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Schubert Varieties, Equivariant Cohomology and Characteristic Classes
 9783037191828, 3037191821

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Jarosław Buczyn´ski, Mateusz Michałek and Elisa Postinghel, Editors

IMPANGA stands for the activities of algebraic geometers at the Institute of Mathematics, Polish Academy of Sciences, including one of the most important seminars in algebraic geometry in Poland. The topics of the lectures usually fit within the framework of complex algebraic geometry and neighbouring areas of mathematics. This volume is a collection of contributions by the attendees of the conference IMPANGA15, organised by participants of the seminar, as well as notes from the major lecture series of the seminar in the period 2010–2015. Both original research papers and self-contained expository surveys can be found here. The articles circulate around a broad range of topics within algebraic geometry such as vector bundles, Schubert varieties, degeneracy loci, homogeneous spaces, equivariant cohomology, Thom polynomials, characteristic classes, symmetric functions and polynomials, and algebraic geometry in positive characteristic.

ISBN 978-3-03719-182-8

www.ems-ph.org

SCR_Impanga 15 | Fonts: Nuri, Helvetica Neue | Farben: Pantone 116, pantone 287 | RB ??? mm

Schubert Varieties, Equivariant Cohomology and Characteristic Classes

Schubert Varieties, Equivariant Cohomology and Characteristic Classes

Jarosław Buczyn´ski, Mateusz Michałek and Elisa Postinghel, Editors

Series of Congress Reports

Series of Congress Reports

Schubert Varieties, Equivariant Cohomology and Characteristic Classes 15 Jarosław Buczyn´ski Mateusz Michałek Elisa Postinghel Editors A tribute to Friedrich Hirzebruch

EMS Series of Congress Reports

EMS Congress Reports publishes volumes originating from conferences or seminars focusing on any field of pure or applied mathematics. The individual volumes include an introduction into their subject and review of the contributions in this context. Articles are required to undergo a refereeing process and are accepted only if they contain a survey or significant results not published elsewhere in the literature. Previously published: Trends in Representation Theory of Algebras and Related Topics, Andrzej Skowron´ski (ed.) K-Theory and Noncommutative Geometry, Guillermo Cortiñas et al. (eds.) Classification of Algebraic Varieties, Carel Faber, Gerard van der Geer and Eduard Looijenga (eds.) Surveys in Stochastic Processes, Jochen Blath, Peter Imkeller and Sylvie Rœlly (eds.) Representations of Algebras and Related Topics, Andrzej Skowron´ski and Kunio Yamagata (eds.) Contributions to Algebraic Geometry. Impanga Lecture Notes, Piotr Pragacz (ed.) Geometry and Arithmetic, Carel Faber, Gavril Farkas and Robin de Jong (eds.) Derived Categories in Algebraic Geometry. Toyko 2011, Yujiro Kawamata (ed.) Advances in Representation Theory of Algebras, David J. Benson, Henning Krause and Andrzej Skowron´ski (eds.) Valuation Theory in Interaction, Antonio Campillo, Franz-Viktor Kuhlmann and Bernard Teissier (eds.) Representation Theory – Current Trends and Perspectives, Henning Krause, Peter Littelmann, Gunter Malle, Karl-Hermann Neeb and Christoph Schweigert (eds.) Functional Analysis and Operator Theory for Quantum Physics, Jaroslav Dittrich, Hynek Kovarˇ ík and Ari Laptev (eds.)

Schubert Varieties, Equivariant Cohomology and Characteristic Classes IMPANGA 15

Jarosław Buczyn´ski Mateusz Michałek Elisa Postinghel Editors

Editors: Jarosław Buczyn´ski Institute of Mathematics Polish Academy of Sciences ul. S´niadeckich 8 00-656 Warszawa, Poland

Mateusz Michałek Institute of Mathematics Polish Academy of Sciences ul. S´niadeckich 8 00-656 Warszawa, Poland

and

and

Faculty of Mathematics, Computer Science and Mechanics of University of Warsaw ul. Banacha 2 02-097 Warszawa, Poland

Max Planck Institute for Mathematics in the Sciences Inselstr. 22 04103 Leipzig, Germany

Email: [email protected]

Email: [email protected]

Elisa Postinghel Department of Mathematical Sciences Loughborough University Loughborough LE11 3TU, United Kingdom Email: [email protected]

2010 Mathematics Subject Classification: Primary 14-06; secondary 32L10, 14M15, 55N91, 14C17, 14G17. Key words: IMPANGA, vector bundles, Schubert varieties and degeneracy loci, homogeneous spaces, equivariant cohomology, Thom polynomials, characteristic classes, symmetric functions and polynomials, quasi-elliptic surfaces

ISBN 978-3-03719-182-8 The Swiss National Library lists this publication in The Swiss Book, the Swiss national bibliography, and the detailed bibliographic data are available on the Internet at http://www.helveticat.ch. This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use permission of the copyright owner must be obtained. © European Mathematical Society 2018

Contact address:



European Mathematical Society Publishing House Seminar for Applied Mathematics ETH-Zentrum SEW A21 CH-8092 Zürich, Switzerland



Email: [email protected] Homepage: www.ems-ph.org

Typeset using the authors’ T E X files: le-tex publishing services GmbH, Leipzig, Germany Printing and binding: Beltz Bad Langensalza GmbH, Bad Langensalza, Germany ∞ Printed on acid free paper 987654321

Contents

Introduction to “Schubert varieties, equivariant cohomology and characteristic classes, IMPANGA15 volume” by Jarosław Buczy´nski, Mateusz Michałek and Elisa Postinghel : : : : : : : : : :

1

Friedrich Hirzebruch – a handful of reminiscences by Piotr Pragacz : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :

9

Pieri rule for the factorial Schur P -functions by Soojin Cho and Takeshi Ikeda : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :

25

Restriction varieties and the rigidity problem by Izzet Coskun : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :

49

On Plücker equations characterizing Grassmann cones by Letterio Gatto and Parham Salehyan : : : : : : : : : : : : : : : : : : : : : : : : :

97

Kempf–Laksov Schubert classes for even infinitesimal cohomology theories by Thomas Hudson and Tomoo Matsumura : : : : : : : : : : : : : : : : : : : : : : :

127

On the multicanonical systems of quasi-elliptic surfaces in characteristic 3 by Toshiyuki Katsura : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :

153

Characteristic classes of mixed Hodge modules and applications by Lauren¸tiu Maxim and Jörg Schürmann : : : : : : : : : : : : : : : : : : : : : : : :

159

On a certain family of U.b/-modules by Piotr Pragacz : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :

203

Equivariant Chern–Schwartz–MacPherson classes in partial flag varieties: interpolation and formulae by Richárd Rimányi and Alexander Varchenko : : : : : : : : : : : : : : : : : : : : :

225

Thom polynomials in A-classification I: counting singular projections of a surface by Takahisa Sasajima and Toru Ohmoto : : : : : : : : : : : : : : : : : : : : : : : : :

237

Schubert polynomials and degeneracy locus formulas by Harry Tamvakis : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :

261

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Contents

Hirzebruch y -genera of complex algebraic fiber bundles – the multiplicativity of the signature modulo 4 by Shoji Yokura : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 315 Pushing-forward Schur classes using iterated residues at infinity by Magdalena Zielenkiewicz : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :

331

List of contributors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

347

Introduction to “Schubert varieties, equivariant cohomology and characteristic classes, IMPANGA15 volume” Jarosław Buczy´nski1 , Mateusz Michałek2 and Elisa Postinghel

The volume This volume is a conclusion of the activities of IMPANGA in the years 2010–2015, which celebrated 15 years of its existence in 2015. It is a follow up to previous books [1, 2, 3] and it contains contributions of the participants of the anniversary conference IMPANGA15. In this introduction we briefly review what IMPANGA is, describe the conference and summarise the content of the volume. Friedrich Hirzebruch passed away in 2012, during the aforementioned period. We dedicate this book to his memory.

1 The seminar IMPANGA IMPANGA is the name of the activities of Algebraic Geometers at the Institute of Mathematics, Polish Academy of Sciences, including one of the most important seminars in algebraic geometry in Poland. The head of the seminar is Piotr Pragacz, and the first seminar was held on 30 October 2000. The topics of the seminar lectures usually fit within the framework of complex algebraic geometry, although talks about positive characteristics, real algebraic geometry, symplectic geometry, complex analysis, singularity theory and other neighbouring areas of mathematics are also welcomed. The acronym IMPANGA stands for the Polish names of the Institute of Mathematics of the Polish Academy of Sciences (IMPAN, Instytut Matematyczny Polskiej Akademii Nauk) and Algebraic Geometry (GA, Geometria Algebraiczna). IMPAN https://www.impan.pl/ is a Polish institute designated to mathematical research. Its headquarters are in Warsaw and it has branches in 6 other major Polish cities. The department of Algebra and Algebraic Geometry of the institute is one the most active research groups and it is chaired by Piotr Pragacz. Its members collaborate closely with other research groups of algebraic geometers in Poland, particularly those at the 1 Buczy´ nski is supported by the research grant from Polish National Science Center, number 2013/11/D/ST1/02580, and by a scholarship of Polish Ministry of Science. 2 Michałek is supported by the research grant from Polish National Science Center, number UMO2016/22/E/ST1/00574 and by the Foundation for Polish Science (FNP).

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Figure 1. Professor Friedrich Hirzebruch at IMPAN, 2009.

University of Warsaw, the Jagiellonian University and the Pedagogical University in Kraków. The seminar IMPANGA usually runs two meetings per month on Fridays. During each meeting (typically in Warsaw) either two talks are delivered by two speakers, or a series of two lectures is presented by a single speaker. The meeting is attended by participants from Warsaw and Kraków and often also by visitors from other cities. The seminar’s participants also organise meetings (schools, workshops, conferences) of various size, topics and impact, see https://www.impan.pl/~pragacz/impanga _miniszkoly.htm. The IMPANGA School 2010 https://www.impan.pl/~impanga/ school/ and the IMPANGA15 conference https://www.impan.pl/~impanga15 are two of the largest events. We hope to organise similar events every 5 years.

2 The conference IMPANGA15 The IMPANGA15 conference took place in B˛edlewo during the week 12–18 April 2015. In addition to hosting an excellent scientific program, we celebrated 15 years of IMPANGA and the 60th birthday of Professor Piotr Pragacz. The conference exposed a rich variety of topics in algebraic geometry, especially: symmetric functions and polynomials, Schubert varieties and degeneracy loci, characteristic classes (particularly of singular varieties), Thom polynomials, characteristic p problems, arithmetic algebraic geometry, moduli problems, tropical geometry. IMPANGA15 gathered about 80 participants from Europe, United States, and Asia. Besides 17 plenary lectures, there were also 9 shorter talks delivered by the participants. During the conference there was a poster session with a range of interesting presentations. Numerous people contributed to the success of the conference. The organising committee consisted of J. Buczy´nski (Polish academy of Sciences; University of Warsaw), M. Donten-Bury (University of Warsaw; Freie Universität Berlin),

Introduction

3

Figure 2. Participants of the IMPANGA15 conference in B˛edlewo, 2015.

G. Kapustka (Polish Academy of Sciences; Jagiellonian University), O. K˛edzierski (University of Warsaw; Polish Academy of Sciences), M. Michałek (Polish Academy of Sciences; University of California, Berkeley), E. Postinghel (University of Leuven) and J. Szpond (Pedagogical University of Kraków). The scientific committee, that helped make the event exceptional, included: Paolo Aluffi (Flordia State University), Bernard Leclerc (Université de Caen), Richárd Rimányi (University of North Carolina at Chapel Hill), Matthias Schütt (Leibniz Universität Hannover) and Ravi Vakil (Stanford University). The invited speakers were: David Anderson (Ohio State University), Anders Buch (Rutgers University), Hélène Esnault (Freie Universität Berlin), Gerard van der Geer (Universiteit van Amsterdam), June Huh (Princeton University and Institute for Advanced Study), Toshiyuki Katsura (Hosei University), Maxim Kazarian (Steklov Institute of Mathematics; Moscow Independent University), JongHae Keum (Korea Institute for Advanced Study), Allen Knutson (Cornell University), Adrian Langer (Polish Academy of Sciences; University of Warsaw), Laurent˛iu Maxim (University of Wisconsin, Madison), Toru Ohmoto (Hokkaido University), Sam Payne (Yale University), Piotr Pragacz (Polish Academy of Sciences), Steven Sam (University of California, Berkeley), Harry Tamvakis (University of Maryland) and Orsola Tommasi (Leibniz Universität Hannover).

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J. Buczy´nski, M. Michałek, E. Postinghel

Figure 3. Professor Piotr Pragacz in B˛edlewo, 2015.

The conference was generously supported by Foundation Compositio Mathematica, Warsaw Centre of Mathematics and Computer Science, Stefan Banach International Mathematical Centre and Polish Academy of Science (DUN initiative). IMPANGA15 was also an opportunity to celebrate 60th birthday of Professor Piotr Pragacz. He was the special honorary guest of the meeting and the session on the afternoon of Tuesday, 14th April, was dedicated to his work. The organisers plannned an excursion to Pozna´n during the afternoon of Wednesday, 15th April. It included a guided tour around the Old Market Square and a piano concert by Maria Wójcik, who played short pieces composed by F. Chopin. The concert took place in the Red Hall of the Działy´nski Palace near the Old Market Square.

3 Contributions in the volume This section contains a short description of the chapters composing this volume. The opening article of the volume is “Friedrich Hirzebruch—a handful of reminiscences” by Piotr Pragacz. It includes personal remarks by the author, some of which have never been exposed to the public before. The article also contains a brief overview of the famous Hirzebruch–Riemann–Roch theorem and other work of Professor Hirzebruch. A fraction of his inspiration and influence on Polish mathematics, and particularly on algebraic geometers, is certified by this historical note. Soojin Cho and Takeshi Ikeda in “Pieri rule for the factorial Schur P-functions” derive formulas for the product of two factorial Schur P-functions as a linear combination of factorial Schur P-functions. In other words, they show how to compute generalisations of famous Littlewood–Richardson coefficients. Further, in the special case of the generalised Pieri rule, they obtain a positive formula for odd maximal orthogonal Grassmannians. Their method relies on a good understanding of the combinatorics of (generalisations of) shifted tableaux.

Introduction

5

Izzet Coskun in “Restriction varieties and the rigidity problem” starts his survey from the basics of Schubert varieties and cohomologies of classical homogeneous spaces, exposing the similarities and differences between the cases A and B–D. A further topic is the restriction problem, which consists in computing an induced map from the cohomologies of full Grassmannian to the cohomologies of an isotropic Grassmannian (orthogonal or symplectic). This is obtained in terms of a combinatorial algorithm using sequences of brackets and braces. Finally, the survey concludes with rigidity problems: are Schubert varieties the only varieties that represent a given Schubert class in cohomology? This article contains lots of exercises and open problems in this area and it is based on a series of lectures by the author at the IMPANGA seminar in Warsaw in 2013. Hasse–Schmidt derivations form a class of homomorphisms from an exterior algebra to the power series ring over this algebra. This class is surveyed in the ”On Plücker equations characterizing Grassmann cones” by Letterio Gatto and Parham Salehyan. A distinguished element of the class is called by authors the Schubert derivation. It allows to write a generating function encoding the quadrics defining the Plücker embedding of the Grassmannians, parametrizing vector subspaces of a fixed finite dimension, all at once. It is shown that an asymptotic expression of the proposed generating function coincides with the celebrated Kadomtsev–Petshiasvily hierarchy. In particular, an interpretation of the hierarchy in terms of Schubert Calculus is presented. Thomas Hudson and Tomoo Matsumura in “Kempf–Laksov Schubert classes for even infinitesimal cohomology theories” prove a generalisation of a determinantal formula of Schubert calculus on Grassmann bundles to the even infinitesimal cohomology theories of Grassmann bundles and Lagrangian Grassmann bundles. This extends and is built upon previous work of the two authors for other algebraic oriented cohomology theories, including K-theory (with Ikeda and Naruse) and Levine–Morel algebraic cobordism. The main tool employed is a formula for Segre classes of vector bundles in terms of Chern classes, that generalises the classical relation between Segre and Chern classes in the case of Chow rings. Quasi-elliptic surfaces are complete two dimensional varieties admitting a map to a smooth curve, such that a generic fiber is a singular irreducible, geometrically reduced curve of arithmetic genus 1. These exist in characteristic 2 and 3 and exhibit a lot of phenomena that happen only in finite characteristics. In the article “On the multicanonical systems of quasi-elliptic surfaces in characteristic 3” Katsura studies quasi-elliptic surfaces S of Kodaira dimension 1 in characteristic 3. The main theorem states that for any m  5 the multicanonical system mKS provides the quasi-elliptic fibration and 5 is the best possible value. The analogous open problem in characteristic 2 of determining when the multicanonical system provides a quasielliptic fibration is also discussed. In the review “Characteristic classes of mixed Hodge modules and applications” Laurent˛iu Maxim and Jörg Schürmann give a beautiful exposition of the topic in the title. They start by providing an introduction to mixed Hodge modules, Hodge– Chern and Hirzebruch classes including most important results and basic operations on them. Various applications and examples are presented, including a generalisation

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J. Buczy´nski, M. Michałek, E. Postinghel

of the classical Riemann–Roch theorem. In the last part, the theory of equivariant characteristic classes for singular varieties is discussed. The survey “On a certain family of U.b/-modules” by Piotr Pragacz contains a introduction on the theory of Kra´skiewicz–Pragacz modules. They are a functorial version of Schubert polynomials, whose existence was conjectured by Lascoux (Oberwolfach, June 1983) and proved by Kra´skiewicz and the author of this manuscript. This survey includes an account on the recent progress on Kra´skiewicz–Pragacz modules and filtrations that have such modules as their subquotients, made by Watanabe by means of the theory of highest weight categories. The manuscript ends with an account on two applications of Kra´skiewicz–Pragacz filtrations and of ample vector bundles to positivity of certain Schur functions, due to Watanabe and Fulton respectively. The article “Equivariant Chern–Schwartz–MacPherson classes in partial flag varieties: interpolation and formulae” by Richárd Rimányi and Alexander Varchenko opens a bridge between characteristic classes and the weight functions. A rational function that appeared in the context of q-hypergeometric solutions of quantum Knizhnik–Zamolodchikov differential equations is shown to be equal to the torus equivariant Chern–Schwartz–MacPherson class of a Schubert cell in a partial flag variety. The key observation is that the interpolation and localisation conditions satisfied by the weight functions are also satisfied by the equivariant Chern–Schwartz– MacPherson classes. In addition, the authors provide explicit combinatorial formulas for the weight functions. In “Thom polynomials in A-classification I: counting singular projections of a surface”, Takahisa Sasajima and Toru Ohmoto study Thom polynomials of mapgerms .Cn ; 0/ ! .Cm ; 0/. These are universal polynomials in Chern classes that are associated to (A-singularity types of) map-germs. They obtain explicit new results for unstable map-germs in low codimension. Furthermore, the formulas for Thom polynomials are applied to solve, both classical and new, enumerative problems for surfaces in P3 and P4 . The survey “Schubert polynomials and degeneracy locus formulas” by Harry Tamvakis is an exposition of the degeneracy loci problems from the point of view of the torus equivariant cohomology. It gives a good account of the bridge from the vector bundle language to the double Schubert polynomials. The author employs the nil-Coxeter algebra approach to construct the double Schubert polynomials introduced by Lascoux and Schützenberger for type A groups, and Ikeda, Mihalcea, and Naruse for type B, C , and D groups. A new proof of the author’s earlier result, called the splitting formula, is given. In this approach, the cases A–D can be treated more uniformly, and the splitting formulas are the only general ones available (that is, they do not treat only very special Weyl group elements, and they are intrinsic to the associated G/P space). The Hirzebruch y genus, where y is a parameter, generalises the Euler–Poincaré characteristic .y D 1/, the Todd genus .y D 0/ and the signature .y D 1/. While the Euler–Poincaré characteristic is multiplicative in fiber bundles, this is not the case for y . Shoji Yokura explains the failure of multiplicativity in general in the article “Hirzebruch y -genera of complex algebraic fiber bundles – the multiplicativity of

Introduction

7

the signature modulo 4”. He provides an explicit formula for the difference between the y genus of the fiber bundle and the product of y for the base and for the fiber. In small dimensions these can be described in terms of the respective differences of signatures and/or Todd genera. As an application he proves that the signature is multiplicative in fiber bundles modulo 4. Magdalena Zielenkiewicz in “Pushing-forward Schur classes using iterated residues at infinity” presents her viewpoint on the equivariant Gysin homomorphism. It contains both a survey of the author’s earlier work on residue formulas and iterated residues at infinity, and a presentation of how to apply these methods. As an example, a simplified proof of formulas of Pragacz and Ratajski for the push-forward of Schur polynomials over the Lagrangian Grassmannian is obtained. Acknowledgements. We are grateful to Piotr Pragacz for his work and guidance while running the seminar IMPANGA since 2000.

