kM ×R ) be the simple object given by Theorem XIII.5.1. Since F is simple we have µhom(F, F )| ≃ kΛ and we deduce, for L, L′ ∈ Loc(kM ), Λ

(XIII.5.1)

−1 −1 ′ µhom(F ⊗ p−1 L, F ⊗ p−1 L′ ) ≃ Hom (πΛ L, πΛ L ),

where p : M × R → − M is the projection. Using (F ⊗ p−1 L)+ ≃ L and the isomorphisms (XII.4.4) and (XIII.5.1), we obtain Hom(L, L′ ) ≃ H 0 (Λ; µhom(F ⊗ p−1 L, F ⊗ p−1 L′ )) −1 −1 ′ ≃ Hom(πΛ L, πΛ L ), −1 which means that πΛ is fully faithful. −1 (ii) We prove that πΛ is essentially surjective. Let L1 ∈ Loc(kΛ ) be given. We view Loc(kΛ ) as a subcategory of DL(kΛ ) of objects concentrated in degree 0, as justified by Remark X.1.3. We recall that the functor µhom(F, ·) induces an equivalence ∼ µSh(kΛ ) −− → DL(kΛ ) (see Proposition X.2.4, where the induced functor is denoted µhom(F, ·)). Hence there exists L1 ∈ µSh(kΛ ) such that µhom(F, L1 ) ≃ L1 . By Corollary XII.3.2 there exists F1 ∈ Db (kM ×R ) such that mΛ (F1 ) ≃ L1 . Then we have an isomorphism in Db (kΛ )

(XIII.5.2)

µhom(F, F1 )| ≃ L1 . Λ

Indeed it holds first in DL(kΛ ). Then, applying the functors H i of Remark X.1.3 to (XIII.5.2), we see that µhom(F, F1 )| is concentrated in degree 0. Hence (XIII.5.2) Λ is an isomorphism in Loc(kΛ ) and then in Db (kΛ ) because Loc(kΛ ) is also a subcategory of Db (kΛ ). ˙ We set L = (F1 )+ ∈ Db (kM ). Then SS(L) = ∅ and, since F+ ≃ kM , we also have L ≃ (F ⊗ p−1 L)+ . Hence (F1 )+ ≃ (F ⊗ p−1 L)+ and Theorem XII.4.5 gives F1 ≃ F ⊗ p−1 L. We deduce −1 µhom(F, F1 )| ≃ µhom(F, F ⊗ p−1 L)| ≃ πΛ L. Λ

ASTÉRISQUE 440

Λ

CHAPTER XIII.5. HOMOTOPY EQUIVALENCE

269

−1 −1 0 Hence L1 ≃ πΛ L. Taking H 0 of both sides we have L1 ≃ πΛ H L. Since −1 H 0 L ∈ Loc(kM ), we have L1 ∈ πΛ (Loc(kM )), as required. (We could see in fact that L is concentrated in degree 0.)

Corollary XIII.5.3. — The projection Λ → − M is a homotopy equivalence. Proof. — As already recalled this follows from Theorem XIII.5.1-(iv) and Proposition XIII.5.2.

SOCIÉTÉ MATHÉMATIQUE DE FRANCE 2023

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2023 439. F. DIAMOND, P. KASSAEI & S. SASAKI – A mod p Jacquet-Langlands relation and Serre filtration via the geometry of Hilbert modular varieties: splicing and dicing 438. SÉMINAIRE BOURBAKI, volume 2021/2022, exposés 1181–1196

2022 437. 436. 435. 434.

A. BORODIN & M. WHEELER – Colored stochastic vertex models and their spectral theory S.-J. OH & D. TATARU – The Yang-Mills heat flow and the caloric gauge R. DONAGI & T. PANTEV – Parabolic Hecke eigensheaves M. BERTOLINI, H. DARMON, V. ROTGER, M. A. SEVESO & R. VENERUCCI – Heegner points, Stark-Heegner points, and diagonal classes 433. F. BINDA, D. PARK & P. A. ØSTVÆR – Triangulated categories of logarithmic motives over a field 432. Y. WAKABAYASHI – A theory of dormant opers on pointed stable curves 431. Q. GUIGNARD – Geometric local ε-factors

2021 430. SÉMINAIRE BOURBAKI, volume 2019/2021, exposés 1166–1180 p 429. E. GWYNNE & J. MILLER – Percolation on uniform quadrangulations and SLE6 on 8/3-Liouville quantum gravity 428. K. PRASANNA – Automorphic cohomology, motivic cohomology, and the adjoint L-function 427. B. DUPLANTIER, J. MILLER & S. SHEFFIELD – Liouville quantum gravity as a mating of trees 426. P. BIRAN, O. CORNEA & E. SHELUKHIN – Lagrangian shadows and triangulated categories 425. T. BACHMANN & M. HOYOIS – Norms in Motivic Homotopy Theory 424. B. BHATT, J. LURIE & A. MATHEW – Revisiting the de Rham-Witt complex