References [1] P. Pragacz (ed.), Topics in cohomological studies of algebraic varieties: Impanga lecture notes. Springer Science & Business Media, 2005. [2] P. Pragacz (ed.), Algebraic Cycles, Sheaves, Shtukas, and Moduli. Impanga lecture notes. Birkhäuser Basel, 2007. [3] P. Pragacz (ed.), Contributions to Algebraic Geometry. Impanga lecture notes. EMS Series of Congress Reports, 2012.

Friedrich Hirzebruch – a handful of reminiscences Piotr Pragacz1

The idea of dedicating an IMPANGA volume to Friedrich Hirzebruch arose a few years ago. A lot has changed since then. First of all, Friedrich Hirzebruch passed away on May 27, 2012. Following his death many conferences, lectures and articles [2, 4, 3, 22] were dedicated to him. The articles [1, 5, 7, 9] and the videointerview [13] appeared while he was still alive. A book by Winfried Scharlau is in preparation. These publications accurately describe the life and work of Professor Friedrich Hirzebruch from the point of view of his close colleagues and coworkers. Therefore, though initially I intended to write an article about him similar to Notes on the life and work of Alexander Grothendieck [16] (1st volume of IMPANGA Lecture Notes) or the one on Józef Maria Hoene-Wro´nski [17] (2nd volume of IMPANGA Lecture Notes), I decided to change my plans, so this essay will be of a different nature. I would like to share a collection of reminiscences about Professor Friedrich Hirzebruch from the vantage of a person, for whom he was a mentor in the years 1993–2006. In addition, I would like to highlight his relations with IMPANGA. I met Friedrich Hirzebruch for the first time in 1988 during the algebraic semester at the Banach Center, at the old residence on Mokotowska street. He came for the conference in algebraic geometry organized as a part of that semester. The main organizer was Wolfgang Vogel. I was helping with organization. Professor Hirzebruch’s lecture was a highly emotional experience for us, the younger participants. He was speaking about links between algebraic geometry and physics. It was a lecture of a Master. I also gave a talk at that conference. I spoke on polynomials in Chern classes supported on determinantal varieties. Professor Hirzebruch was present at my talk and asked a couple of interesting questions. I had an impression that he liked what I was doing. In 1992 Adam Parusi´nski encouraged me to apply, following himself, for a scholarship from the Humboldt Foundation. For that reason I had to visit the Max Planck Institute in Bonn for two weeks. The main purpose of the visit was my talk at the Oberseminar, where I presented a development of my theory of polynomials supported on degeneracy loci. I recall that besides Professor Hirzebruch in the audience there were Robert MacPherson, Christian Okonek, Don Zagier and others. I had an impression that my talk was met with a positive reception. Shortly after I received the This article was translated by Masha Vlasenko with the editorial assistance of the author. He also made a couple of additions. The article originally appeared as [18] in the Polish journal Wiadomo´sci Matematyczne, and we thank the Editors of this journal for permission to publish this English translation. 1 Supported by National Science Center (NCN) grant no. 2014/13/B/ST1/00133.

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P. Pragacz

Figure 1. F. Hirzebruch during the Leonhard Euler Congress in Saint Petersburg in 2007.

Humboldt Scholarship under which I spent years 1993–1995 at the Max Planck Institute in Bonn, where my host professor was Professor Friedrich Hirzebruch. During that visit Professor Hirzebruch became my mentor, and remained so in the following years. To be honest, it was already after my habilitation, but it was Professor Hirzebruch who helped me to find my place in mathematics. Let us be more systematic. We will start with a short biography. Friedrich Hirzebruch was born on October 17, 1927 in Hamm in North RhineWestphalia. His father was a teacher of mathematics and director of a gymnasium. He was also a teacher of the famous mathematician Karl Stein (the terms Stein space and Stein factorization are named after him). In the years 1945–48 Friedrich Hirzebruch studied mathematics at the University of Münster, which was greatly affected by the war. There he mostly studied complex analysis under the guidance of Heinrich Behnke. Hans Grauert and Reinhold Remmert were also alumni of the Münster School. The school had connections with Henri Cartan in Paris. In the years 1949–50 Friedrich Hirzebruch studied topology under Heinz Hopf in Zurich. There he prepared a doctoral thesis on the resolution of singularities of 2-dimensional complex spaces. Working with Hopf and Beno Eckmann in Zurich, he encountered for the first time Stiefel–Whitney and Chern classes in topology. Chern classes accompanied him for his entire life, see [12]. Studying in Zurich, Hirzebruch realized that results of algebraic topology and algebraic geometry could be applied in order to progress in complex analysis. The stay in Zurich was also fruitful in establishing his first international contacts. Friedrich Hirzebruch defended his doctoral thesis (advisers: Behnke and Hopf) in Münster in 1950.

Friedrich Hirzebruch

11

He spent the years 1950–1952 at the University of Erlangen. In the years 1952–1954 he visited the Institute of Advanced Study (IAS) in Princeton. The visit greatly affected his research and mathematical mastery. Shortly upon arrival, Hirzebruch established a collaboration with Kunihiko Kodaira and Donald C. Spencer, and also with Armand Borel. He was corresponding with Jean-Pierre Serre and René Thom. He studied algebraic geometry: sheaves, bundles, sheaf cohomology, Chern classes, characteristic classes, and also cobordism theory. In Princeton Hirzebruch achieved a turning point in his work on the signature theorem and on the Riemann–Roch theorem for projective manifolds of arbitrary dimensions (in December 1953). The IAS in Princeton enchanted him. Hirzebruch started to cherish the idea of creating something similar in his homeland. He succeeded in 1980, when the Max Planck Institute for Mathematics (MPIM) was established in Bonn. Naturally, Hirzebruch became the first director of that institute. On returning to Münster in 1954 he wrote the book Neue topologische Methoden in der algebraischen Geometrie. The book contained a complete proof of the Riemann–Roch theorem and constituted his habilitation thesis (see [10]). Hirzebruch visited Princeton once again in 1955/56, this time hosted by the University. During that visit he met Atiyah, Borel, Bott, Chern, Lang, Milnor, Serre, Singer and many others. He also lectured on his habilitation thesis. In 1956 he was made a professor at the university of the beautiful town of Bonn.

Figure 2. Bonn seen from Venusberg; picture received as a gift from the Bouvier bookshop in Bonn

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P. Pragacz

Hirzebruch worked at the University of Bonn until 1993. Though he traveled a lot around the world, there was only one period of long absence: the sabbatical (research leave) at Princeton in 1959/60. Hirzebruch’s favorite mathematical objects were: manifolds, surfaces, singularities (his “first love”) and characteristic classes. Manifolds: topological, algebraic, complex (though his last lecture was about real curves). Many results were obtained in cooperation with Michael Atiyah who was a close friend of Hirzebruch (cf. [22, Sect. 9]). Surfaces: doctoral thesis devoted to the resolution of singularities of 2-dimensional complex spaces with the help of Hopf’s  -processes; Hilbert modular surfaces were studied jointly with Don Zagier, Antonius Van de Ven and Gerard van der Geer. Singularities: exotic structures on manifolds arising from singularities studied jointly with Egbert Brieskorn (see [7]). Characteristic classes: Stiefel–Whitney classes, Chern classes (Shiing-Shen Chern was a close friend of Hirzebruch, see [12]), Pontryagin classes (important in the signature theorem) and the Todd genus (crucial for the Riemann–Roch theorem). Hirzebruch wrote three papers jointly with Borel on characteristic classes of homogeneous spaces. Divisibility of Chern classes and polynomials of Chern classes were among the topics which fascinated Hirzebruch. At the beginning of his study of Chern classes, Hirzebruch was intrigued by the fact that for a smooth algebraic surface c12 C c2 was always divisible by 12. Similarly: cn .E/, where E is a bundle over the 2n-dimensional sphere, is divisible by .n  1/Š. At the Arbeitstagung in 1958 he called his talk “cn divisible by .n  1/Š over the 2n-sphere” (cf. [5]). In 1953 in Princeton Hirzebruch discovered and proved two fundamental theorems: on the signature and Riemann–Roch formula in arbitrary dimension. Let E be a vector bundle of rank r over a (smooth) manifold X . Let x1 ; : : : ; xr be the Chern roots of E. The series p r Y xi p tanh xi i D1

is symmetric in x1 ; : : : ; xr , and therefore it is a series L in the Chern classes c1 .E/; : : : ; cr .E/. This series L.E/ D L.c1 .E/; : : : ; cr .E// is called the Hirzebruch L-class of E. When E is a real vector bundle over a differentiable manifold X , then Pontryagin classes are defined as pi .E/ D .1/i c2i .E ˝ C/ 2 H 4i .X; Z/

for i D 1; 2; : : : :

If X is a closed oriented manifold of even dimension n, then the intersection form on H n=2 .X / is nondegenerate due to Poincaré duality and it is symmetric when n is divisible by 4. Then we define  .X / (the signature of X ) as the signature of this form, that is the number p  q if the form can be written as 2 2      zpCq z12 C    C zp2  zpC1

Friedrich Hirzebruch

13

for a basis z1 ; : : : ; zpCq in H n=2 .X; R/. We refer the reader to [8] for an excellent account to signatures in algebra and topology. Hirzebruch’s signature theorem tells us that for a compact oriented (smooth) manifold X of dimension 4n one has Z  .X / D L.p1 .X /; : : : ; pn .X // ; X

where pi .X / D pi .TX / are called the Pontryagin classes of X and TX is the tangent bundle of X . For example, Z 1 p1 .X / ; when dim X D 4 ;  .X / D 3 X Z 1 p2 .X /  p1 .X /2 ; when dim X D 8 ;  .X / D 45 X Z 1 62p3 .X /  13p2 .X /p1 .X / C 2p1 .X /3 ; when dim X D 12 :  .X / D 945 X An important role in the proof of this theorem is played by Thom’s results on cobordism. The expository article [11] tells about this and also about the history of the discovery of the Riemann–Roch theorem in higher dimensions. Now let X be a (smooth) projective complex algebraic manifold of dimension n and E ! X be a vector bundle of rank r. In a letter to Kodaira and Spencer, Serre conjectured that the Euler number X .X; E/ D .1/i dim H i .X; E/ i

can be expressed through the Chern classes of X and E. This was known for curves. Namely, André Weil showed that for a smooth curve X of genus g Z .X; E/ D c1 .E/ C r.1  g/ : X

More explicitly, when E D O.D/ is the line bundle attached to a divisor D, then .X; O.D// D deg D C 1  g ; which is the classical Riemann–Roch formula for linear systems on curves. Many mathematicians tried to generalize these formulas to varieties of higher dimensions. Max Noether, Guido Castelnuovo, Francesco Severi and Oscar Zariski obtained some results for surfaces. Hieronymus G. Zeuthen and Corrado Segre generalized the notion of genus. A crucial step was made by Hirzebruch. He had at his disposal ingenious computations of John A. Todd, an outstanding British algebraic geometer (he was the advisor of Michael Atiyah). In the 1930s, Todd determined polynomials in the Chern classes of manifolds which give the arithmetic genus for the cases of dimension d  6. The arguments consisted essentially of inverting an N  N matrix, when N is the number

14

P. Pragacz

of partitions of d . Using these partial results, Todd also verified for d  6 the multiplicative property of the arithmetic genus, and conjectured that it should hold more generally. Hirzebruch understood how to generalize the computations of Todd, and defined the Todd genus: r Y i D1

xi D td.c1 .E/; : : : ; cr .E// D td.E/ : 1  exi

And this Todd genus appeared to be the key to the proof of the Serre conjecture. (Hirzebruch reports in [10, p. 16] that the Todd polynomials, as formal algebraic pieces, appeared already in the work of Nörlund (1924), and were called Bernoulli polynomials of higher order.) In [10, 11] Hirzebruch notices that the only power series f .x/ with the constant term 1 that satisfies the condition for every m ; the coefficient of x m in f .x/mC1 is equal to 1 is the series

x x X x 2k D 1 C B C ; 2k 1  ex 2 .2k/Š 1

f .x/ D

kD1

where B2k are the Bernoulli numbers: B2 D

1 ; 6

B4 D 

1 ; 30

B6 D

1 ;::: : 42

(In [10, 11] a different notation for the Bernoulli numbers was used: B1 D 16 ; B2 D 1 1 ; B3 D 42 ; : : :). This characterization was helpful to define the Todd genus. With 30 ci D ci .E/ we have 1 1 1 c1 C .c12 C c2 / C c1 c2 C    : 2 12 24 The Hirzebruch–Riemann–Roch theorem (HRR) tells us that Z ch.E/td.X / ; .X; E/ D td.E/ D 1 C

X

(0.1)

P where ch.E/ D riD1 exi is the Chern character, and td.X / D td.TX / means the Todd genus. We then have 1 1 ch.E/ D r C c1 C .c12  2c2 / C .c13  3c1 c2 C 3c3 / C    : 2 6 Write ti D ci .TX /. If dim X D 2, then (0.1) reads Z r 1 c 2  2c2 C t1 c1 C .t12 C t2 / : .X; E/ D 2 X 1 6 For the line bundle E D O.D/ attached to a divisor D, we get Z 1 1 ŒD2 C t1 ŒD C .t12 C t2 / : .X; O.D// D 2 X 6

15

Friedrich Hirzebruch

If dim X D 3, then (0.1) reads Z 1 1 r 1 3 .c1  3c1 c2 C 3c3 / C t1 .c12  2c2 / C .t12 C t2 /c1 C t1 t2 : .X; E/ D 4 12 24 X 6 The nature of the left-hand side in (0.1) is holomorphic, while the right-hand side is of a topological nature. We give a sketch of proof from Hirzebruch’s book [10]. Let us introduce the notation Z ch.E/td.X / : T .X; E/ D X

We denote by  W F .E/ ! X the bundle of flags of subspaces of dimensions 1; : : : ; r 1 in the fibers of E. The bundle F .E/ is often called the bundle of complete flags associated to E. If X is a point, and so E D Cr , then we get a nonsingular projective variety F .r/ of complete flags in Cr . The study of properties of the flag bundle F .E/ plays a key role in Hirzebruch’s proof. One has ([10, Theorem 14.3.1]) T .X; E/ D T .F .E/;   E/ : Let .X / D .X; OX / be the arithmetic genus (cf. [10, p. 123 and pp. 1–2]). It is known that one has .F .r// D 1. By [10, Appendix II,Theorem 8.1] .X; E/ D .F .E/;   E/  .F .r// D .F .E/;   E/ : By the splitting principle, the bundle   E splits on F .E/   E D L1 ˚    ˚ Lr into a sum of line bundles Li . We then have 

T .F .E/;  E/ D

r X

T .F .E/; Li /

i D1

and ([10, Theorem 16.1.2]) .F .E/;   E/ D

r X

.F .E/; Li / :

i D1

Observe that to conclude the proof, it suffices to show HRR for line bundles. Indeed, then we have .F .E/; Li / D T .F .E/; Li /

for

1i r;

and .X; E/ D T .X; E/. To achieve HRR for line bundles, we proceed as follows. By [10, Theorem 20.2.2], we have a fundamental equation for the arithmetic genus: .X / D td.X / : This is a consequence of three facts:  .F .E// D .X / by [10, Theorem 20.2.1], hence .F .TX // D .X /;  td.F .E// D td.X / by [10, Theorem 14.3.1], hence td.F .TX // D td.X /;  .F .TX // D td.F .TX // due to [10, Theorem 20.1.1].

16

P. Pragacz

For a line bundle L on X we define .L/X D .X /  .X; L / ;

T .L/X D td.X /  T .X; L /

to be the virtual characteristic and virtual genus respectively. One can prove that for any line bundle L, the virtual characteristic and genus coincide: .L/X D T .L/X (see [10, Theorem 20.3.1]). A functional equation used in the proof also shows up at several other places in [10]: Theorem 11.3.1, Theorem 12.3.2, Theorem 17.3.1, Theorem 19.3.1 and Theorem 19.3.2. (We shall return to Theorem 11.3.1 in the identity (0.2).) Substituting L by L we obtain T .X; L/ D .X; L/, which completes the sketch of the proof. The Hirzebruch–Riemann–Roch theorem was a starting point for several further results:  Grothendieck’s generalization of HRR for a proper morphism between algebraic varieties. This generalization is expressed by means of commutativity of a certain diagram involving homology groups and K-groups in algebraic geometry. This led to  development of topological K-theory (Atiyah, Hirzebruch), which in turn served as an important instrument for  the index theorem for elliptic operators (Atiyah, Singer). Hirzebruch considered generalizations of .X; E/. Introducing an (auxiliary) variable y, he defined the following y -characteristic y .X; E/ D

n X pD0

p



p

.X; E ˝ ^ .TX / /y D

n X n X

.1/q hp;q .X; E/y p ;

pD0 qD0

where hp;q .X; E/ D dimC H q .X; E ˝ ^p .TX / /. The Todd genus, virtual characteristic and other notions were generalized in a similar manner. In [10, Theorem 21.3.1] Hirzebruch gave a generalization of HRR for y .X; E/ (though he considered HRR, and not this latter generalization, as the central theorem in his book [10]). In the article [14] in the volume Algebraic Geometry: Hirzebruch 70 Alain Lascoux demonstrated that, applied to flag varieties, y -characteristic becomes an adequate tool for studying algebra and combinatorics of the Macdonald polynomials. Let K0 .var=X / be the relative Grothendieck group of complex algebraic varieties over a variety X . In [6] Jean-Paul Brasselet, Jörg Schürmann and Shoji Yokura defined a natural transformation Ty W K0 .var=X / ! H .X / ˝ QŒy commuting with proper pushdown, which generalizes the corresponding Hirzebruch characteristic. It is a homology class version of the motivic measure corresponding to a suitable specialization of the well-known Hodge polynomial. This transformation unifies the Chern class transformation of MacPherson and Schwartz (for y D 1), the Todd class transformation in the singular Riemann–Roch theorem of Baum–Fulton–MacPherson (for y D 0) and the L-class transformation of Cappell–Shaneson (for y D 1).

Friedrich Hirzebruch

17

The Hirzebruch–Riemann–Roch theorem has found many applications. In our work with Vishwambhar Pati and Vasudevan Srinivas on the diagonal property, which says that on X  X there exists a bundle of rank dim X together with a section vanishing along the diagonal, HRR appeared to be very useful for studying the 3dimensional quadric X D Q3 . Let ŒQ2 ; ŒL and ŒP  (the 2-dimensional quadric, line and point) be generators of the Chow groups A1 .Q3 /; A2 .Q3 / and A3 .Q3 / respectively. The total Chern class of a vector bundle E on Q3 has the form 1 C d1 .E/ ŒQ2 C d2 .E/ ŒL C d3 .E/ ŒP  ; where di .E/ 2 Z. It appears (see [21]) that to show absence of the diagonal property for Q3 it is sufficient to prove that there is no bundle E of rank 3 over Q3 with d3 .E/ D 1 D d1 .E/. The latter follows from the fact that substituting these numbers into HRR 1 3 13 .Q3 ; E/ D .2d13  3d1 d2 C 3d2 / C .d12  d2 / C d1 C 3 ; 6 2 6 where di D di .E/, we obtain .Q3 ; E/ D 15  2d2 , which is a contradiction. 2 In fact, the book [10] contains much more than a proof of HRR. In our work [15] with Adam Parusi´nski on the topological Euler characteristic of possibly singular complex hypersurfaces, at first we proved a formula for the Euler characteristic of the

Figure 3. (from left) F. Hirzebruch, the author and A. Parusi´nski during Hirzebruch 70

18

P. Pragacz

zero locus of a section of a very ample line bundle. In order to generalize this formula to an arbitrary line bundle L, our strategy was to take a very ample line bundle M such that L ˝ M is again very ample. But we were not able to make this idea work for several days. . . For a vector bundle E of rank r over a smooth complex manifold X let us define Z c.E/1 cr .E/ c.X / : .X jE/ D X

It is well known that this expression gives the (topological) Euler characteristic of the zero locus of a section which is transverse to the zero section of E. A few days after that, we found in [10] that there is an identity 2 .X jL˚M /C.X jL˝M / D .X jL/C.X jM /C.X jL˚M ˚L˝M / : (0.2) This is a very particular case of [10, Theorem 11.3.1], proved for Hirzebruch’s virtual Ty -genus. With the help of this identity a proof of our formula for the Euler characteristic of a projective complex hypersurface went smoothly (see [15, p. 349]). Several years later we proved this formula for complete (compact) hypersurfaces – but this is a subject for another story. . .

Friedrich Hirzebruch

19

Figure 4. Though one cannot see Michael Atiyah in this picture, he participated in the conference and gave a talk on the links between physics and algebraic geometry in the 1980s and 1990s, and also on the role of Friedrich Hirzebruch and his Arbeitstagung in the development of these links.

As I write about the Hirzebruch–Riemann–Roch theorem, I am using the book Topological methods in algebraic geometry which was given by Professor Hirzebruch to me and my coorganizers, Michał Szurek and Jarosław Wi´sniewski, as thanks for the organization of the conference Hirzebruch 70 in Warsaw, in 1998. The conference took place at the Banach Center residence on Mokotowska street. A lot of prominent algebraic geometers came to Warsaw on that occasion. The proceedings of that conference were published as the volume Algebraic Geometry: Hirzebruch 70 of the AMS series Contemporary Mathematics. Since 2000 I have been running the IMPANGA algebraic geometry seminar at the Institute of Mathematics of the Polish Academy of Sciences (see the webpage [19], describing more broadly the IMPANGA algebraic geometry enviroment at this institute). Professor Hirzebruch was the first foreign speaker at the seminar. On May 7, 2001 he gave a talk Old and new applications of characteristic classes. The talk took place at his favorite place, i.e., at the Banach Center residence on Mokotowska street. The speaker received the traditional IMPANGA cup. A lot of advertisements of his talk were posted around over the institute; there was one in the elevator. In the evening I went to the National Opera with Fritz and his wife Inge. There, in the elevator, Fritz smiled: —In every second elevator in Warsaw there is an advertisement of my talk.

20

P. Pragacz

When I along with colleagues was organizing in 2010 the school and conference IMPANGA 10, Professor Hirzebruch agreed to be the Honourable Chairman of the Scientific Committee. Unfortunately, for a health reason he couldn’t come in person to the Banach Center in B˛edlewo, but he took an active part in choosing conference speakers. Professor Hirzebruch’s support was very important for the young participants of IMPANGA. In token of our gratitude, a speaker at IMPANGA 10, Masha Vlasenko, brought him a conference cup. This way there were now two IMPANGA cups at his office. Earlier the participants of IMPANGA took an active part in the symposium Manifolds in mathematics and in other fields (IM PAN 2002) organized by Friedrich Hirzebruch and Stanisław Janeczko. A large group of Polish algebraic geometers benefited from the hospitality of Professor Hirzebruch at the Max Planck Institute in Bonn: Grzegorz Banaszak, Andrzej Dabrowski, ˛ Wojciech Gajda, Piotr Kraso´n, Adrian Langer, Adam Parusi´nski, Tomasz Szemberg, Jarosław Wi´sniewski, Jarosław Włodarczyk and the author. Those few years spent at MPIM were the most mathematically productive ones in my life. The cheerful atmosphere at the institute led by Professor Hirzebruch was steadily converted into advances at work. After the first year of my stay at MPIM he handed me his Collected Papers with an appropriate dedication and a humorous request: —May I please ask you to not just read my Collected Papers but also quote them; this way more mathematicians will get interested and the books will be selling better. The bibliography to [20] shows that I took his advice heartily. One of the topics I was working on in Bonn along with my doctoral student Jan Ratajski were Lagrangian, symplectic and orthogonal degeneracy loci. This is a beautiful part of Schubert calculus where geometry intertwines with algebra. We discovered an intriguing identity for the Schur functions in the variables xi and their squares, (0.3) s .x12 ; : : : ; xn2 / s .x1 ; : : : ; xn / D sC2 .x1 ; : : : ; xn / ; where  D .n; n  1; : : : ; 1/ and . C 2/i D i C 2i . This identity allowed us to prove a new Gysin formula for Lagrangian Grassmann bundles, but we kept feeling that we didn’t understand it in sufficient depth. I confided to Professor Hirzebruch, and he suggested that we look at (0.3) from the point of view of quaternionic manifolds, pointing to several articles by himself and a paper by Peter Slodowy. And indeed with the use of quaternionic flag manifolds identity (0.3) becomes natural (see [20, (10.7)]). As I already mentioned, HRR appeared very helpful to show that the 3-dimensional quadric does not have the diagonal property. Being asked, Professor Hirzebruch gave us a few pieces of advice on how the right-hand side of HRR can be effectively computed for complete intersections.

Friedrich Hirzebruch

21

Fritz cared a lot that the mathematicians hosted by the Max Planck Institute felt comfortable. They were often invited by him for lunch. When the institute was located in Beuel, he would invite his guests to the Italian restaurant Tivoli. When the institute moved to the center of Bonn, they would go to a small restaurant over the confectionery Fassbender located very close the new building of the institute. Fat Thursday is the culmination of the carnival at MPIM. This tradition was brought from the former building, because in Beuel the carnival procession takes place on this day. Being a guest of MPIM about the carnival time in 2001, I was amazed to find a necktie in my pigeonhole on Wednesday before Fat Thursday. I was told that there was a tradition to cut ties during the Thursday celebration. I also learned that it was Fritz that put the tie in my pigeonhole. The next day the tie was cut. . . —but not THAT ONE! The tie from Fritz remained a precious souvenir, and has been serving me ever since then. Bonn has been a very attractive mathematical center. I was attending Professor Egbert Brieskorn’s lectures at the university. These beautiful lectures (in German) highlighted relations of singularities and. . . art. In 2006 I had an opportunity to give two talks at Brieskorn’s seminar on singularities. I started with expressing my gratitude for being able to speak at the seminar on singularity theory. Brieskorn replied instantly: —Pardon me, there is no singularity theory, there are only singularities. I benefited a lot from his numerous clever comments. Every Friday at 17:15 there was a colloquium at the university with very interesting talks. Professor Brieskorn told me that in order to encourage people to participate in the colloquium (at the initial stage) Hirzebruch used to say: —As Catholics weekly attend the mass, in a similar manner mathematicians should attend the colloquium. Telling about his father, Professor Hirzebruch recalled that he was greatly respected by his pupils. As he was entering the class, it was sufficient for him to say: —Jungen! (—Boys!), and the class would fall silent. Friedrich Hirzebruch was also gifted at working with people. When a problem arose, he would always manage the situation in a way that no conflict was possible.

22

P. Pragacz

When he spoke on anything, he would concentrate on positive sides (though we knew there were negatives as well). He had a brilliant sense of humor. One could spot the following practice at the institute. Only one person would come to work in a suit and necktie, others were more casual: sweaters, . . . Once that person—it was Fritz— was asked, why to torture oneself that much. His answer was: —If someone from outside were to visit our institute, they would notice without asking anyone, who is the director here! At the institute we felt safe under Professor Hirzebruch’s wing. We received enormous intellectual, spiritual and also financial support. He was continually advising, keeping our spirits up, helping. Friedrich Hirzebruch was a fantastic mathematician. He solved the Riemann– Roch problem in all dimensions. His organizational skills made Bonn into an incredibly attractive mathematical center. A lot of Polish mathematicians, and particularly many IMPANGA participants, benefited and continue to benefit from it. He was open to everyone, whom he would help as much as he could. In one of Don Zagier’s texts on Friedrich Hirzebruch I found the words I would like to finish this article with: “In many almost invisible ways, he made the people around him slightly better people, and the world around him a slightly better world.” Acknowledgements. My sincere thanks go to Masha Vlasenko for her translation and valuable comments. I am greatly indebted to Michael Atiyah for telling me about John A. Todd and his work on the arithmetic genus. I am also grateful to Grzegorz Banaszak, Gerard van der Geer, Adam Parusi´nski, Thomas Peternell and an anonymous referee for careful reading of the paper, and pointing out some corrections and improvements. I would like to thank Andrea Kohlhuber and Renata Podgórska for their help with pictures. Finally, I wish to thank Brian Harbourne for his linguistic help.

References [1] M. Atiyah, Friedrich Hirzebruch – an appreciation, Proceedings of the Hirzebruch 65 Conference in Algebraic Geometry (Ramat Gan, 1993) (M. Teicher, ed.). Israel Math. Conf. Proc. 9 (1996), 1–5. [2] M. Atiyah, Friedrich Ernst Peter Hirzebruch, 17 October 1927–27 May 2012. Biogr. Mem. Fellows R. Soc. 60 (2014), 229–247. [3] M. Atiyah, C. Bär, J.-P. Bourguignon, G.-M. Greuel, Y. I. Manin, and M. Sanz-Solé, Friedrich Hirzebruch Memorial Session at the 6th European Congress of Mathematics (Kraków, July 5th, 2012). Eur. Math. Soc. News 85 (September 2012), 12–20. [4] M. Atiyah and D. Zagier (coordinating editors), Friedrich Hirzebruch (1927–2012). Notices Amer. Math. Soc. 61 (2014), 2–23. [5] J.-P. Bourguignon, Questions au Professeur Hirzebruch. Gaz. Math. 53 (Juin 1992), 11–19.

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[6] J.-P. Brasselet, J. Schürmann, and S. Yokura, Hirzebruch classes and motivic Chern classes for singular spaces. J. Topol. Anal. 2 (2010), 1–55. [7] E. Brieskorn, Singularities in the work of Friedrich Hirzebruch. Surv. Differ. Geom. 7 (2007), 17–60. [8] É. Ghys and A. Ranicki, Signatures in algebra, topology and dynamics. Ensaios Mat. 30 (2016), 1–173. [9] H. Grauert, G. Harder, and R. Remmert, Curriculum vitae mathematicae. Math. Ann. 278 (1987), iii–viii. [10] F. Hirzebruch, Topological Methods in Algebraic Geometry. 3rd edition, Springer 1966. [11] F. Hirzebruch, Prospects in Mathematics. The signature theorem: reminiscences and recreation. Ann. Math. Stud. 70 (1971), 3–31. [12] F. Hirzebruch, Why do I like Chern, and why do I like Chern classes? Notices Amer. Math. Soc. 58 (2011), 1231–1234. [13] M. Kreck, Video interview with Friedrich Hirzebruch. Simons Foundation website (2011). [14] A. Lascoux, About the “y” in the y -characteristic of Hirzebruch. In Algebraic Geometry: Hirzebruch 70 (P. Pragacz, M. Szurek and J. Wi´sniewski, eds.), Contemp. Math. 241, Amer. Math. Soc., Providence RI, 1999, 285–296. [15] A. Parusi´nski and P. Pragacz, A formula for the Euler characteristic of singular hypersurfaces. J. Alg. Geom. 4 (1995), 337–351. [16] P. Pragacz, Notes on the life and work of Alexander Grothendieck. In Topics in Cohomological Studies of Algebraic Varieties (P. Pragacz, ed.), Trends in Mathematics, Birkhäuser, Basel 2004, xi–xxviii. [17] P. Pragacz, Notes on the life and work of Józef Maria Hoene-Wro´nski. In Algebraic Cycles, Sheaves, Shtukas and Moduli (P. Pragacz, ed.), Trends in Mathematics, Birkhäuser, Basel 2007, 1–20. [18] P. Pragacz, Friedrich Hirzebruch – gar´sc´ reminiscencji, Wiadom. Mat. 52(1) (2016), 37–51. [19] https://www.impan.pl/pragacz/impanga.htm (accessed: 07.02.2017).

e-poly[20] P. Pragacz and J. Ratajski, Formulas for Lagrangian and orthogonal degeneracy loci; Q nomial approach. Compositio Math. 107 (1997), 11–87. [21] P. Pragacz, V. Srinivas, and V. Pati, Diagonal subschemes and vector bundles. Pure Appl. Math. Q. 4(4), part 1 of 2, Special Issue in honor of Jean-Pierre Serre (2008), 1233–1278. [22] D. Zagier, Life and work of Friedrich Hirzebruch. Jahresber. Dtsch. Math.-Ver. 117 (2015), 93–132.

Pieri rule for the factorial Schur P-functions Soojin Cho1 and Takeshi Ikeda2 Contents 1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . 2 Factorial Schur P -functions . . . . . . . . . . . . . . . . . . . . . . . 2.1 Definition of factorial Schur P -functions . . . . . . . . . 2.2 Reverse marked shifted tableaux . . . . . . . . . . . . . . . 2.3 Switchable entries . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Reverse decomposition tableaux . . . . . . . . . . . . . . . 2.5 Mixed insertion algorithm . . . . . . . . . . . . . . . . . . . 2.6 Factorial Schur P -functions as a ratio . . . . . . . . . . . 3 `-essential reverse decomposition tableaux . . . . . . . . . . . . . 3.1 Barred reverse decomposition tableaux . . . . . . . . . . 3.2 A lemma on factorial powers . . . . . . . . . . . . . . . . . 3.3 `-essential reverse decomposition tableaux . . . . . . . . 3.4 Separation of variables . . . . . . . . . . . . . . . . . . . . . 3.5 Proof of Theorem 3.8 . . . . . . . . . . . . . . . . . . . . . . 4 Pieri rule for the factorial Schur P -functions . . . . . . . . . . . . 4.1 -good reverse decomposition tableaux . . . . . . . . . . 4.2 Pieri rule for the factorial Schur P -functions . . . . . . . 4.3 On reverse hookwords . . . . . . . . . . . . . . . . . . . . . . 4.4 Hatted reverse decomposition tableaux . . . . . . . . . . . 4.5 Involutions on hatted reverse decomposition tableaux . 4.6 Proof of positivity . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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1 Supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education, Science and Technology(2011-0012398). 2 This work is supported by KAKENHI 15K04832, 16H03920.

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1 Introduction The factorial Schur P -function was introduced by Ivanov [6]. A geometric interpretation of this function as the torus equivariant Schubert class of the maximal orthogonal Grassmannian was established in [3, 5]. The aim of this article is to determine the equivariant Littlewood–Richardson coefficients, i.e. the multiplicative structure constants of the factorial Schur P -functions. We first prove an identity expressing the product of two factorial Schur P -functions as a linear combination of factorial Schur P -functions (Theorem 3.8). The coefficients are explicitly defined polynomials in the deformation parameter. Our basic combinatorial language is the decomposition tableaux introduced by Serrano [9]. In fact, this combinatorial object was successfully used in describing a new Littlewood– Richardson rule [1] for the ordinary Schur P -functions. Our formula involves sign factors and there are large possibility of cancellation of terms. Thus it would be desirable to analyze how those cancellations happen precisely. Natural expectation is that such process would lead to a positive formula in the sense that it can be written as a sum of products of positive roots. This notion was introduced by Peterson and proved by Graham in [2] for the Kac–Moody flag variety. The main result of this paper is a Pieri rule for the factorial P -functions. We proved that our Pieri formula is positive when we apply our result to odd maximal orthogonal Grassmannians (see Cor. 4.23 for precise statemant). It is remarkable that recently Li and Ravikumar proved in [8] a manifestly positive equivariant Pieri rule of the isotropic Grassmannians. Our strategy for deriving the structure constants is to generalize the proof of a Littlewood–Richardson rule for Schur P -functions in [1]. Our rule is expressed in terms of barred (reverse) decomposition tableaux. We are also greatly inspired by the work [7] of Kreiman who proved an equivariant Littlewood–Richardson rule for the Grassmannian by generalizing a short proof by Stembridge in [10] for the ordinary Littlewood–Richardson rule. 1.1 Organization. In Section 2, we first define the factorial Schur P -function and recall its basic properties. We also give several expressions for the function and explain the related combinatorial objects: reverse marked shifted tableaux and reverse decomposition tableaux etc. In Section 3, we prove Theorem 3.8. In Section 4, we prove the Pieri rule for the factorial Schur P -functions, and then discuss the positivity. 1.2 Acknowledgements. We express deep thanks to Dongho Moon and Shyamashree Upadhyay for useful discussions in the early stage of this work. We are grateful to Maki Nakasuji for valuable discussions, and for making Sage programs on the reverse decomposition tableaux. Conversations with Tomoo Matsumura on related joint projects greatly helped us.

Pieri rule for the factorial Schur P -functions

27

2 Factorial Schur P-functions We fix a positive integer n, the number of variables throughout the paper. 2.1 Definition of factorial Schur P-functions. By a strict partition, we mean a strictly decreasing sequence  D .1 ; : : : ; ` / of positive integers with `  n: Let SP denote the set of all strict partitions. We call ` the length of , which is denoted by `./: Define the generalized factorial .xjt/k of any variable x and a positive integer k by .xjt/k D .x  t1 /    .x  tk / .k  1/; where t D .t1 ; t2 ; : : :/ is an infinite sequence of indeterminates. Let x1 ; : : : ; xn be a finite set of variables. For each strict partition  of length `./; the factorial Schur P -function is defined by 3 2 `./ `./ n Y Y Y X 1 x C x i j 5; P .xjt/ D (2.1) w 4 .xi jt/i .n  `.//Š xi  xj w2Sn

i D1

i D1 j Di C1

where Sn is the symmetric group and w 2 Sn permutes the variables x1 ; : : : ; xn : It can be shown that this is actually a symmetric polynomial in x1 ; : : : ; xn with coeffiP cients in ZŒt D ZŒt1 ; t2 ; : : :. Let us denote by jj the sum `./ i D1 i : One sees that P .xjt/ is homogeneous of degree jj:  The aim of this paper is to compute the coefficients f .t/ 2 ZŒt that appear in the expansion of the product of two factorial Schur P -functions: X  P .xjt/P .xjt/ D f .t/P .xjt/; 

where the summation is over all strict partition  such that j j  jj C j j: The  .t/ 2 ZŒt is homogeneous of degree jjCj jjj: If jjCj jjj D polynomial f;   .0/ is a non negative integer, which is the structure constant 0 then f; .t/ D f; for Schur P -functions. 2.2 Reverse marked shifted tableaux. For a strict partition  D .1 ; : : : ; ` /, the shifted diagram S./ is S./ D f.i; j / 2 Z2 j 1  i  `; i  j  i C i  1g:

(2.2)

We depict such a diagram by square boxes with coordinates .i; j /, arranged like matrices, that the first coordinate i (the row index), the second coordinate j (the column index). For example, the shifted diagram of .6; 4; 1/ is (2.3) with the square of thick lines has the coordinate .i; j / D .2; 3/.

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Let P denote the ordered alphabet f1; 10 ; : : : ; n; n0 g with the order 1 < 1 0 < 2 < 2 0 <    < n < n0 :

(2.4)

The elements 10 ; 20 ; : : : are said to be primed, and we denote by jaj the unprimed version of any a 2 P: Let  be a strict partition. A reverse marked shifted tableau R of shape  is a filling of each box of S./ with an element in P in such that (1) The entries weakly decrease along any row from left to right, and along any column from top to bottom. (2) Each column contains at most one k, for each k  1: (3) Each row contains at most one k 0 , for k  1: (4) The entry of any box on the main diagonal is unprimed. Here is an example of a reverse marked shifted tableau of shape .6; 4; 1/: 3 3 2 2 10 1 2 10 1 1 1

(2.5)

Let T be a reverse marked shifted tableau of shape  and a be an element of P. Then, we use the notation a 2 T to indicate both the position of a in S./ and the value assigned to the position. The row and column indices of a are denoted by r.a/ and c.a/ respectively. Define 8 ˆ if r.a/ < c.a/ and a is primed, jaj. Then, we have the following inequalities from (4.8). Note, in this case, that N d D ıNd  1 for d  jaj. We have jajC1 C    C jbj C jbjC1  jaj  jbjC1 C ıjaj C ıjajC1 C    C ıjbj  .jbj  jaj C 1/ ; jbjC2 C    C `C1  jbjC1 C ıjbjC1 C    C ı`  .`  jbj/ : Since jaj C prec.a/ C 1 < c.a/, where prec.a/ D jaj and c.a/ D jaj C jajC1 C    C `C1 C N jaj C N jajC1 C    C N `C1 , we also have jaj C 1 < jajC1 C    C `C1 C N jaj C N jajC1 C    C N `C1  .jaj C jaj  1  `/ C ıjaj C N jaj C    C ı` C N ` C N `C1  jaj C .jaj  1  `/ C .`  jaj C 1/ C 1 D jaj C 1 ; which is a contradiction. This shows that ".b/ D 1. We now can assume that ".b/ D 1 and hence c.b/ 6D 1; that is b is in BQ # . Assume on the contrary to the claim that .tjbj Cprec.b/C1 C ".b/tc.b/ / is ‘nonzero’. Then jbj C prec.b/ C 1 > c.b/, due to the definition of a and b. We let d be the Pjbj1 number of d ’s between b and a for jaj  d  jbj, and D d Djaj d . We also let ˇ be the number of jbj’s in B# that precedes b. Then, for d D jaj; jaj C 1; : : : ; jbj  2, we have (4.9) d C1 C d C1 C ıd C1  d C ıd  1 : Moreover, we have jbj C jbj C ˇ C ıjbj  jbj1 C ıjbj1  1 :

(4.10)

From (4.9) and (4.10), we obtain the following; .jbj  jaj/ C C jbj C ˇ  .jaj  jbj / C ıjaj  ıjbj

(4.11)

Note that prec.a/ D ıjaj C jaj and prec.b/ D ıjbj C ˇ: Moreover, by Lemma 4.9, we have c.a/  c.b/ D C jbj C

jbj1 X

.ıd C ıNd //  C jbj C .jbj  jaj/ :

d Djaj

We therefore have .jaj  jbj / C .prec.a/  prec.b// D .jaj  jbj / C ıjaj C jaj  ıjbj  ˇ  .jbj  jaj/ C C jbj C ˇ C jaj  ˇ D .b  a/ C C jbj C jaj  c.a/  c.b/ C jaj : This is a contradiction, since we are assuming that jbj C prec.b/ C 1 > c.b/ and jaj C prec.a/ C 1 < c.a/, hence c.a/  c.b/ > .jaj  jbj / C .prec.a/  prec.b// C 1. 