2020 423. K. ARDAKOV – Equivariant D-modules on rigid analytic spaces 422. SÉMINAIRE BOURBAKI, volume 2018/2019, exposés 1151–1165 421. J.H. BRUINIER, B. HOWARD, S.S. KUDLA, K. MADAPUSI PERA, M. RAPOPORT & T. YANG – Arithmetic divisors on orthogonal and unitary Shimura varieties 420. H. RINGSTRÖM – Linear systems of wave equations on cosmological backgrounds with convergent asymptotics 419. V. GORBOUNOV, O. GWILLIAM & B. WILLIAMS – Chiral differential operators via quantization of the holomorphic σ-model 418. R. BEUZART-PLESSIS – A local trace formula for the Gan-Gross-Prasad conjecture for unitary groups: the Archimedean case 417. J.D. ADAMS, M. VAN LEEUWEN, P.E. TRAPA & D.A. VOGAN, JR. – Unitary representations of real reductive groups 416. S. CROVISIER, R. KRIKORIAN, C. MATHEUS & S. SENTI (eds.) – Some aspects of the theory of dynamical systems: A tribute to Jean-Christophe Yoccoz, II 415. S. CROVISIER, R. KRIKORIAN, C. MATHEUS & S. SENTI (eds.) – Some aspects of the theory of dynamical systems: A tribute to Jean-Christophe Yoccoz, I

2019 414. SÉMINAIRE BOURBAKI, volume 2017/2018, exposés 1136–1150 413. M. CRAINIC, R. LOJA FERNANDES & D. MARTÍNEZ TORRES – Regular Poisson manifolds of compact types 412. E. HERSCOVICH – Renormalization in Quantum Field Theory (after R. Borcherds) 411. G. DAVID – Local regularity properties of almost- and quasiminimal sets with a sliding boundary condition 410. P. BERGER & J.-C. YOCCOZ – Strong regularity 409. F. CALEGARI & A. VENKATESH – A torsion Jacquet-Langlans correspondence 408. D. MAULIK & A. OKOUNKOV – Quantum groups and quantum cohomology 407. SÉMINAIRE BOURBAKI, volume 2016/2017, exposés 1120–1135

2018 406. L. FARGUES & J.-M. FONTAINE – Courbes et fibrés vectoriels en théorie de Hodge p-adique (Préface par P. COLMEZ) 405. J.-F. BONY, S. FUJIIÉ, T. RAMOND & M. ZERZERI – Resonances for homoclinic trapped sets 404. O. MATTE & J. S. MØLLER – Feynman-Kac formulas for the ultra-violet renormalized Nelson model 403. M. BERTI, T. KAPPELER & R. MONTALTO – Large KAM tori for perturbations of the defocusing NLS equation 402. H. BAO & W. WANG – A new approach to Kazhdan-Lustig theory of type B via quantum symmetric pairs 401. J. SZEFTEL – Parametrix for wave equations on a rough background III: space-time regularity of the phase 400. A. DUCROS – Families of Berkovich Spaces 399. T. LIDMAN & C. MANOLESCU – The equivalence of two Seiberg-Witten Floer homologies 398. W. TECK GAN, F. GAO, W. H. WEISSMAN – L-groups and the Langlands program for covering groups 397. S. RICHE & G. WILLIAMSON – Tilting modules and the p-canonical basis

2017 396. 395. 394. 393. 392. 391. 390. 389. 388. 387. 386.

Y. SAKELLARIDIS & A. VENKATESH – Periods and harmonic analysis on spherical varieties V. GUIRARDEL & G. LEVITT – JSJ decompositions of groups J. XIE – The Dynamical Mordell-Lang Conjecture for polynomial endomorphisms of the affine plane G. BIEDERMANN, G. RAPTIS & M. STELZER – The realization space of an unstable coalgebra G. DAVID, M. FILOCHE, D. JERISON & S. MAYBORODA – A Free Boundary Problem for the Localization of Eigenfunctions S. KELLY – Voevodsky motives and l dh-descent SÉMINAIRE BOURBAKI, volume 2015/2016, exposés 1104–1119 S. GRELLIER & P. GÉRARD – The cubic Szegő equation and Hankel operators T. LÉVY – The master field on the plane R. M. KAUFMANN, B. C. WARD – Feynman Categories B. LEMAIRE, G. HENNIART – Représentations des espaces tordus sur un groupe réductif connexe p-adique

2016 385. A. BRAVERMAN, M. FINKELBERG & H. NAKAJIMA – Instanton moduli spaces and W -algebras 384. T. BRADEN, A. LICATA, N. PROUDFOOT & B. WEBSTER – Quantizations of conical symplectic resolutions 383. S. GUILLERMOU, G. LEBEAU, A. PARUSIŃSKI, P. SCHAPIRA & J.-P. SCHNEIDERS – Subanalytic sheaves and Sobolev spaces 382. F. ANDREATTA, S. BIJAKOWSKI, A. IOVITA, P. L. KASSAEI, V. PILLONI, B. STROH, Y. TIAN & L. XIAO – Arithmétique p-adique des formes de Hilbert

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The aim of this paper is to apply the microlocal theory of sheaves of Kashiwara-Schapira to the symplectic geometry of cotangent bundles, following ideas of Nadler-Zaslow and Tamarkin. We recall the main notions and results of the microlocal theory of sheaves, in particular the microsupport of sheaves. The microsupport of a sheaf F on a manifold M is a closed conic subset of the cotangent bundle T ∗ M which indicates in which directions we can modify a given open subset of M without modifying the cohomology of F on this subset. An important theorem of Kashiwara-Schapira says that the microsupport is coisotropic and recent works of Nadler-Zaslow and Tamarkin study in the other direction the sheaves which have for microsupport a given Lagrangian submanifold Λ, obtaining information on Λ in this way. Nadler and Zaslow made the link with the Fukaya category but Tamarkin only made use of the microlocal sheaf theory. We go on in this direction and recover several results of symplectic geometry with the help of sheaves. In particular we explain how we can recover the Gromov nonsqueezing theorem, the GromovEliashberg rigidity theorem, the existence of graph selectors. We also prove a three cusps conjecture of Arnol’d about curves on the sphere. In the last sections we recover more recent results on the topology of exact Lagrangian submanifolds of cotangent bundles.