Pieri rule for the factorial Schur P -functions

47

N Example 4.22. Let B D 123443N 3322 2N and  D .10; 5; 3/ then B is -good and for N the corresponding factor in c;B is t9  t10 . If b is the second 3N then the the entry 2, corresponding factor in c;B is t5  t5 D 0: Thus c;B is zero. Consider the specialization of parameters t D .t1 ; t2 ; : : :/ defined by t1 D 0 and ti C1 D tQi for i  1. Under this specialization the factorial Schur P -function P .xj0; tQ1 ; tQ2 ; : : :/ gives the torus equivariant Schubert class of odd maximal orthogonal Grassmannian ([4] Theorem 6.1 (2), see also [5] Theorem 8.7 (2)). Thus the   .tQ/ WD f; .0; tQ1 ; tQ2 ; : : :/ are the torus equivariant Schubert structure coefficients f; constants for the odd maximal Grassmannian. In this context, the corresponding root system is of type B, and the positive roots (cf. [4] §3) are given by tQi ˙ tQj

.i > j /;

tQi

.i  1/:

Corollary 4.23. Let ;  be strict partitions. The type B specialization of Pieri forgood mula (4.2) is non-negative, in the sense that if B is in BRDT.k/ , then c;B .0; tQ1 ; tQ2 ; : : :/ is zero or a product of positive roots of type B. Positivity issue in type D is a little cumbersome. As we have seen above, no negative root of the form ti  tj .i < j / appears in the Pieri rule. Other possibility we have to care is 2ti , which is a sum .ti C tj / C .ti  tj / of positive roots for i > 1 with any j < i: The only remaining case is 2t1 which cannot be a sum of positive roots. This can really appear in our formula, however, we can easily show that if ` C n is odd, this does not happen. We have no explanation of what this phenomena means geometrically.

References [1] S. Cho, A new Littlewood-Richardson rule for Schur P -functions. Trans. Amer. Math. Soc. 365 (2013), 939–972. [2] W. Graham, Positivity in equivariant Schubert calculus. Duke Math. J. 109, 3 (2001), 599– 614. [3] T. Ikeda, Schubert classes in the equivariant cohomology of the Lagrangian Grassmannian. Adv. Math. 215 (2007), 1–23. [4] T. Ikeda, L.C. Mihalcea, and H. Naruse, Double Schubert polynomials for the classical groups. Adv. Math. 226 (2011) 840–886. [5] T. Ikeda and H. Naruse, Excited Young diagrams and equivariant Schubert calculus. Trans. Amer. Math. Soc. 361 (2009), 5193–5221. [6] V. N. Ivanov, Interpolation analogues of Schur Q-functions. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. 307 (2004), 99–119. [7] V. Kreiman, Equivariant Littlewood-Richardson skew tableaux. Trans. Amer. Math. Soc. 362 (2010), 2589–2617.

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[8] C. Li and V. Ravikumar, Equivariant Pieri rules for isotropic Grassmannians. Math. Ann. 365(1–2) (2016), 881–909. [9] L. Serrano, The shifted plactic monoid. Math. Z. 266 (2010), 363–392. [10] J. R. Stembridge, A concise proof of the Littlewood-Richardson rule. Electron. J. Combin. 9 (2002), Note 5, 4 pp. (electronic).

Restriction varieties and the rigidity problem Izzet Coskun1 Contents 1 2 3 4

Introduction . . . . . . . . . . . . . . . . . . . Preliminaries on Schubert varieties . . . The Golden Rules of quadric geometry Restriction problem . . . . . . . . . . . . . 4.1 The symmetric case . . . . . . . . 4.2 The skew-symmetric case . . . . 5 The rigidity problem . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . .

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1 Introduction Homogeneous varieties are ubiquitous in mathematics. Especially the Grassmannians and flag varieties associated to the classical Lie groups play a central role in geometry, representation theory and combinatorics. In this paper, we survey recent developments in two important problems, the restriction and rigidity problems, in the geometry and cohomology of homogeneous varieties following [C2, C3, C4] and [C5]. These notes grew out of lectures I gave at IMPAN in December 2013. The lectures were organized around the following three themes: (1) Develop a concrete geometric theory of isotopic flag varieties in the spirit of the classical theory of Grassmannians, reducing the theory to a few simple principles of quadric geometry. (2) Construct explicit rational equivalences between subvarieties of homogeneous varieties and unions of Schubert varieties. (3) Use explicit rational equivalences and intersection theory to study rigidity of Schubert classes.

1 I would like to thank Piotr Pragacz, Jarek Buczy´ nski, the participants of the seminar and IMPAN for their hospitality and giving me the opportunity to deliver the lectures that gave rise to these notes. I thank the referee for many useful suggestions. During the preparation of this article the author was partially supported by the NSF CAREER grant DMS-0950951535 and the NSF grant DMS 1500031.

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Let V be an n-dimensional C vector space. Let G.k; n/ denote the Grassmannian parameterizing k-dimensional subspaces of V . Let Q be a nondegenerate symmetric or skew-symmetric form. A subspace W is isotropic with respect to Q if w T Qv D 0 for every v; w 2 W . The isotropic Grassmannians OG.k; n/ (respectively, S G.k; n/) parameterize k-dimensional subspaces of V isotropic with respect to the symmetric (respectively, skew-symmetric) form Q. The isotropic Grassmannians naturally include in the Grassmannian G.k; n/. The restriction problem asks to compute the induced map in cohomology in terms of the Schubert bases of the corresponding Grassmannians. The restriction problem was solved for the symmetric case in [C3] and the skewsymmetric case in [C4]. The idea is to give a sequence of explicit rational equivalences that specialize the intersection of a Schubert variety in G.k; n/ with OG.k; n/ or S G.k; n/ into a union of Schubert varieties in OG.k; n/ or S G.k; n/. A similar strategy led to geometric Littlewood–Richardson rules for Grassmannians (see [V1, C1]) and two-step flag varieties (see [C1, CV]). The goal was to understand the cohomology of OG.k; n/ and S G.k; n/ in equally concrete terms. The varieties that occur during the specialization process are called restriction varieties and have since found several other applications. We will introduce restriction varieties and give the solution of the restriction problem in §4. In §5, we will focus on the rigidity problem which asks to classify the Schubert classes in the cohomology of a homogeneous variety that can be represented by subvarieties other than Schubert varieties. This problem dates back at least to Borel and Haefliger [BH] in the 1960s and has recently been answered in the cominuscule case ([Ro, RT], see also [CR, HM] and [MZ]). We will explain geometric approaches to the rigidity problem and discuss the rigidity of Schubert classes in Grassmannians and isotropic Grassmannians. Restriction varieties play an important role here as well by providing explicit deformations of Schubert varieties in some cases. I have tried to preserve the informal nature of the lectures by focusing on examples and important special cases, referring the reader to the literature for proofs and details. I have included many exercises throughout the text. These vary considerably in difficulty. I have also included a variety of open problems. I have used the heading ‘Problem’ to distinguish them from exercises. Organization of the paper. In §2, we will review the background on Grassmannians, isotropic Grassmannians and their cohomology. In §3, we will discuss 4 basic principles that govern the geometry of quadratic forms. In §4, we introduce restriction varieties and describe the solution of the restriction problem. In §5, we introduce the rigidity problem and discuss recent progress towards its solution.

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2 Preliminaries on Schubert varieties In this section, we review basic facts concerning Schubert varieties and the cohomology of homogeneous varieties. We refer the reader to [Bri, EH2] and [GH] for more detailed introductions to the subject. The Grassmannian. Let V be an n-dimensional C vector space. The Grassmannian G.k; n/ parameterizes k-dimensional subspaces of V . It is a smooth, projective varin ety of dimension k.n  k/ and embeds in P.k/1 under the Plücker embedding given by G.k; n/ ! P.^k V / W 7! Œ^k W : The ideal of the Grassmannian under the Plücker embedding is generated by an explicit set of quadratic relations called Plücker relations. The cohomology of G.k; n/ has an additive Z-basis given by the classes of Schubert varieties. An admissible partition for G.k; n/ is a partition  with k parts such that n  k  1  2      k  0: It is customary to omit the parts that are equal to 0 from the notation and we will P follow this custom whenever it is unambiguous. Let jj D kiD1 i denote the weight of the partition. The Young diagram associated to the partition  is an array of k left-justified rows of unit squares with i squares in the i th row. Young diagrams provide a convenient pictorial representation of partitions. A partition is admissible for G.k; n/ if and only if its Young diagram fits in a k  .n  k/ rectangle. Given an admissible partition  and a flag F W F1 F2    Fn D V; we define the Schubert variety † .F / as fW 2 G.k; n/j dim.W \ FnkCi i /  i; 1  i  kg: The Schubert variety † .F / contains a Zariski dense open subset †0 .F / isomorphic to affine space Ak.nk/jj called a Schubert cell. In particular, the Schubert variety † .F / is irreducible of dimension k.n  k/  jj. The Schubert cell †0 .F / parameterizes fW 2 G.k; n/ j dim.W \ FnkCi i / D i; dim.W \ FnkCi i 1 / D i  1; 1  i  kg:

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Exercise 2.1. Fix an ordered basis e1 ; : : : ; en of V . Let Fi be the span of e1 ; : : : ; ei and let ai D n  k C i  i . Show that any subspace W 2 †0 .F / admits a unique basis of the form v1 ; : : : ; vk , where X cj;i ei : vj D eaj C i 

The cohomology class  of † .F / depends only on the partition  and not on the flag F . Furthermore, the Schubert cells with respect to a fixed flag give a cell decomposition of G.k; n/. Consequently, as  varies over all admissible partitions, the cohomology classes  form an additive Z-basis for the cohomology of G.k; n/. Example 2.2. The Schubert varieties in G.2; 4/ (other than all of G.2; 4/ and a point) are †1 .F /; †1;1 .F /, †2 .F / and †2;1 .F /. Interpreting G.2; 4/ as the space of lines in P3 , these Schubert varieties parameterize the following geometric loci:  †1 .F / parameterizes lines that intersect a fixed line PF2 ,  †1;1 .F / parameterizes lines that are contained in a fixed plane PF3 ,  †2 .F / parameterizes lines that contain a fixed point PF1 ,  †2;1 .F / parameterizes lines that contain the fixed point PF1 and are contained in the fixed plane PF3 . In the Plücker embedding, Schubert varieties are cut out on the Grassmannian by linear equations. The image of G.2; 4/ in the Plücker embedding is the quadric hypersurface p12 p34  p13 p24 C p14 p23 D 0: If F is the flag generated by the standard basis, then †1 .F / is defined on the Grassmannian by p34 D 0. Notice that this Schubert variety is isomorphic to a cone over a quadric surface with vertex Œ1 W 0 W 0 W 0 W 0 W 0 corresponding to the line PF2 . Hence, Schubert varieties are in general singular. Similarly, †1;1 .F / and †2 .F / are defined by the linear equations p14 D p24 D p34 D 0 and p23 D p24 D p34 D 0, respectively. Their intersection is †2;1 .F /. Exercise 2.3. Show that in the Plücker embedding the Schubert variety † .F / is cut out on G.k; n/ by the linear equations pj1 ;j2 ;:::;jk D 0, where j1 < j2 <    < jk and there exists at least one index l such that jl > n  k C l  l .

Restriction varieties and the rigidity problem

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Exercise 2.4. Show that linear subspaces on G.k; n/ are Schubert varieties corresponding to partitions with k  n  k  1 or k1 D n  k. Geometrically, these are Schubert varieties that parameterize subspaces that either contain a fixed .k  1/dimensional subspace or are contained in a fixed .k C 1/-dimensional subspace. A Schubert variety † .F / is smooth if and only if it is isomorphic to a subGrassmannian of G.k; n/ parameterizing k-planes that contain a fixed linear space Fs and are contained in a fixed linear space Fm for s  k  m. Equivalently, the complement of the Young diagram of  in the k  .n  k/ rectangle is itself a rectangle (see [BiL, BiC, C2, LS] for more on singularities of Schubert varieties). More generally, we can describe the singular locus of a Schubert variety and give an explicit i resolution. Express the partition  by . i11 ; : : : ; jj /, where 1 > 2 >    > j and the part l occurs with multiplicity il . A singular partition associated to  is a partition whose Young diagram is obtained from the Young diagram of  by adding a single hook. These are the partitions of the form i

lC1 . i11 ; : : : ; . l C 1/il C1 ; lC1

1

i

; : : : ; jj /:

Example 2.5. The singular partitions associated to  D .4; 4; 3; 1/ in G.4; 9/ are the partitions .5; 5; 5; 1/, .4; 4; 4; 4/. Theorem 2.6 ([LS]). The singular locus of the Schubert variety † .F / is [ † .F /; 

where varies over all singular partitions associated to . Theorem 2.6 is easy to deduce from an explicit resolution of singularities. Set P bs D slD1 il . Consider the variety parameterizing the following partial flags fWb1 Wb2    Wbj jWbl FnkCbl l ; 1  l  j g; where dim.Wbl / D bl for 1  l  j . This variety may be constructed as an iterated bundle of Grassmannians, hence, it is smooth. The projection to Wbj defines a surjective birational map onto † .F /. The exceptional locus of this map has codimension at least 2. Therefore, the image of the exceptional locus, which is easy to identify with [ † .F /, is precisely the singular locus of † .F / (see [C2] for further details). Isotropic Grassmannians. Let Q be a nondegenerate symmetric or skew-sym- metric form on V . A symmetric form over C can be diagonalized and up to conjugacy is determined by its rank. In a suitable basis, we can write Q as x12 C    C xn2 :

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The rank of a skew-symmetric form is always even. Hence, if Q is a nondegenerate skew-symmetric form on V , we must necessarily have n D 2m. Skew-symmetric forms can be written in the normal form x1 ^ x2 C    C x2m1 ^ x2m : When Q is symmetric, the equation Q D 0 defines a smooth, quadric hypersurface in PV . We will frequently refer to the geometry of this hypersurface. When Q is skew-symmetric, the form is harder to visualize. A subspace W V is isotropic with respect to Q if v T Qw D 0 for every v; w 2 W . The locus in G.k; n/ parameterizing subspaces W that are isotropic with respect to Q is called the isotropic Grassmannian and denoted by OG.k; n/ and S G.k; n/ depending on whether Q is symmetric or skew-symmetric, respectively. Proposition 2.7. The Grassmannian OG.k; n/ is a smooth variety of dimension dim.OG.k; n// D

k.2n  3k  1/ : 2

If n 6D 2k, then OG.k; n/ is irreducible. When n D 2k, OG.k; n/ has two isomorphic connected components. The Grassmannian S G.k; n/ is a smooth irreducible variety of dimension k.2n  3k C 1/ dim.S G.k; n// D : 2 Exercise 2.8. Consider the incidence correspondence I D f.v; W /jv 2 W g, where W is an isotropic subspace and v is a vector in W . Show that I is isomorphic to an OG.k  1; n  2/ bundle over the quadric hypersurface Q D 0 in the symmetric case and to an S G.k  1; n  2/ bundle over PV in the skew-symmetric case. Deduce Proposition 2.7 by induction on k and n. See [C3] and [C4] for more details. When n D 2k, an individual component of OG.k; 2k/ is called the spinor variety. Two k-dimensional subspaces W1 and W2 belong to the same connected component if and only if dim.W1 \ W2 / D k modulo 2. Example 2.9. A quadric surface in P3 has two one-parameter families of lines. Two lines belong to the same connected component if and only if they are disjoint. Similarly, a quadric fourfold in P5 has two three-parameter families of planes. Two distinct planes belong to the same connected component if and only if they intersect in a point. Exercise 2.10. Using the fact that G.2; 4/ is a quadric fourfold in P5 under the Plücker embedding show that OG.3; 6/ parameterizes planes in G.2; 4/. Conclude that each component of OG.3; 6/ is isomorphic to P3 . The cohomology of isotropic Grassmannians has an additive Z-basis given by the classes of Schubert varieties. We now describe the Schubert varieties in each case.

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Schubert varieties in OG.k; n/. First, we consider OG.k; 2m C 1/. Let 0  s  k be an integer. Let  and denote strictly decreasing sequences of integers m  1 > 2 >    > s > 0;

m  1  sC1 >    > k  0

such that i C j 6D m for any i; j . Schubert varieties in OG.k; 2m C 1/ are indexed by sequences .I /. Fix an isotropic flag ? F W F1    Fm Fm1

   F1? V;

where Fi? D fw 2 V j w T Qv D 0 for all v 2 Fi g denotes the orthogonal to Fi under the form Q. In geometric terms, PFi? is the linear space everywhere tangent to the hypersurface Q D 0 along the linear space PFi . The Schubert variety †I .F / is the Zariski closure of the locus fƒ 2 OG.k; 2m C 1/ j dim.ƒ \ FmC1i / D i for 1  i  s; dim.ƒ \ F?j / D j for s C 1  j  kg: The class I of †I .F / is independent of the isotropic flag and depends only on the partitions. Given  there is an associated sequence m  1  Q sC1 >    > Q m  0 of strictly decreasing integers defined by requiring that there does not exist an index 1  i  s and an index s C 1  j  m such that i C Q j D m. In other words, Q is the sequence obtained from m  1; m  2; : : : ; 1; 0 by omitting the integers m  Q hence we have that l D Q i . s ; : : : ; m  1 . The sequence is a subsequence of , l Define the discrepancy of the pair .; / by dis.; / D .m  k/s C

k X

.m  k C l  il /:

lDsC1

Then the codimension of †I .F / in OG.k; 2m C 1/ is s X

i C dis.; /:

i D1

When k D m, is uniquely determined by  and authors often omit in this case. We will not follow this convention. Example 2.11. The Schubert varieties in OG.2; 5/ (other than the whole space and a point) are †1I0 and †2I1 . They parameterize lines on a quadric threefold that intersect a fixed line PF2 and lines that contain a point PF1 (and are automatically contained in the tangent hyperplane PF1? at that point), respectively.

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Exercise 2.12. By induction on P k, calculate the dimension of †I OG.k; n/ and verify that its codimension is siD1 i C dis.; /: Next, we consider OG.k; 2m/. In this case, m-dimensional linear spaces form two connected components. We can specify incidence conditions with respect to mdimensional linear spaces in either component. Our notation needs to reflect this difference. We denote the linear spaces in one connected component by Fm and the ? . Technically, the interseclinear spaces in the other connected component by Fm1 tion of a linear space tangent along Fm1 intersects Q D 0 in the union of two half-dimensional linear spaces belonging to different connected components. However, this abuse will simplify the notation greatly. Let 0  s  k be an integer and let  and denote sequences of strictly decreasing integers m  1  1 >    > s  0

m  1  sC1 >    > k  0

such that i C j 6D m  1 for any i; j . Schubert varieties in OG.k; 2m/ are indexed by such sequences .I /. In order to isolate the Schubert varieties in the spinor variety, in addition we need to assume that when k D m and m is even (respectively, odd), s is even (respectively, odd). The Schubert variety †I .F / is defined as the Zariski closure of the locus fƒ 2 OG.k; 2m/ j dim.ƒ \ Fmi / D i for 1  i  s; dim.ƒ \ F?j / D j for s < j  kg: Example 2.13. In OG.2; 6/, the Schubert varieties (other than the whole space or a point) are as follows: (1) The Schubert varieties †I2;0 ; †0I0 parameterize lines on a quadric fourfold that intersect a fixed plane. The class depends on the type of plane. (2) The Schubert varieties †I2;1 ; †0I1 parameterize lines that are contained in the linear space PF1? and intersect a plane. (3) The Schubert variety †1I0 parameterizes lines that intersect a fixed line. (4) The Schubert varieties †1I2 , †1;0I parameterize lines that are contained in a fixed plane. (5) The Schubert varieties †2;0I , †2I2 parameterize lines that are contained in a plane and contain a fixed point PF1 . As in the previous case, given , there is a corresponding partition m  1  Q sC1 >    > Q m  0 satisfying the condition that there does not exist indices 1  i  s and s C1  j  m such that i C Q j D m  1. The sequence Q is obtained from m  1; m  2; : : : ; 1; 0

Restriction varieties and the rigidity problem

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by removing the integers m  1  s ; : : : ; m  1  1 . The partition .; / is a Q of total length m. Hence, l D Q i . Define the discrepancy subpartition of .; / l P dis.; / D s.m  k/ C klDsC1 .m  k C l  il /. The codimension of †I in OG.k; 2m/ is given by s X i C dis.; /: i D1

Schubert varieties in SG.k; n/. Since Q is a nondegenerate skew-symmetric form, we must have n D 2m. Let 0  s  k be an integer. Let  and be strictly decreasing sequences m  1 >    > s > 0 m > sC1 >    > k  0 such that i C j 6D m for any i; j . Let F be an isotropic flag ?

   F1? V: F1    Fm Fm1

Then the Schubert variety †; .F / is the Zariski closure of the locus fƒ 2 S G.k; n/ j dim.ƒ \ FmC1i / D i for 1  i  s; dim.ƒ \ F?j / D j for s < j  kg: The discrepancy is defined as in OG.k; 2m C 1/ and the codimension of the Schubert variety in S G.k; n/ is given by s X

i C dis.; /:

i D1

Cell decompositions. In all three cases, there is a partial ordering on the sequences .I /. For S G.k; n/ and OG.k; 2m C 1/, a sequence .0 I 0 /  .I / if and only if s 0  s, 0i  i for 1  i  s and 0j  j for s 0 < j  k. For OG.k; 2m/ in addition we need to require that if sC1 D m  1 and s 0 D s C 1, then 0sC1 > 0. A Schubert variety †0 I0 .F / †I .F / if and only if .0 I 0 /  .I /. The complement [ †I .F /  †0 I0 .F / .0 I0 />.I/

is called a Schubert cell and is isomorphic to affine space. The Schubert cells give a cell decomposition of OG.k; n/ and S G.k; n/. Consequently, Schubert classes give an additive Z-basis of the cohomology of OG.k; n/ and S G.k; n/. Exercise 2.14. When n D 2m C 1, after a change of variables, write Q as x1 x2 C 2    C x2m1 x2m C x2mC1 . Show that the complement of the locus x1 D x3 D    D x2i C1 D 0 in the locus x1 D x3 D    D x2i 1 D 0 is isomorphic to affine space. Generalize this to show that Schubert cells give a cell decomposition of OG.k; 2m C 1/. Further generalize to OG.k; 2m/ and S G.k; n/.

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Flag varieties. Let k1 < k2 <    < kt < n be a sequence of t increasing positive integers. For notational convenience, set k0 D 0 and kt C1 D n. The flag variety F .k1 ; : : : ; kt I n/ parameterizes partial flags Wk1 Wk2    Wkt V; where Wki has dimension ki . For any set of subindices i1 ; : : : ; il of f1; : : : ; tg, the flag variety admits a forgetful morphism i1 ;:::;il W F .k1 ; : : : ; kt I n/ ! F .ki1 ; : : : ; kil I n/ .Wk1 ; : : : ; Wkt / 7! .Wki1 ; : : : ; Wkil /: In particular, the projection t W F .k1 ; : : : ; kt I n/ ! G.kt ; n/ realizes the partial flag variety F .k1 ; : : : ; kt I n/ as a F .k1 ; : : : ; kt 1 I kt / bundle over the Grassmannian G.kt ; n/. By induction, dim.F .k1 ; : : : ; kt I n// D

t X

ki .ki C1  ki /:

i D1

The cohomology of a partial flag variety is generated by the classes of Schubert varieties. We use a notation for Schubert varieties in F .k1 ; : : : ; kt I n/ which is well-adapted for the forgetful morphism t . A coloring c associated to a sequence k1 ; : : : ; kt is a sequence of kt integers c1 ; : : : ; ckt such that exactly ki  ki 1 of the integers in the sequence are equal to i . Let  be an admissible partition for G.kt ; n/ and c a coloring for the sequence k1 ; : : : ; kt . Schubert varieties in F .k1 ; : : : ; kt I n/ are parameterized by colored partitions .I c/. Let F be a complete flag, then the Schubert variety †Ic .F / is the Zariski closure of the locus parameterizing Wk1

   Wkt 2 F .k1 ; : : : ; kt I n/ such that dim.Wkj \ Fnkt Ci i / D #fcl jcl  j; l  i g for 1  j  t; 1  i  kt : For 1  u < t, define the codimension of the color u cdim.u/ by X cdim.u/ D #fj > i jcj D u C 1g: 1i kt ; ci u

Define the codimension of the coloring cdim.c/ by cdim.c/ D

t 1 X

cdim.u/:

uD1

Exercise 2.15. By analyzing the projection t show that the codimension of the Schubert variety †Ic in the flag variety is given by jj C cdim.c/: Exercise 2.16. Show that F .1; 3I 4/ is isomorphic to OG.2; 6/. Describe the correspondence between the Schubert varieties under this isomorphism.

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The isotropic partial flag varieties OF .k1 ; : : : ; kt I n/ and SF .k1 ; : : : ; kt I n/ parameterize isotropic partial flags Wk1    Wkt V; where Wki is an isotropic subspace of dimension ki with respect to a symmetric, respectively, skew-symmetric, non-degenerate quadratic form. The forgetful morphisms t W OF .k1 ; : : : ; kt I n/ ! OG.kt ; n/ and t W SF .k1 ; : : : ; kt I n/ ! S G.kt ; n/ realize these varieties as F .k1 ; : : : ; kt 1 ; I kt /-bundles over the isotropic Grassmannians. Consequently, we have kt .2n  3kt  1/ X ki .ki C1  ki /; C dim.OF .k1 ; : : : ; kt I n// D 2 t 1

i D1

kt .2n  3kt C 1/ X ki .ki C1  ki /: C dim.SF .k1 ; : : : ; kt I n// D 2 t 1

i D1

Schubert varieties in these varieties can be parameterized by triples .I I c/, where .I / is a pair of partitions admissible for OG.kt ; n/ or S G.kt ; n/ and c is a coloring of the sequence k1 ; : : : ; kt . The Schubert variety †IIc .F / with respect to the isotropic flag F is the Zariski closure of the locus of flags satisfying the following incidence conditions dim.Wkh \ Fd n2 ei / D #fcl jcl  h; l  i g dim.Wkh \ F?j / D #fcl jcl  h; l  j g: As Ps before, the codimension of the Schubert variety in the flag variety is given by i D1 i C dis.; / C cdim.c/: Dual Schubert classes. Every Schubert class in a rational homogeneous variety has a dual Schubert class. Two Schubert classes of complementary dimension have intersection number equal to one if and only if they are dual Schubert classes. Otherwise, their intersection is zero. For the reader’s convenience, we will explicitly describe the dual Schubert class in every classical case. The dual classes for G.k; n/ have the following description. Given a partition , the dual partition  is defined by i D n  k  ki C1 . The dual partition is the partition corresponding to the 180 degree rotation of the complement of  in the k  .n  k/ rectangle. Then the dual of  is  . Example 2.17. The classes 3;3;2;1 and 4;3;2;2 are dual in G.4; 9/. Similarly, in OG.k; 2m C 1/ or S G.k; 2m/ the dual of the Schubert class I is given by  ; , where i D m  ki C1 and j D m  kj C1 . Example 2.18. The classes 2I0 and 3;1 are duals of each other in OG.2; 7/.

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Exercise 2.19. Describe the dual Schubert classes for OG.k; 2m/. (Hint: If m is even or if s 6D 0 and sC1 6D m  1, then define i D m  1  ki C1 and j D m  1  kj C1 . When m is odd, you need to modify the half dimensional linear spaces.) Given a coloring c for k1 ; : : : ; kt , the dual coloring c  has ci D ckt i C1 . In other words, the dual coloring reverses the order of c. Given a Schubert class Ic in a partial flag variety or a Schubert class IIc in an isotropic flag variety, the dual class is given by  Ic  and  I Ic  , respectively. Proposition 2.20. Let Y be a subvariety of a rational homogeneous variety. Then the cohomology class of Y is a nonnegative linear combination of Schubert classes. In particular, for rational homogeneous varieties, the cones of effective and nef cycles coincide and are generated by Schubert classes. Proof. Since Schubert classes give an additive basis of the cohomology, Pwe can express the class of Y as a linear combination of Schubert classes ŒY  D a  . For each Schubert class  , we can pair ŒY  by the dual Schubert class  . By Kleiman’s Transversality Theorem [Kl], a general translate of a Schubert variety with class  intersects Y transversely in finitely many points. Hence,   ŒY  D a  0:  As in the proof of Proposition 2.20, we can compute the classes of subvarieties of rational homogeneous varieties by pairing with dual Schubert classes. Exercise 2.21. Compute the cohomology class of OG.k; n/ and S G.k; n/ in G.k; n/. Exercise 2.22. Compute the intersection products in the cohomology of G.2; 4/ and G.2; 5/. Exercise 2.23. Compute the intersection products in the cohomology of OG.2; 5/, OG.3; 6/ and OG.2; 6/.

3 The Golden Rules of quadric geometry There are four general principles that govern the geometry of isotropic linear spaces with respect to a quadratic form. In this section, we will recall these principles. In the next section, these principles will dictate the geometry of restriction varieties and their specializations. We begin by discussing symmetric forms. Let Q be a nondegenerate symmetric form. Let Qdr denote a d -dimensional linear space such that the restriction of Q is a quadratic form of corank r, equivalently rank d  r. Up to conjugacy, the form in Qdr is x12 C    C xd2 r . The singular locus of the corresponding quadric hypersurface in PQdr is defined by x1 D    D xd r D 0. This is the kernel or vertex K of the quadratic form restricted to Qdr .

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The corank bound. Let Qdr22 Qdr11 be two linear spaces such that K1 K2 . Then r2  r1  d2  d1 : In particular, the corank r of a linear section of a nondegenerate quadric hypersurface is bounded by its codimension n  d . Exercise 3.1. Verify that the tangent hyperplane section Tp Q\Q to a smooth quadric hypersurface Q is singular only along p. More generally, let Q be a quadric hypersurface of corank r and vertex W . Let p be a smooth point on Q. Then the tangent hyperplane section Tp Q \ Q is singular along the span of p and W . Using this verify the corank bound. The linear space bound. The largest dimensional isotropic subspace contained in Qdr has dimension b d Cr c. Furthermore, an isotropic subspace of dimension j in2 tersects the kernel of the quadratic form in Qdr in a subspace of dimension at least max.0; j  b d r c/. 2 p c and xd r D 0 Exercise 3.2. By taking x2j 1 D 1 x2j for j D 1; : : : ; b d r 2 if d  r is odd, show that there are isotropic subspaces of dimension b d Cr c in Qdr . 2 Either by induction on dimension or by the Lefschetz hyperplane theorem, show that a smooth quadric hypersurface does not contain any linear spaces of more than half the dimension. Let Qd0 r be the linear space of Qdr defined by xd rC1 D    D xd D 0. Notice that the quadratic form restricted to Qd0 r is nondegenerate, therefore, the largest dimensional isotropic subspace in Qd0 r has dimension b d r c. Deduce the 2 linear space bound. Irreducibility. The quadratic form restricted to Qdd 2 is a product of two linear forms, which define two isotropic subspaces of dimension d  1. If n D 2k, then the two linear spaces belong to distinct connected components of OG.k; 2k/. The quadratic form restricted to Qdd 1 is the square of a linear form. The variation of tangent spaces. Let X denote the quadric hypersurface in PQdr defined by the restriction of the quadratic form Q to Qdr . Assume that the kernel K of Qdr intersects an isotropic linear space L in a subspace of codimension j in L. Then the image of the Gauss map of X restricted to PL has dimension at most j  1. In other words, the tangent spaces to the quadric hypersurface X at the points of PL where X is smooth vary in at most a .j  1/-dimensional family. Exercise 3.3. Consider the quadric hypersurface xy z 2 in P3 . Show that the tangent space to the surface along the line x  az D y  a1 z D 0 is constant. Generalize the calculation to arbitrary dimension and rank to deduce the principle of variation of tangent spaces.

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There are similar basic principles for skew-symmetric forms. We explain them next. Let Qdr denote a d -dimensional subspace of V such that the restriction of the skew-form Q to Qdr has corank r. Let Ker.Qdr / denote the kernel of the restriction of Q to Qdr . Evenness of rank. The rank of a skew-symmetric form is even. Consequently, d  r is even. In particular, after a change of variables, we can write the restriction of the form Q to Qdr as x1 ^ x2 C    C xd r1 ^ xd r : in these coordinates, the kernel is given by x1 D    D xd r D 0. The corank bound. Let Qdr22 Qdr11 and let r10 D dim.Ker.Qdr11 / \ Qdr22 /. Then r2  r10  d1  d2 : In particular, d C r  n for any Qdr . The linear space bound. The dimension of an isotropic subspace of Qdr is bounded above by b d Cr c. A j -dimensional isotropic linear subspace of Qdr satisfies dim.L \ 2 Ker.Qdr //  j  b d r c. 2 The kernel bound. Let L be a j -dimensional isotropic subspace such that dim.L \ Ker.Qdr // D j  1. Then an isotropic linear subspace that intersects L  Ker.Qdr / is contained in L? . Exercise 3.4. Prove the corank bound, the linear space bound and the kernel bound following a similar strategy to the symmetric case.

4 Restriction problem Let X denote OG.k; n/ or S G.k; n/. Let i W X ! G.k; n/ denote the natural inclusion. The restriction problem asks to compute the induced map in cohomology i  W H  .G.k; n/; Z/ ! H  .X; Z/. In particular, Schubert classes provide additive Z-bases for both H  .G.k; n/; Z/ and H  .X; Z/. Given a Schubert class  2 H  .G.k; n/; Z/, X  i  . / D a  2 H  .X; Z/;  are nonnegative, integer coefficients. The restriction problem asks for poswhere a  itive, geometric algorithms for computing the coefficients a . In this section, we will explain the solution of the restriction problem. We call  of the map induced by these inclusions the restriction cothe structure constants a efficients. By Poincaré duality, the restriction problem can be described in concrete

Restriction varieties and the rigidity problem

63

geometric terms. Let † be a general Schubert variety in G.k; n/. Then the class i   is the cohomology class of the intersection † \ i.X /. We will compute the class of this intersection via a sequence of specializations. We refer the reader to [FP, Pr1] and [Pr2] for alternative approaches to the restriction problem. There is a close connection between the restriction problem and the branching problem in representation theory. The reader may consult [BS, C2] and [Pu] for more details. The basic strategy is to give explicit rational equivalences between † \ i.X / and a union of Schubert varieties in X . This strategy has been very fruitful in obtaining geometric Littlewood–Richardson rules for Grassmannians and two-step flag varieties (see [C1, CV] and [V1]) and for computing Gromov–Witten invariants and quantum cohomology (see [C1, C7, C8] and [V2]). We will not discuss these rules here and refer the reader to the literature. We start with two fundamental examples which capture the main aspects of the algorithm for computing restriction coefficients. Example 4.1. Consider a general hyperplane section H \ Q of a quadric in P3 . This is a smooth conic curve and is not a Schubert variety in OG.1; 4/. Vary H in a oneparameter family Ht until H0 becomes tangent to Q. Then H0 \ Q D L [ L0 is a union of two lines one from each ruling. These are Schubert varieties in OG.1; 4/. The class of the conic H \ Q is equal to the sum of the classes of L and L0 . Example 4.2. Consider the locus of lines in a quadric Q in P6 such that the lines intersect a fixed line L and are contained in a general hyperplane section H \ Q containing L. This is not a Schubert variety in OG.2; 7/. Take a one-parameter family of hyperplanes Ht such that H0 is tangent to Q along a point p 2 L. Consider the varieties of lines that are contained in Ht and intersect L. In the limit t D 0, the lines may contain p. In that case, the lines are contained in H0 and contain p. This is a Schubert variety with class 3I1 . Otherwise, the lines intersect L away from p. In that case the lines must be contained in L? by the principle of variation of tangent spaces. This locus is a Schubert variety with class 2I2 . We conclude that the original variety has class 3I1 C 2I2 . Exercise 4.3. Verify the details of Example 4.2. 4.1 The symmetric case. We will compute the restriction coefficients by a sequence of specializations. We begin with the intersection of a general Schubert variety † .F / with OG.k; n/. Initially, every linear space in the flag F defining † is transverse to the quadric Q. We will successively change the flag by making the linear spaces less and less transverse to Q until the flag becomes isotropic. In the process, the variety will break into a union of Schubert varieties for OG.k; n/. Restriction varieties are the components of the limits that occur during the process. The order of specialization and the limits are dictated by the principles discussed in the previous section. We will now introduce restriction varieties and discuss their basic geometric properties. Notation 4.4. Let Q be a symmetric nondegenerate quadratic form on an n-dimensional vector space V . Let Lni denote an isotropic subspace of dimension ni . When

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2m D n, let Lm and L0m denote isotropic subspaces in different connected compor nents. Let Qdjj denote a dj -dimensional subspace of V such that the restriction of Q r

to Qdj has corank rj with kernel Kj . Note that the dimension of Kj is rj . j

Definition 4.5. An orthogonal sequence r

   Qdr11 .L ; Q / D Ln1    Lns Qdks ks is a sequence of isotropic subspaces Lni of dimension ni (or possibly L0ns if 2ns D n) r r and linear spaces Qdj of dimension dj such that the restriction of Q to Qdj has j j kernel Kj and corank rj satisfying the following conditions. r

(1) The singular loci of Qdj are ordered by inclusion Kj Kj C1 for 1  j < ks. j

(2) For every 1  i  s and 1  j  k  s, dim.Kj \ Lni / D min.ni ; rj /. (3) If rj D rj C1 > 0 for some j , either r1 D rj and nr1 D r1 ; or rl  rl1 D dl1  dl for every l  j and dj  dj C1 D 1. r

Remark 4.6. The fact that Lni is an isotopic subspace contained in Qdjj implies certain inequalities among ni ; dj and rj . The corank bound implies that rj C dj  rj 1 C dj 1  n

for 1 < j  k  s:

The linear space bound implies that 2ns  dks C rks : We will always implicitly assume these inequalities. The first two conditions say that the kernels Kj are in as special a position as possible. They are totally ordered by inclusion and they have the maximal possible intersection with the isotropic subspaces in the sequence. The third condition is a consequence of the order of specialization. r We will not specialize a linear space Qdj until the kernels Kl for l > j are as large j as possible given the corank bound. Notation 4.7. Let xj denote the number of isotropic subspaces in the sequence that are contained in Kj , equivalently, xj is the number of ni for 1  i  s such that ni  rj . Definition 4.8. We call the sequence .L ; Q / admissible if it satisfies the following conditions. (A1) For every 1  j  k  s, we have xj  k  j C 1 

dj  rj : 2

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Restriction varieties and the rigidity problem

(A2) There does not exist integers i; j such that rj C 1 D ni . (A3) We have rks  dks  3. Definition 4.9. The restriction variety V .L ; Q / associated to an admissible orthogonal sequence .L ; Q / is the subvariety of OG.k; n/ defined as the Zariski closure of the following locus r

V .L ; Q /0 DfW 2 OG.k; n/ j dim.W \ Lni / D i; dim.W \ Qdj / D k  j C 1; j

dim.W \ Kj / D xj g Remark 4.10. Condition (A1) is a consequence of the linear space bound. The isotropic subspace of dimension k  j C 1 has to intersect the kernel Kj in a subspace d r of dimension at least k  j C 1  j 2 j . Condition (A3) guarantees that the rer striction of Q to Qdks is not a nonreduced isotropic space or the union of two ks isotopic subspaces. Condition (A2) is a consequence of variations of tangent spaces. If rj C 1 D ni , then any linear space intersecting Lni away from Kj is contained in L? ni . Exercise 4.11. The restriction varieties do not need to be irreducible. Show that V .Q30 Q40 / in OG.2; 5/ has two connected components. We introduced the notion of a marked sequence to describe the irreducible components of restriction varieties. Definition 4.12. Let .L ; Q / be an admissible orthogonal sequence. An index j such that dj  rj xj D k  j C 1  2 is called a special index. A marking m of .L ; Q / for each special index j desigd Cr nates one of the irreducible components of the . j 2 j /-dimensional isotropic subrj spaces of Qd as even and the other one as odd such that: j

 If dj1 C rj1 D dj2 C rj2 for two special indices j1 < j2 , then the component containing a linear space W is assigned the same parity for both indices.  If 2ns D dj C rj for a special index j , the component which contains Lns is assigned the parity of s.  If n D 2k, then the component containing Lk is assigned the parity that characterizes the spinor variety containing Lk . A marked restriction variety V .L ; Q ; m / is the subvariety of V .L ; Q / parameterizing k-dimensional isotropic subspaces W such that for each special index j , d Cr W intersects isotropic subspaces of dimension j 2 j designated even (respectively, odd) by m in a subspace of even (respectively, odd) dimension.

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Exercise 4.13. Show that for the restriction variety in Exercise 4.11 there are two possible markings and these distinguish the two irreducible components. Proposition 4.14. The marked restriction variety V .L ; Q ; m / is an irreducible subvariety of OG.k; n/ of dimension dim.V .L ; Q ; m // D

s X i D1

.ni  i / C

ks X

.dj C xj  2s  2j /

j D1

Sketch of proof. The dimension and the irreducibility can be checked by induction on k. When k D 1, V .L ; Q ; m / is isomorphic to either a projective space of dimension n1  1 or an irreducible quadric surface of dimension d1  2. In this case, the proposition is true. Suppose that the proposition holds for k < k0 . Suppose V .L ; Q ; m / is a marked restriction variety in OG.k0 ; n/. If s D k0 , then it is isomorphic to a Schubert variety in G.k0 ; Lns /. Consequently, it is irreducible of the claimed dimension. Otherwise, we can define a new sequence .L0; Q0 ; m0 / by omitting Qdr11 . The reader can easily check that the resulting sequence is still an admissible orthogonal sequence. Sending W 2 V .L ; Q ; m /0 to W \ Qdr22 defines a morphism onto V .L0 ; Q0 ; m0 /0 , where the generic fiber over W 0 consists of choosing W 0 W Qdr11 . The proposition follows from this description and the Theorem on the Dimension of Fibers [Sh, Theorem I.6.7].  Remark 4.15. The dimension of a marked restriction variety does not depend on the marking. Example 4.16. Set u D d n2 e. If the sequence .L ; Q / satisfies dj C rj D n for 1  j  k  s, then .L ; Q / is an isotropic flag. In that case the restriction variety V .L ; Q / is simply the Schubert variety †un1 ;:::;uns Irks ;:::;r1 defined with respect to this isotropic flag. Example 4.17. The intersection of a general partial flag Fa1    Fak leads to the sequence Qa01    Qa0k . If ai  2i  1 for some i , then the intersection is empty. If ai  2i for every 1  i  k, then the intersection is nonempty of the expected dimension. Restriction varieties are in general singular. S. Adalı [Ad] has given an explicit resolution and described their singular loci. Quadric diagrams. It is convenient to record admissible sequences in terms of combinatorial objects called quadric diagrams. Consider a sequence of n integers written from left to right. The place of the integer is its order in the sequence counted from the left. We will allow brackets  and braces g between certain integers in the sequence. We say that a bracket or a brace is in position i if i of the integers are to the left of the bracket or brace.

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Definition 4.18. An orthogonal sequence of brackets and braces of type .k; n/ is a sequence of n natural numbers, s  k right brackets  and k  s right braces g such that  Every bracket or brace occupies a positive position and each position is occupied by at most one bracket or brace. If n D 2m, a bracket at position m may be decorated by a prime 0 .  Every number i in the sequence satisfies 0  i  k  s. The positive integers are nondecreasing from left to right and are to the left of every zero in the sequence.  Every bracket is to the left of every brace. Notation 4.19. By convention, the brackets are indexed from left to right and the braces are indexed from right to left. Let i and gi denote the i th brace and bracket and let p.i / and p.gi / denote their positions, respectively. In a sequence of brackets and braces of type s for OG.k; n/, we make the convention that gksC1 denotes s and k  s C 1 should be read as 0. This convention will allow us to state certain combinatorial rules more succinctly. Let l.i / and l. i / denote the number of positive integers in the sequence that are equal to i and less than or equal to i , respectively. For i > j > 0, let .i; j / denote the number of integers between gi and gj . Let .j; 0/ denote the number of integers to the right of gj . For example, for 111223300000g00g00g0000 we have p.1 / D 1, p.2 / D 2, p.3 / D 5, p.4 / D 7, p.g3 / D 12, p.g2 / D 14, p.g1 / D 16. We also have l.1/ D 3, l.2/ D 2, l.3/ D 2, .3; 2/ D 2, .2; 1/ D 2, .1; 0/ D 4. When we are discussing more than one sequence, we will write pD , D and lD to indicate that we are referring to the invariants of D. Definition 4.20. An orthogonal quadric diagram for OG.k; n/ is an orthogonal sequence of brackets and braces of type .k; n/ with s brackets such that the following conditions hold. (D1) l.j /  .j; j  1/ for 1  j  k  s. (D2) 2p.s /  p.gks / C l. k  s/. (D3) Suppose that integer 1  j < k  s occurs in the sequence. If j C 1 does not occur in the sequence, either j D 1 and every position after a 1 is occupied with a ; or l.i / D .i; i  1/ for every j C 1 < i  k  s and .j C 1; j / D 1. Definition 4.21. An orthogonal quadric diagram is admissible if it satisfies the following additional conditions. (A1) Let xj denote the number of brackets such that p.i /  l. j /. Then xj  k  j C 1 

p.gj /  l. j / : 2

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(A2) The two integers to the left of a bracket are equal. If there is only one integer to the left of a bracket and s < k, then the integer is 1. (A3) There are at least three zeros to the left of gks . Example 4.22. In the diagrams 12000g00g00 and 00000g0 condition (A2) is violated. In the diagram 200g00g00 condition (A3) is violated. The translation between admissible sequences .L ; Q / and quadric diagrams is straightforward. Given an admissible sequence .L ; Q /, the corresponding quadric diagram D.L ; Q / is obtained as follows. The sequence of integers starts with r1 integers equal to 1, followed by ri  ri 1 integers equal to i in increasing order. The sequence ends with n  rks integers equal to 0. The are s brackets in positions ni and k  s braces in positions dj . Conversely, given an admissible quadric diagram D, the associated admissible sequence .L ; Q /.D/ has an isotropic linear space Lp.i / of dimension p.i / for l.j / j each bracket i in D and a linear space Qp.g j / of dimension p.g / of corank l. j /

for each brace gj in D. Exercise 4.23. Show that admissible sequences correspond to admissible quadric diagrams (see [C3]). Definition 4.24. An admissible quadric diagram is saturated if l.j / D .j; j  1/ for every 1  j  k  s. Exercise 4.25. Show that Example 4.16 translates to the following lemma. Lemma 4.26. A saturated admissible quadric diagram represents a Schubert variety. Now we will describe a combinatorial process for computing the class of a restriction variety in terms of Schubert classes. The algorithm will be recorded in terms of quadric diagrams. Every quadric diagram will be the root of a tree of quadric diagrams where each leaf terminates in a saturated admissible quadric diagram. The class of the restriction variety will be the sum of the Schubert classes summed over the leaves of this tree. We begin by defining two new quadric diagrams. Definition 4.27. Fix an admissible quadric diagram D. Assume that D is not saturated, then there exists an index j such that l.j / < .j; j  1/. Let

D maxfj j l.j / < .j; j  1/g: Define D a to be the sequence of brackets and braces obtained from D by changing the .l. / C 1/st integer in D to . If pDa .s / > lDa . /, let  D minfi j pDa .i / > lDa . /g:

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Define D b to be the sequence of brackets and braces obtained from D a by moving the bracket  to position pDa . /. Otherwise, D b is not defined. Example 4.28. Let D D 000000g00g00. Then D 2 and D a D 200000g00g00 D b D 200000g00g00: Let D D 2233000g00g00g00. Then D 1 and D a D 1233000g00g00g00 D b D 1233000g00g00g00: Let D D 000g00g00. Then D 2, D b is not defined and D a D 200g00g00. Let D D 0000g00. Then D 1, D a D 1000g00 and D b D 1000g00. We see that D a and D b may fail to be admissible. Notice that D a may fail all three conditions in Definition 4.21, whereas D b may fail only (A2). We first give an algorithm that will replace D a and D b with a set of admissible quadric diagrams. Algorithm 4.29. Input: A quadric diagram D. (1) If D does not satisfy condition (A1), discard D and do not return any diagrams. If D satisfies condition (A1), proceed to Step 2. (2) If D does not satisfy condition (A2), let i0 be the index of the minimal index bracket for which (A2) fails and let  be the integer immediately to the left of i0 . If  > 0 (resp.  D 0), replace  by   1 (resp. k  s), move g1 (resp. gks ) one unit to the left provided that the position p.g1 /  1 (resp. p.gks /  1) is unoccupied. Return the resulting diagram to Step 1. If position p.g1 /  1 (resp. p.gks /  1) is occupied, discard the diagram and do not return any diagrams. If condition (A2) is satisfied, proceed to Step 3. (3) If D does not satisfy (A3), replace D with two identical diagrams obtained from D by replacing gks in position pD .gks / with sC1 in position p D pD .gks /  1. If n is even and 2p D n, then in one of the diagrams decorate sC1 with a prime. Return the resulting diagrams to Step 1. If D satisfies (A3), output the diagram. Example 4.30. We apply Algorithm 4.29 to the diagrams in Example 4.28 which are not admissible. 200000g00g00 ! 22000g000g00 200000g00g00 ! 100000g0g000: 1233000g00g00g00 ! 1133000g00g0g000: 2

200g00g00 ! 00000g00: In order to save space, when we replace a diagram with two identical diagrams we will write 2 over the arrow rather than drawing the diagram twice. 1000g00 ! The diagram is discarded since it does not satisfy (A1).

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Exercise 4.31. Run Algorithm 4.29 on the following diagrams 0000g00g0 22330000g000g00g00: Exercise 4.32. Show that Algorithm 4.29 terminates in a (possibly empty) set of admissible quadric diagrams. (Hint: At each stage either the diagram is discarded, the number of braces decreases or a brace moves to the left. None of these steps can be repeated infinitely often; see [C3]). The set of admissible quadric diagrams derived from D a (resp. D b ) are the set of quadric diagrams output by Algorithm 4.29 with input D a (resp. D b ). Definition 4.33. Let D be a sequence of brackets and braces such that p.s / > l. /. If l. i / < p.x C1 / for 1  i  k  s, then set yx C1 D k  s C 1. Otherwise, let yx C1 D maxfi j l. i /  p.x C1 /g: Recall that x C1 is the first bracket in a position greater than l. /. If the integer to the immediate left of x C1 is 0, then yx C1 D k  s C 1. Otherwise, yx C1 is the integer to the immediate left of x C1 . We are now ready to state the main algorithm. Algorithm 4.34. Input: An admissible quadric diagram D. (1) If D is saturated, return D and stop. Otherwise, proceed to Step 2. (2) If p.s /  l. / or p.x C1 /  l. /  1 > yx C1  , then return the quadric diagrams derived from D a to Step 1. Otherwise, return the quadric diagrams derived from both D a and D b to Step 1. Successive runs of Algorithm 4.34 either decrease the number of braces or increase the number of positive integers in the sequence. Since neither of these can go on for ever, the algorithm terminates in a set of saturated, admissible quadric diagrams. It is easy to check that if D is an admissible quadric diagram, then each run of the algorithm outputs at least one admissible quadric diagram. Definition 4.35. A degeneration path is a sequence of admissible quadric diagrams D1 ! D 2 !    ! D r such that Di C1 is one of the outputs of running Algorithm 4.34 on Di for 1  i < r. Theorem 4.36 ([C3, Theorem 5.12]). Let V .L ; Q / be a restriction variety and let D be the corresponding admissible quadric diagram. Then X ŒV .L ; Q / D cI I ; where cI is the number of degeneration paths starting with D and ending with the quadric diagram associated to the Schubert class I .

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The proof of Theorem 4.36 is obtained by interpreting Algorithm 4.34 as a specialization. Consider the one-parameter family of partial flags .L ; Q /.t/, where all the flag elements are fixed except for Qdr and Qdr varies in a pencil that becomes tangent to Q in one larger dimension at t D 0. The algorithm describes the flat limit of the restriction varieties defined with respect to the flags .L ; Q /.t/ at t D 0. Exercise 4.37. Carry out this specialization for the Examples 4.1 and 4.2. Show that the descriptions in these examples agree with Algorithm 4.34. We now discuss several consequences of Theorem 4.36. The first corollary is a solution of the restriction problem. The pullback of a Schubert class in G.k; n/ can be expressed as the class of a sum of restriction varieties. Suppose that the Schubert variety in G.k; n/ is defined with respect to a general partial flag Fa1 DnkC11 Fa2 DnkC22    Fak Dnk : The restriction of Q to the flag elements Fi is a non-degenerate quadratic form Qa0i . If ai  2i  1 for some i , then i   D 0. Suppose that ai  2i for 1  i  k. We associate the sequence Qa01 Qa02    Qa0k to the class  . Running Algorithm 4.29 on the corresponding quadric diagram produces a collection of admissible quadric diagrams. The class i   is the sum of the Schubert classes obtained by running Algorithm 4.34 on this collection of admissible quadric diagrams. We conclude the following theorem. Theorem 4.38. Algorithm 4.34 gives a positive, geometric rule for computing restriction coefficients. Example 4.39. Our first example computes the class i  .2;1 / D ŒV .Q30 Q50 / D 22I1 C 21;0I C 21I2 in OG.2; 6/. 2

000g00g0 ! 00000g0 ! 10000g0 . & 000000 0000000 Example 4.40. The next example calculates i  .3;2;1/ D ŒV .Q40 Q60 Q80 / D 24I3;1 C 23I3;2 C 42;1I1 in OG.3; 9/. We run the algorithm on 0000g00g00g0 and obtain the following. 2

2

3000g00g00g0 ! 000000g00g0 ! 200000g00g0 ! 00000000g0 ! 10000000g0 # 220000g00g0 ! 120000g00g0 ! 122000g00g0 # 110000g0g00 ! 112000g0g00

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Exercise 4.41. Compute all the restriction coefficients for the inclusions i W OG.2; 5/ ! G.2; 5/;

i W OG.2; 6/ ! G.2; 6/ i W OG.3; 7/ ! G.3; 7/:

As an another application of Theorem 4.36, one can compute the classes of the moduli space of vector bundles of rank two on hyperelliptic curves in OG.g 1; 2g C 2/. Let Q1 and Q2 be two general quadric hypersurfaces in P2gC1 and consider the pencil they generate. Let I be the incidence correspondence consisting of pairs .Q; L/, where Q is a quadric of this pencil and L is a connected component of the space of g-dimensional projective linear spaces on Q. Since the half-dimensional linear spaces on Q have two components, I is a double cover of P1 ramified over the points in the pencil where the rank of the quadric drops. Since the discriminant is a hypersurface of degree 2g C 2 and the pencil is general, we conclude that I is ramified over 2g C 2 points. An easy local calculation shows that I is a hyperelliptic curve of genus g. Furthermore, every hyperelliptic curve arises via this construction. Let C be a smooth hyperelliptic curve of genus g. Let M2;o .Cg / denote the moduli space of vector bundles of rank 2 with a fixed odd-degree determinant on C . Realize C as above. By a celebrated theorem of Desale and Ramanan [DR], M2;o .Cg / is isomorphic to the space of .g  2/-dimensional projective linear spaces contained in the pencil of quadrics. In particular, ŒM2;o .Cg / D 2g1 i  g1;g2;:::;2;1 . Using this description, Theorem 4.36 allows us to compute the class of M2;o .Cg / in OG.g  1; 2g C 2/. The first four cases are given in the following table. (1) ŒM2;o .C2 / D 2I1 (2) ŒM2;o .C3 / D 40I1 C 4I3;1 (3) ŒM2;o .C4 / D 162I3;1 C 161;0I1 C 161I4;1 (4) ŒM2;o .C5 / D 643;1I3;1 C 642;1;0I1 C 642;1I5;1 C 323;0I4;1 C 323I5;4;1 C 325;0I3;1C 325I5;3;1 C 324;0I3;2 C 324I5;3;2. When g D 2, M2;o .C2 / is a complete intersection of two quadric fourfolds. A consequence of this computation is that for g > 2, these moduli spaces are not complete intersections of ample divisors in OG.g  1; 2g C 2/ (see [C3]). Exercise 4.42. Compute ŒM2;o .C6 / and ŒM2;o .C7 /. An algorithm similar to Algorithm 4.34 computes the restriction coefficients for isotropic flag varieties. We refer the reader to [C3] for details. There are other natural maps that one can define between orthogonal flag varieties and ordinary flag varieties. Perhaps the most interesting is the following  W OG.k; n/ ! F .k; n  kI n/

W 7! .W; W ? /

and more generally  W OF .k1 ; : : : ; kt I n/ ! F .k1 ; : : : ; kt ; n  k1 ; : : : ; n  kt I n/ .Wk1 ; : : : ; Wkt / 7! .Wk1 ; : : : ; Wkt ; Wk?1 ; : : : ; Wk?t /:

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Problem 4.43. Generalize Algorithm 4.34 to obtain a positive geometric algorithm for computing the map induced on cohomology by  in terms of the Schubert bases of these flag varieties. 4.2 The skew-symmetric case. The discussion for S G.k; n/ is similar, but the definitions have to be adapted for alternating forms. The fact that skew-symmetric forms have even rank constrains the geometry. Consequently, this case is more delicate. We preserve the notation from the previous section. r

Notation 4.44. Let Lni denote an isotropic subspace of dimension ni . Let Qdjj denote a dj -dimensional subspace of V such that the restriction of the skew-symmetric r form Q to Qdj has corank rj . Let Kj denote the kernel of the restriction of Q to j

r

Qdj . The dimension of Kj is rj . j

It is no longer possible to ensure that the kernels of the restriction of Q to linear spaces are nested. This complicates the definition of a symplectic sequence. Throughout let .L ; Q / be a partial flag r

   Qdr11 Ln1    Lns Qdks ks satisfying  dim.Kj \ Kl /  rj  1 for 1  j < l  k  s,  dim.Lni \ Kj /  min.ni ; dim.Kj \ Kks /  1/ for every 1  i  s and 1  j  k  s. Definition 4.45. We say that the partial flag .L ; Q / is in order if  Kj \ Kl D Kj \ Kj C1 for 1  j < l  k  s,  dim.Lni \Kj / D min.ni ; dim.Kj \Kks // for 1  i  s and 1  j  k s. The sequence is in perfect order if  Kj Kj C1 for 1  j < k  s,  dim.Lni \ Kj / D min.ni ; rj / for 1  i  s and 1  j  k  s. Definition 4.46. The partial flag .L ; Q / is a symplectic sequence if it satisfies the following conditions (1) Either the sequence is in order or there exists at most one index 1    k  s such that Kj Kl for l  j >  and Kj \ Kl D Kj \ Kj C1 for j < ; l > j: Furthermore, if K Kks , then dim.Lni \ Kj / D min.ni ; dim.Kj \ Kks // for 1  i  s and 1  j <  dim.Lni \Kj / D min.ni ; dim.Kj \Kks /1/ for 1  i  s and   j  ks: If K 6 Kks , then dim.Lni \ Kj / D min.ni ; dim.Kj \ Kks // for 1  i  s and 1  j < k  s.

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(2) If ˛ D dim.Kj \ Kks / > 0, then either j D 1 and n˛ D ˛ or there exists an index 1  j0  k  s such that for j0 6D l > min.j; /, we have rl  rl1 D dl1  dl , dj0 1  dj0  rj0  rj0 1 C 2  dim.Kj01 / C dim.Kj01 \ Kj0 / and K 6 Kj0 . Remark 4.47. As in the orthogonal case, ni , rj and dj automatically satisfy certain inequalities. The corank bound implies that rj  dim.Kj \ Kj 1 /  dj 1  dj for every 1 < j  k  s . The linear space bound implies that 2.ns C rj  dim.Kj \ Lns //  rj C dj for every 1  j  k  s. These inequalities will be implicitly assumed. More importantly, dj rj has to be even since the rank of a skew-symmetric form is even. In the orthogonal case, we could require the kernels to be nested. Due to the parity restrictions we cannot do this in symplectic case, which leads to the more convoluted condition (1) in the definition of symplectic sequences. Condition (2) is a consequence of the order of specialization. These conditions will automatically be satisfied in practice, so the reader can ignore them in a first reading. Notation 4.48. Let xj denote the number of isotropic subspaces in the sequence that are contained in Kj . Definition 4.49. We call the sequence .L ; Q / admissible if it satisfies the following conditions. (SA1) For every 1  j  k  s, we have xj  k  j C 1 

dj  rj : 2

(SA2) There does not exist integers i; j such that ni D dim.Kj \ Lni / C 1. Definition 4.50. The symplectic restriction variety V .L ; Q / associated to an admissible symplectic sequence .L ; Q / is the subvariety of S G.k; n/ defined as the Zariski closure of the following locus r

V .L ; Q /0 DfW 2 S G.k; n/ j dim.W \ Lni / D i; dim.W \ Qdj / D k  j C 1; j

dim.W \ Kj / D xj g Remark 4.51. As in the orthogonal case, condition (SA1) is a consequence of the linear space bound. The isotropic subspace of dimension k  j C 1 has to intersect d r the kernel Kj in a subspace of dimension at least k  j C 1  j 2 j . Condition (SA2) is a consequence of variations of tangent spaces. If rj C 1 D ni , then any linear space intersecting Lni away from Kj is contained in L? ni . Exercise 4.52. As in the orthogonal case, use induction on k to prove the following proposition.

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Proposition 4.53. The symplectic restriction variety V .L ; Q / is an irreducible subvariety of S G.k; n/ of dimension dim.V .L ; Q // D

s ks X X .ni  i / C .dj C xj  1  2k C 2j / i D1

j D1

Definition 4.54. A symplectic sequence .L ; Q / is saturated if rj C dj D n for 1  j  k  s. Schubert varieties in S G.k; n/ are examples of restriction varieties. As in the orthogonal case, we have the following. Lemma 4.55. Let .L ; Q / be a saturated, admissible symplectic sequence in perfect order and let n D 2m. Then V .L ; Q / is a Schubert variety with class mn1C1;:::;mns C1Irks ;:::;r1 . Conversely, every Schubert variety arises as such a symplectic restriction variety. Let ai D n  k C i  i > 2i  1 for 1  i  k. Then the intersection of a general Schubert variety † in G.k; n/ with S G.k; n/ is also a symplectic restriction variety r associated to the sequence Qar11    Qakk ; where 0  ri  1 satisfies ai D ri mod 2 and Kj \ Kj 1 D 0 for 1 < j  k  s. Hence, symplectic restriction varieties interpolate between these two kinds of varieties. Symplectic diagrams. We can record symplectic restriction varieties by symplectic diagrams. Their definition and properties are similar to orthogonal diagrams. Definition 4.56. Let 0  s  k be an integer. A sequence of brackets and braces of type s for S G.k; n/ consists of a sequence of natural numbers of type s, s brackets and k  s braces such that  Every bracket or brace occupies a positive position and each position is occupied by at most one bracket or brace.  Every bracket is to the left of every brace.  Every positive integer greater than or equal to j is to the left of the j th brace.  The total number of integers equal to zero or greater than j to the left of the j th brace is even. Notation 4.57. We use the same conventions as in the case of OG.k; n/. We count the brackets from left to right and denote the i th bracket by i . We count the braces from right to left and denote the j th brace by gj . Definition 4.58. Two sequences of brackets and braces are equivalent if the lengths of their sequence of numbers are equal, the brackets and braces occur at the same positions and the collection of digits that occur between any two consecutive brackets

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and/or braces are the same. We can depict an equivalence class of sequences by the representative where the digits are listed so that between any two consecutive brackets and/or braces the positive integers precede the zeros and are listed in non-decreasing order. We will always use this canonical representative and blur the distinction between the equivalence class and this representative. Example 4.59. The sequences 122101000g001g00 and 112210000g100g00 are equivalent. The latter is the canonical representative. Due to the evenness of rank, in the symplectic case it is not possible to keep the positive integers in increasing order and all at the beginning of the sequence. Definition 4.60. A sequence of brackets and braces is in order if the sequence of numbers consists of non-decreasing positive integers followed by zeros except possibly for one j immediately to the right of gj C1 for 1  j < k  s. Otherwise, we say that the sequence is not in order. A sequence is in perfect order if the sequence of integers consists of non-decreasing positive integers followed by zeros. As in the orthogonal case, we call a sequence of brackets and braces saturated if l.j / D .j; j  1/ for every 1  j  k  s. Example 4.61. The sequence 22000000g100g0 is in order. The sequence 11220000g00g00 is in perfect order. The sequences 2210000g00g0 and 1223300g100g00g00 are not in order. In the first sequence the 1 in place 3 and in the second sequence the 1 in place 8 violate the order. Definition 4.62. A symplectic diagram for S G.k; n/ is a sequence of brackets and braces that satisfies the following conditions. (1) l.j /  .j; j  1/ for 1  j  k  s. (2) Let j be the sum of p.s / and the number of positive integers between s and gj . Then 2j  p.gj / C rj for 1  j  k  s: (3) Either the sequence is in order or there exists at most one integer 1    k  s such that the sequence of integers is non-decreasing followed by a sequence of zeros except for at most one occurrence of  between s and g C1 and at most one occurrence of i after gi C1 . (4) Let j denote the number of positive integers between gj and gj 1 . If an integer i occurs to the left of all the zeros, then either i D 1 and there is a bracket in the position following it, or there exists at most one index j0 such that .j; j  1/ D l.j / for j0 6D j > min.i; / and .j0 ; j0  1/  l.j0 / C 2  j0 . Moreover, any integer  violating order occurs to the right of gj0 .

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Definition 4.63. A symplectic diagram is called admissible if it satisfies the following two conditions. (SA1) Let xj denote the number of brackets i such that every integer to the left of i is positive and less than or equal to j . Then xj  k  j C 1 

p.gj /  rj : 2

(SA2) The two integers to the left of a bracket are equal. If there is only one integer to the left of a bracket and s < k, then the integer is 1. Exercise 4.64. Choose a symplectic basis of V and define a correspondence between symplectic diagrams and symplectic sequences. Show that every admissible symplectic sequence can be represented by an admissible symplectic diagram. Conversely, every admissible symplectic diagram is the diagram associated to an admissible symplectic sequence (see [C4]). As in the orthogonal case, we have the following easy lemma. Lemma 4.65. Admissible, saturated symplectic diagrams in perfect order correspond to Schubert cycles. We are now ready to describe the algorithm. The goal is to transform an admissible symplectic diagram into a collection of admissible, saturated symplectic diagrams in perfect order. The fact that the rank of a skew-symmetric form is even constrains the possible degenerations. Consequently, describing D a in this case is trickier. Definition 4.66. We make the convention that an integer equal to k  s C 1 is 0 and gksC1 refers to s . Let D be an admissible symplectic diagram. (1) If D is not in order, let  be the integer in condition (3) violating the order. (a) If every integer  < j  k  s occurs to the left of , let  be the leftmost integer equal to  C 1 in the sequence of D. Let D a be the canonical representative of the diagram obtained by interchanging  and . (b) If an integer  < j  k  s does not occur to the left of , let  be the leftmost integer equal to j C 1. Let D a be the canonical representative of the diagram obtained by swapping  with the leftmost 0 to the right of gj C1 not equal to  and changing  to j . (2) If D is in order but is not a saturated admissible diagram in perfect order, let be the largest index j for which l.j / < .j; j  1/. (a) If l. / < . ;  1/, let  be the leftmost digit equal to C 1. Let D a be the canonical representative of the diagram obtained from D by changing  and the leftmost 0 to the right of g C1 not equal to  to .

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(b) If l. / D . ;  1/, let  be the integer equal to  1 immediately to the right of g . (i) If occurs to the left of , let  be the leftmost integer equal to in the sequence of D. Let D a be the canonical representative of the diagram obtained from D by changing  to  1 and  to 0. (ii) If does not occur to the left of , let  be the leftmost integer equal to

C 1. Let D a be the canonical representative of the diagram obtained by swapping  with the leftmost zero to the right of g C1 not equal to  and changing  to . Let p be the position in D immediately to the right of . If there exists a bracket at a position p0 > p, let q > p be the minimal position occupied by a bracket. Let D b be the diagram obtained from D a by moving the bracket in position q to position p. Otherwise, D b is not defined. Example 4.67. Let D D 2300g10g0g0, then  D 1 violates the order and  D 2 and 3 occur to the left of it. Hence, we are in case (1)(i) and D a D 1300g20g0g0 is obtained by swapping 1 and 2. Similarly, let D D 200200g00g, then the second 2 violates the order and D a D 220000g00g, D b D 220000g00g. Let D D 124400g00g1g0g00, the 1 in the ninth place violates the order and 3 does not occur to its left, so we are in case (1)(ii) and D a D 123400g10g0g0g00. Let D D 2200g00g00, then D is in order and D 1. Since l.1/ D 0 < .1; 0/1, we are in case (2)(i) and D a D 1200g10g00 and D b D 1200g10g00. Let D D 3300g200g0g, then D is in order and D 3. Since l.3/ D 2 D .3; 2/  1, we are in case (2)(ii)(a) and D a D 2300g000g0g. Finally, let D D 330000g00g1g0, then D is in order and D 2. Since l.2/ D 0 D .2; 1/  1 and 2 does not occur in the sequence, we are in case (2)(ii)(b) and D a D 230000g10g0g0. It may be that neither D a nor D b is admissible. As in the orthogonal case, there are algorithms for transforming them into admissible diagrams. Algorithm 4.68. Input: A symplectic diagram D. Step 1. If D satisfies condition (SA1), proceed to the next step. Otherwise, let j be the largest index for which (SA1) fails. Let 1 < 2 be the places of the two rightmost integers equal to j . Let D c be the diagram obtained from D as follows. Delete gj and the j in place 2 . Move the brackets, braces and integers in positions 1 < p < 2 to position p C 1 and add a j in place 1 C 1. Add a bracket in position 1 C 1. Subtract one from the integers j < h  k  s; and if j D k  s, change the integers equal to k  s to 0. Return D c to the beginning. Step 2. If D satisfies condition (SA2), output D. Otherwise, let i be the smallest index bracket for which (SA2) fails. Let j be the integer immediately to the left of i . Let D c be the symplectic diagram obtained from D as follows.

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Replace this j with j  1 (k  s if j D 0) and move gj 1 (gks if j D 0) one position to the left unless that position is occupied. If the position is occupied, discard the diagram and stop. Otherwise, return D c to Step 2. We say that D is a diagram derived from D a (respectively, D b ) if D is an output of running Algorithm 4.68 on D a (respectively, D b ). Example 4.69. Let D D 223300g00g00g00. Then the diagram D a D 123300g00g10g00 fails condition (SA1) since x1 D 0 < 1 D 5  .10  2/=2. Hence, according to Step 1 of Algorithm 4.68, we replace D a with the admissible diagram 1112200g00g000. Let D D 00g00g00. Then D a D 22g00g00 fails condition (SA1) since x2 D : Hence, Step 1 of Algorithm 4.68 replaces D a with 0000g00 which is 0 < 1  22 2 admissible. Similarly, if D D 113300g00g00g00, then the diagram D a D 112300g20g00g00 fails condition (SA1) since x2 D 1 < 2. Hence, according to Step 1 of Algorithm 4.68, we replace D a with 1122200g000g00, which is admissible. If D D 222200g0000g00, then the diagram D a D 122200g1000g00 is not admissible since it fails condition (SA2) for 1 . Step 2 of Algorithm 4.68 replaces D a first with 112200g100g000. Note that this diagram fails condition (SA2) for 2 . Hence, Step 2 replaces it with 111200g10g0000. This diagram is admissible, hence it is the diagram derived from D a . Exercise 4.70. Show that Algorithm 4.68 terminates and replaces D a or D b derived from an admissible symplectic diagram D with a collection of admissible symplectic diagrams. Finally, we can state the main algorithm for computing the classes of symplectic restriction varieties. Let D be an admissible symplectic diagram and let  be as in Definition 4.66. Let ./ denote the place of  in the sequence of integers. If p.s / > ./, then ˛ be the first bracket to the right of . If the integer to the immediate left of ˛ is positive, let y be this integer. Otherwise, let y D k  s C 1. Algorithm 4.71. Input: Let D be an admissible symplectic diagram. (1) If D is saturated and in perfect order, return D and stop. Otherwise, proceed to Step 2. (2) If p.s /  ./ or p.˛ / 1 > y  in D, return the admissible symplectic diagrams derived from D a to Step 1. Otherwise, return the admissible symplectic diagrams derived from both D a and D b to Step 1.

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Definition 4.72. A degeneration path is a sequence of admissible symplectic diagrams D 1 ! D 2 !    ! Dr such that Di C1 is one of the outputs of running Algorithm 4.71 on Di for 1  i < r. As in the orthogonal case, we have the following theorem. Theorem 4.73 ([C4, Theorem 3.33]). Let D be an admissible symplectic diagram and let V .D/ be the corresponding symplectic restriction variety. Then X ŒV .D/ D cI I ; where cI is the number of degeneration paths starting with D and ending in the symplectic diagram D.I /. As a corollary, we obtain a positive geometric rule for computing the restriction coefficients for S G.k; n/. Theorem 4.74 ([C4, Theorem 5.2]). Theorem 4.73 provides a positive, geometric rule for computing the restriction coefficients for the inclusion i W S G.k; n/ G.k; n/. There is a generalization of the algorithm to symplectic flag varieties [C6]. As in the orthogonal case, the following variant is an interesting open problem. Problem 4.75. Let  W SF .k1 ; : : : ; kt I n/ ! F .k1 ; : : : ; kt ; n  k1 ; : : : ; n  kt I n/ be the map given by .Wk1 ; : : : ; Wkt / 7! .Wk1 ; : : : ; Wkt ; Wk?1 ; : : : ; Wk?t /: Compute the map induced in cohomology in terms of the Schubert bases via a similar sequence of specializations. We conclude this section with several examples. Example 4.76. 00g00g00 ! 0000g00 ! 000000 # 1100g00 We conclude that i  3;2 D 2;1I C 3I2 in S G.2; 6/. Finally, we give a larger example in S G.3; 10/ that illustrates the inductive structure of the algorithm.

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Example 4.77. 300g20g10g000 ! 200g00g10g000 ! 20000g10g000 ! 10000g00g000 #

#

10000g00g000 12200g00g000 .

&

#

1000000g000 11200g0g0000 11200g10g000 . &

#

#

0000000000 1100100g000 111100g0000 11100g00g000 . & 1110000g000 111100g0000

# 1110000g000

This calculation shows that i  5;4;3 D 3;2;1I C 4;3I3 C 24;2I4 C 5;1I3 in H  .S G.3; 10/; Z/.

5 The rigidity problem In 1961, Borel and Haefliger [BH] asked whether Schubert classes in rational homogeneous varieties can be represented by projective subvarieties other than Schubert varieties. In this section, we discuss recent progress in answering this problem. A Schubert class  is called rigid if Schubert varieties are the only projective subvarieties representing  . More generally, given a positive integer m, the class m can be represented by the unions of m Schubert varieties. The class  is called multi rigid if unions of m Schubert varieties are the only closed algebraic sets representing the class m . As we have seen in §2, Schubert varieties are often singular. If the class of a singular Schubert variety is rigid, then that class cannot be represented by a smooth subvariety. The rigidity of Schubert classes has been studied by many authors [Br, C2, C5, CR, Ho1, Ho2, Ro, RT, Wa]. The multi rigid Schubert classes have been classified in cominuscule homogeneous varieties. We recall that the cominuscule homogeneous varieties coincide with the Compact Complex Hermitian Symmetric Spaces and are the following varieties: (1) Grassmannians G.k; n/, (2) Smooth quadric hypersurfaces in Pn , (3) The isotropic Grassmannians S G.m; 2m/,

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(4) The spinor varieties, i.e., the irreducible components of OG.m; 2m/, (5) Two exceptional varieties: The Cayley plane OP2 (E6 =P6 ) and the Freudenthal variety G.O3 ; O6 / (E7 =P7 ). Robles and The in [RT] building on the work of R. Bryant, J. Hong and M. Walters identified an obstruction O in Lie algebra cohomology and showed that if O vanishes, then  is multi rigid. They classified the Schubert classes in cominuscule homogeneous varieties with O D 0. Furthermore, for classes with O 6D 0, Robles [Ro] constructed explicit irreducible representatives for either 2 or 4 . Robles and the author in [CR] proved the following sharpening. Theorem 5.1 ([CR, Theorem 1.1]). Let  be a Schubert class with O 6D 0 in the cohomology of a cominuscule homogeneous variety X and let m be a positive integer. Then there exists an irreducible subvariety of X that represents m . Hong and Mok [HM] (see also [MZ]) have an alternative approach to proving the rigidity of certain smooth Schubert varieties in homogeneous varieties of Picard rank one. Their method uses the varieties of minimal rational tangents. The complete classification of rigid and multi rigid Schubert classes for more general rational homogeneous varieties is largely open. In this section, we discuss the problem in greater detail, give several applications of restriction varieties and pose several open problems. Rigidity in G.k; n/. We begin by describing rigid and multi rigid Schubert classes in G.k; n/. In §2, we described the smooth Schubert varieties of G.k; n/ as the linearly embedded sub-Grassmannians. Even when a Schubert variety is singular, there may be representatives of  which are smooth. Example 5.2. A Schubert variety with class 1 in G.k; n/ is singular if k and n  k are both greater than 1. However, 1 is the class of a hyperplane under the Plücker embedding of G.k; n/. By Bertini’s Theorem, we can always find smooth hyperplane sections that represent 1 . Hence, the class 1 is not rigid and can be represented by smooth subvarieties of G.k; n/. On the other hand, the theory of minimal degree varieties can be used to prove the rigidity of certain classes. Exercise 5.3. Show that the degree d of an irreducible, nondegenerate variety of dimension m in Pn satisfies the inequality d  n  m C 1. The varieties where the equality d D n  m C 1 holds are called varieties of minimal degree. The varieties of minimal degree have been classified by Bertini [Be]. Eisenbud and Harris have given a modern account of the classification in [EH1].

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Theorem 5.4. [EH1] A variety of minimal degree is one of the following: (1) A quadric hypersurface. (2) The Veronese surface in P5 . (3) A rational normal scroll, that is the projectivization PE of a vector bundle E on P1 embedded by the complete linear series jOPE .1/j. (4) A cone over one of the previous varieties. Example 5.5. The Schubert class 2 in G.2; 5/ is rigid. Let X be a projective subvariety of G.2; 5/ representing 2 . Then in the Plücker embedding X has dimension 4 and degree 3. We claim that X must span a P6 and be a variety of minimal degree. Otherwise, X would span a P5 and be a cubic hypersurface in this P5 . The Grassmannian G.2; 5/ is cut out by quadratic Plücker relations. By Bezout’s Theorem, G.2; 5/ cannot contain X without containing its linear span. However, the largest dimensional linear space in G.2; 5/ has dimension 3. This is a contradiction. Hence, X must be a variety of minimal degree. From the classification of these varieties, we conclude that X must be a cone with vertex p over a cubic scroll. Hence, X must be the intersection of G.2; 5/ with its tangent plane at p. Since this is a Schubert variety with class 2 , we conclude that X is a Schubert variety. The following fundamental theorem of Thom, Grauert-Kerner, Schelssinger, Kleiman-Landolfi allows one to prove the rigidity of many other Schubert classes in G.k; n/. Theorem 5.6 ([KL, Theorem 2.2.8]). Let K be a field and let X D PnK  Pm K be embedded in projective space by the Segre morphism. If n  1 and m  2, then the cone over X is rigid over K in the sense that any small deformation is isomorphic to the cone over X . Exercise 5.7. Let  be the partition 1 D    D k1 D n  k  1 and k D 0. Show that in the Plücker embedding a Schubert variety † is a cone over the Segre embedding of Pk1  Pnk1 . Deduce that the Schubert class  is rigid unless n D 2k D 4. By induction building on Example 5.5 and Exercise 5.7, [C2] characterizes rigid Schubert classes in G.k; n/. The Schubert subvarieties contained in a variety Y

G.k; n/ carry a lot of geometric information about Y (see [AC] for a related discussion). Let Y be a representative of a Schubert class. The singular locus of Y and the intersection of Y with various Schubert varieties are often enough to force Y to be a Schubert variety. For our purposes, it is convenient to record a partition i  by grouping the parts that are equal. More concretely, write  D . i11 ; : : : ; jj / with 1 > 2 >    > j and s occurs is times in the partition . In particular, Pj sD1 is D k.

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Definition 5.8. A partition  D . i11 ; : : : ; jj / is a rigid partition for G.k; n/ if there does not exist an index 1  s < j with is D 1 and n  k > s D sC1 C 1. Theorem 5.9 ([C2, Theorem 1.3]). A Schubert class  in G.k; n/ is rigid if and only if  is a rigid partition for G.k; n/. A multi rigid class by definition is rigid. However, a rigid class does not have to be multi rigid. Example 5.10. The class 2 in G.2; 5/ is rigid (see Example 5.5) but not multi rigid. To describe deformations of m2 , it is more convenient to work projectively. Fix an irreducible curve C of degree m in P4 . Consider the subvariety X of G.2; 5/ parameterizing lines that intersect C . Consider the incidence correspondence I D f.p; l/ j p 2 l; p 2 C g C  G.2; 5/: The variety I is irreducible by the theorem on the dimension of fibers and dominates X . Hence, X is irreducible and has dimension 4. We claim that the class of X is m2 . To determine the class, we can intersect with complementary dimensional Schubert cycles. A general Schubert variety †3;1 intersects X in m points. The variety †3;1 parameterizes lines in P4 that pass through a point q and lie in a hyperplane P containing q. The hyperplane P intersects C in m points p1 ; : : : ; pm . Hence, the lines parameterized by X \ †3;1 are the m lines spanned by q and pi . On the other hand, a general †2;2 is disjoint from X . The Schubert variety †2;2 parameterizes lines contained in a plane …. Since a general plane … is disjoint from C , X \ †2;2 D ;. We conclude that X is an irreducible variety with class m2 . Hong [Ho1, Ho2] and Robles and The [RT] have classified the multi rigid Schubert classes in G.k; n/. Theorem 5.11 ([Ho2, RT]). A Schubert class  is multi rigid if and only if  D i . i11 ; : : : ; jj / satisfies the following conditions  is  2 for 1 < s < j ,  s1  s  2 for 1 < s  j ,  i1  2 if 1 < n  k; and ij  2 if j > 0. If a Schubert class  is rigid and the corresponding Schubert variety is singular, then  cannot be represented by a smooth subvariety of G.k; n/. However, even when  is not rigid, it might not be possible to represent  by a smooth subvariety of G.k; n/. Example 5.12. Consider the class 3;2 in G.3; 7/. This class is not rigid. Fix a fourdimensional subspace V 0 V . Let Y be a smooth hyperplane section of the variety G.2; V 0 / (in the Plücker embedding). Consider the incidence variety X D f.W2 ; W3 /jW2 2 Y; W2 W3 g G.2; V 0 /  G.3; 7/:

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85

The second projection is a variety in G.3; 7/ with class 3;2 . To see this, we can pair the variety with complementary dimensional Schubert cycles. Consider a general Schubert cycle †a;b;c such that a C b C c D 7 and b  3. Then the linear spaces parameterized by this Schubert cycle have two-dimensional subspaces contained in a linear space W of dimension at most 3. Since W \ V 0 D 0, we conclude that †a;b;c \ X D ;. Similarly, the linear spaces parameterized by a †3;2;2 are contained in a 5dimensional linear space U . For a general such Schubert variety the two-dimensional U \ V 0 will not be a point of Y . Hence, †3;2;2 \ X D ;. Finally, for a general †4;2;1 , the intersection †4;2;1 \ X consists of one point. We conclude that X represents 3;2 . It is clear that X is not a Schubert variety. On the other hand, 3;2 cannot be represented by a smooth subvariety of G.3; 7/. Suppose it were represented by a smooth subvariety Y . Then the intersection of Y with a general Schubert variety 1;1;1 would be smooth by Kleiman’s transversality theorem. However, this intersection has class 4;3;1 , which is a rigid and singular class. We conclude that 3;2 cannot be represented by a smooth subvariety. i

Definition 5.13. A partition  D . i11 ; : : : ; jj / is a non-smoothable partition for G.k; n/ if either there exists an index 1  s < j such that is 6D 1 and n  k > s or there exists an index 1  s < j such that n  k > s 6D sC1 C 1. Theorem 5.14 ([C2, Theorem 1.6]). If  be a non-smoothable partition for G.k; n/, then  cannot be represented by a smooth subvariety of G.k; n/. A complete characterization of smoothable Schubert classes is not known except in G.2; n/ and G.3; n/ (see [C2, Corollaries 4.5 and 4.6]). Problem 5.15. Classify smoothable Schubert classes in G.k; n/. For example, can 3;2;1;0 be represented by smooth subvarieties of G.4; 8/? There are several other notions of smoothability. We will not say anything about them other than giving a few examples and raising some questions. Problem 5.16. When is (a multiple of) a Schubert class a positive linear combination of classes of smooth subvarieties of G.k; n/? P Since Schubert classes span extremal rays of the effective cone, if m D ai ŒZi  with Zi smooth subvarieties and ai > 0, then each Zi must have class proportional to  . Consequently, if  is multi rigid and a Schubert variety with class  is not smooth, then the class  cannot be a positive linear combination of classes of smooth subvarieties of G.k; n/. More generally, one can ask the following more interesting problem. Problem 5.17. When is (a multiple of) a Schubert class a linear combination of classes of smooth subvarieties of G.k; n/?

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Example 5.18. The class 2 in G.2; 5/ can be expressed as 2 D 12  1;1 . A Schubert variety †1;1 is isomorphic to the Grassmannian G.2; 4/ and is smooth. On the other hand, the class 12 can be represented by a codimension 2 linear section of the Grassmannian. By Bertini’s Theorem, such a linear section can be chosen to be smooth. Hence, 2 can be represented as a linear combination of classes of smooth subvarieties of G.2; 5/, even though it is rigid. Remark 5.19. In contrast, Hartshorne, Rees and Thomas [HRT] have shown that 2 in G.3; 6/ cannot be represented as a linear combination of the classes of closed smooth submanifolds of G.3; 6/. In particular, it is not a linear combination of classes of smooth subvarieties of G.k; n/. It is also interesting to ask these problems for classes other than Schubert classes. Problem 5.20. Given an effective cohomology class c in G.k; n/, determine when c can be represented by an irreducible subvariety. Determine when c can be represented by a smooth subvariety. Determine when c is a positive linear combination of classes of smooth subvarieties. Determine when c is a linear combination of classes of smooth subvarieties. Using geometric constructions it is possible to give examples of classes that can be represented by irreducible or smooth subvarieties of G.k; n/. However, at present a complete classification is far from known. One may ask the representative cycles to have other geometric properties. Problem 5.21. When can a cohomology class c in G.k; n/ be represented by a rational subvariety? When can it be represented by a rationally connected subvariety? When can it be represented by a smooth, rational or rationally connected subvariety? Remark 5.22. Projective space has the remarkable property that every effective cohomology class can be represented by a smooth irreducible subvariety. To represent m times the class of a linear space of dimension k, simply take a smooth hypersurface of degree m in a linear space of dimension k C 1. Similarly, every effective cohomology class of Pn can be represented by a rational subvariety by taking the hypersurface to have an ordinary .m  1/-fold point. These are false for other Grassmannians since a large multiple of a multi rigid Schubert class cannot be represented by an irreducible subvariety. In particular, it cannot be represented by a rational subvariety. Rigidity in OG.k; n/. In this subsection, we summarize some of the results on the rigidity and multi rigidity of Schubert classes in OG.k; n/ following [C5]. The next example describes the classification for OG.1; n/.

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87

Example 5.23. The variety OG.1; n/ is a smooth quadric hypersurface Q in Pn1 . The Schubert classes are (1) Isotropic linear spaces PLj for 0  j < n2 , (2) If n is even, isotropic linear spaces PL n2 and PL0n , 2

PQdnd

n 2

for n  d > C 1. (3) The quadric sections Q \ The linear spaces are smooth and their classes are rigid. When d < n, the quadric sections Q \ PQdnd are singular with the singular locus isomorphic to Pnd 1 . The cohomology class of the Schubert variety Q \ PQdnd is the same as the cohomology class of any linear section Q \PQdr with r  nd . The singular locus of this variety is isomorphic to Pr1 . Hence, this variety is not isomorphic to a Schubert variety if r < n  d . Therefore, these classes are not rigid. In particular, since Q \ PQd0 is a smooth quadric, every Schubert class in OG.1; n/ can be represented by a smooth , subvariety of OG.1; n/. The classes of the linear spaces PLj , for 1 < j  n1 2 are rigid but not multi rigid. For example, twice the class of PLj can be represented by a smooth quadric of the same dimension. If 2k D n, the classes of the Schubert varieties PLk and PL0k are multi rigid [Ho1]. Example 5.23 shows that by deforming the quadrics to less singular quadrics, one can obtain deformations of Schubert varieties. Similarly, while the isotropic spaces are rigid, their multiples may be deformed to quadrics. One can use these two facts systematically to prove the failure of rigidity or multi rigidity for many Schubert classes. Therefore, restriction varieties play an important role in the study of rigidity. The next example illustrates the point. Example 5.24. The orthogonal Grassmannian OG.2; 5/ is isomorphic to P3 . The codimension two Schubert varieties †1I1 in OG.2; 5/ are lines; however, not all lines in OG.2; 5/ are Schubert varieties. A Schubert variety †1I1 is determined by specifying a point p on Q P4 . Projectively, the Schubert variety parameterizes lines in Q that contain the point p. In particular, the space of Schubert varieties with class 1I1 is a quadric threefold. Let Q0 Q be a codimension one smooth quadric and let l Q0 be a line. Then the space of lines that are contained in Q0 and intersect l is also a line in OG.2; 5/. Note that this is the restriction variety V .L2 Q40 /. The lines parameterized by the Schubert variety †1I1 sweep out the singular quadric surface Tp Q \ Q, whereas the lines parameterized by the restriction variety sweep out the smooth quadric surface Q0 . Since OG.2; 5/ is isomorphic to P3 , the space of lines in OG.2; 5/ is isomorphic to the Grassmannian G.2; 4/ parametrizing lines in P3 . The Grassmannian G.2; 4/ admits a map to .P4 / sending a point q 2 G.2; 4/ to the hyperplane in P4 spanned by the linear spaces parameterized by the line corresponding to q. This is a two-to-one map branched over the locus of Schubert varieties. This is one of the first examples where restriction varieties provide an explicit deformation of Schubert varieties. This example also shows that a Schubert class may be represented by a variety that is isomorphic, even projectively equivalent (under GL.n/ but not SO.n/), to a Schubert variety but is not a Schubert variety.

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Exercise 5.25. Generalize the previous example to show that the Schubert classes m;m1;:::;2Im1 are not rigid in OG.m; 2m C 1/. Describe the space of lines in OG .m; 2m C 1/. Discuss the case of higher dimensional linear spaces. We now use restriction varieties to give deformations of Schubert varieties more systematically. Definition 5.26. A Schubert class I in OG.k; n/ is of Grassmannian type if s D k (equivalently, has length 0) and k > 0. A Schubert class I is of quadric type if s D 0 and 1 < n2  1. Notation 5.27. For discussing rigidity, it is more convenient to record the partitions .I / for OG.k; n/ slightly differently. Given  define a sequence a./ by setting ai D b n1 ci i . Record the sequence a./ by grouping the equal terms .˛1i1 ; : : : ; 2 ˛tit / so that ˛1 < ˛2 <    < ˛t and ˛j occurs with multiplicity ij in the sequence a./. Similarly, given define a sequence b. / by setting bj D n  j  j . Record the sequence b. / also by grouping the equal terms .ˇ1j1 ; : : : ; ˇuju / so that ˇ1 <    < ˇu and ˇl occurs with multiplicity jl in the sequence b. /. Example 5.28. Given the partitions .4; 2; 1I 3; 0/ for OG.5; 13/, we have a./ D .1; 2; 2/ D .11 ; 22 / and b. / D .6; 8/. The next two theorems characterize the rigidity of Schubert classes of Grassmannian and quadric type. Theorem 5.29 ([C5, Theorem 1.4]). Let I be a Schubert class of Grassmannian type in the cohomology of OG.k; n/. Express the associated partition a./ by grouping the equal terms as .˛1i1 ; : : : ; ˛tit /. Then: (1) The class I is rigid if and only if there does not exist an index 1  u < t such that iu D 1 and 0 < ˛u D ˛uC1  1. (2) The class I is multi rigid if and only if iu  2 for every 2  u  t, i1  2 unless ˛1 D 0, and ˛u  ˛uC1  2 for every 1  u < t. (3) The class I is not smoothable if there exists an index 1  u < t such that 0 < ˛u < ˛uC1  1, or an index 1  u < t such that ˛u > 0 and iu > 1. Exercise 5.30. Let i W OG.k; n/ ! G.k; n/ denote the natural inclusion. Identify the image of a Schubert variety of Grassmannian type in OG.k; n/ as a Schubert variety in G.k; n/ under i . Show that the class of Grassmannian type I in OG.k; n/ is rigid (respectively, multi rigid) if and only if the corresponding class in G.k; n/ is rigid (respectively, multi rigid). Deduce parts (1) and (2) of the Theorem 5.29 from Theorems 5.9 and 5.11. Show that if the class is smoothable in OG.k; n/, then the corresponding class is smoothable in G.k; n/ and deduce part (3).

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89

Theorem 5.31 ([C5, Theorem 1.5]). Let I be a Schubert class of quadric type in the cohomology of OG.k; n/. Express the partition b. / by grouping the equal terms as .ˇ1i1 ; : : : ; ˇtit /. Then: (1) The class I is not rigid unless t D 1 and ˇ1 D n  k. (2) The class I is not smoothable if there exists an index 1  u < t such that b nC1 c < ˇu < ˇuC1  1 or an index 1  u < t such that b nC1 c < ˇu and 2 2 iu > 1. Sketch of proof. Let I be a Schubert class of quadric type in OG.k; n/. We then have 1 < n2  1. Consequently, the dimension of F?1 is greater than n2 C 1. Since r1 , we have the corank of a quadric is bounded by its codimension, for a quadric Qn 1 r1 that r1  1  n 1 3. In particular, Qn1 is irreducible. Let V be the restriction variety defined by the sequence .Q / 0 0 0 Qn

Qn

   Qn : 1 2 k

The i -th linear space in this sequence has the same dimension as the i -th linear space in the sequence defining the Schubert variety †I but the restriction of the quadratic form is nondegenerate instead. Exercise 5.32. Show that the sequence .Q / is admissible and V is a restriction variety. Using Algorithm 4.34 compute the class of V to see that it is I . The exception t D 1 and ˇ1 D n  k in the statement of the theorem corresponds to the fundamental class of OG.k; n/. In all other cases, we now show that the restriction variety constructed in the previous paragraph gives a non-trivial deformation of the Schubert variety. The linear spaces parameterized by V sweep out the quadric 0 . Hence, if k 6D 0, the restriction variety cannot be projectively equivalent to Qn k a Schubert variety since for a Schubert variety the linear spaces sweep out a quadric of corank k . If k D 0, since t 6D 1, there exists u such that u > k  u. Let v be maxfujbu > k  ug. The smallest dimensional quadric that contains a v-dimensional subspace of every linear space parameterized by a Schubert variety †b has corank bv . In the restriction variety this quadric has the same dimension and has full rank. Therefore, we conclude that the restriction variety cannot be projectively equivalent to a Schubert variety. This concludes the proof that unless t D 1 and k D 0, a Schubert cycle of quadric type is not rigid. In fact, we have proved that such a class can always be represented by the intersection of a general Schubert variety in G.k; n/ with the orthogonal Grassmannian OG.k; n/. Exercise 5.33. Consider the Schubert variety † in OG.k; n/ which parameterizes isotropic subspaces contained in a maximal isotropic subspace. Notice that this Schubert variety is smooth and isomorphic to a Grassmannian G.k; b n2 c/. Deduce part (2) of the theorem from Theorem 5.29 by intersecting a representative of I by a general translate of †. (Hint: if I can be represented by a smooth subvariety, then, by Kleiman’s Transversality Theorem, this intersection would be smooth.) 

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Exercise 5.34. Using the fact that Schubert classes of quadric type in OG.k; n/ can be represented by the intersection of a Schubert variety in G.k; n/ with OG.k; n/ deduce the following corollary of the proof. Corollary 5.35. Let I be a Schubert class of quadric type in OG.k; n/. If t D 1, then I is smoothable. More generally, show that if the corresponding Schubert class in G.k; n/ is smoothable, then so is the Schubert class of quadric type in OG.k; n/. We now turn our attention to more general cohomology classes. Example 5.36. The quadric diagram associated to the Schubert class 4I3;1 in OG.3; 9/ is 122000g00g0. By Algorithm 4.34, the restriction variety associated to the sequence 100000g00g0 has the same class but is not isomorphic to a Schubert variety. Similarly, the quadric diagram associated to the Schubert class 5I5;4;3;0 in OG.5; 13/ is 22234000g0g0g000g. The restriction variety associated to the quadric diagram 22340000g0g0g000g has the same cohomology class but is not isomorphic to a Schubert variety. The quadric diagram associated to the class 2I1 in OG.2; 5/ is 1000g0: The restriction variety associated to the sequence 0000g0 also has the same class. More generally, the quadric diagram associated to the Schubert class 5;3;1I3;1 in OG.5; 11/ is 12200000g00g0. The restriction variety associated to the sequence 12000000g00g0 has the same class. The latter varieties are not isomorphic to Schubert varieties. By analyzing Algorithm 4.34, we can find restriction varieties that have the same class as (or a multiple of) a Schubert variety but are not isomorphic to a Schubert variety. The next two theorems use this strategy to show that the class is not rigid or multi rigid. Theorem 5.37 ([C5, Theorem 1.7]). Let I be a Schubert class in OG.k; n/. Express the sequences a./; b. / by grouping the equal terms .˛1i1 ; : : : ; ˛tit I ˇ1j1 ; : : : ; ˇuju /: Assume that one of the following conditions holds for .I /: (1) ˇ1 < n  k and sCj1 6D b n1 c  i for any 1  i  s. 2 (2) There exists an index 1  u < t such that iu D 1, 0 < ˛u D ˛uC1  1 and c  i1 CCiu 6D j for any s < j  k. b n1 2 c  i  n  sC1 g D s C sC1  (3) #fi jb n1 2 1  h  s such that b n1 c  h D n  sC1 . 2 Then I is not rigid.

n3 2

and there exists an index

Sketch of proof. The idea is to use restriction varieties to obtain deformations of Schubert varieties. Let I be a Schubert class in the cohomology of OG.k; n/. For

Restriction varieties and the rigidity problem

91

ci . First, assume that ˇ1 < nk and sCj1 6D i for any simplicity, set i D b n1 2 1  i  s. By assumption, we have that sCj1 D sCj1 1 1 D    D sC1 j1 C1. We must have that either i < bsCj1 or i > bsC1 C 1 for every 1  i  s. Define an admissible sequence .L ; Q / by 

1



sCj 1

1

sC1 sC2 k 1

Qn

   QnsCj

   Qn : L1    Ls Qn sC1 sC2 k 1

This sequence differs from the sequence defining the Schubert variety with class I sCi 1

sCj 1

1 1 only in that the ranks of the quadrics QnsCi ; : : : ; QnsCj are one more than the 1 1 corresponding quadrics in the sequence associated to the Schubert variety.

Exercise 5.38. Show that this sequence is admissible and using Algorithm 4.34 compute that the cohomology class of the restriction variety V .L ; Q / is I . Show that V .L ; Q / is not isomorphic to the Schubert variety †I and conclude that the class I is not rigid. Next, assume that there exists an index 1  u < t such that iuPD 1, ˛u D ˛uC1 1 and i1 CCiu 6D bj for any s < j  k. For simplicity, set h D ulD1 il . There exists a subvariety Y of G.h C 1; FhC1 / parameterizing .h C 1/-dimensional subspaces ƒ FhC1 that satisfy dim.ƒ \ Fi /  i for i < h but is not a Schubert variety. Let Z be the Zariski closure of the following quasi-projective variety fW 2 OG.k; n/ j W \FhC1 2 Y; dim.W \Fi / D i; for i 6D h; dim.W \F?j / D j g: Then the class of Z is I since specializing Y to a Schubert variety specializes Z to a Schubert variety. Furthermore, Z is not a Schubert variety. Therefore, the class I is not rigid. Finally, assume that Condition (3) of the theorem holds. Consider the restriction variety V associated to the sequence 

1

sC1 k

   Qn : L1    Lh1 Lh C1 LhC1    Qn sC1 k

Notice that this sequence differs from the sequence defining the Schubert variety in that the dimension of the h-th isotropic linear space is one larger and the corank of the smallest dimensional quadric is one smaller. Exercise 5.39. Show that this sequence is admissible. Using Algorithm 4.34 compute that V has class I . Deduce that I is not rigid.  One can use similar arguments to conclude the following. Theorem 5.40 ([C5, Theorem 1.8]). Let I be a Schubert class in OG.k; n/. Express the sequences a./; b. / by grouping the equal terms .˛1i1 ; : : : ; ˛tit I ˇ1j1 ; : : : ; ˇuju /: Assume that either one of the conditions in Theorem 5.37 or one of the following conditions holds for .I /:

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(1) There exists an index 1  u  t such that b n1 c  i1 CCiu 6D j for any 2 s C 1  j  k and either iu D 1 with 0 < ˛u < n2 or ˛u D ˛uC1  1. (2) s1 > s C 1 > 1 and either as C s > sC1 or as C s D j for some s C 1  j  k and j D j 1 C 2 D sC1 C j  s  1. Then I is not multi rigid. Example 5.41. Consider the Schubert variety 4;2I2;0 in OG.4I 11/ with quadric diagram 220000000g00g. Then the variety corresponding to the sequence 22000g0000g00g has class 24;2I2;0. The corresponding Schubert variety is not multi rigid. It is also possible to prove the rigidity of certain Schubert classes. We refer the reader to [C5, Theorem 1.10]. As a consequence one can characterize rigid Schubert classes in OG.2; n/ if n > 8. Finally, Robles and The have classified the multi rigid Schubert classes in the spinor variety. Recall that in this case the sequence is uniquely determined from , so it suffices to specify the conditions on . Theorem 5.42 ([RT, Theorem 8.1]). Let I be a Schubert class in the cohomology of the spinor variety Sp.m; 2m/. Express the associated sequence a./ by grouping the equal terms .˛1i1 ; : : : ; ˛tit /: Then I is multi rigid if and only if (1) il  2 and ˛l1  l  2 for all 1 < l  t, (2) i1  2 if ˛1 > 0 and s > 1 if s > 0. The following problem remains open in its full generality. Problem 5.43. Characterize rigid and multi rigid Schubert classes in all orthogonal Grassmannians OG.k; n/. Rigidity in SG.k; n/. As in the case of orthogonal Grassmannians, restriction varieties give deformations of Schubert varieties in S G.k; n/ under suitable numerical assumptions. Recall that in this case n D 2m is even. The following example is typical. Example 5.44. The Grassmannian S G.1; n/ is isomorphic to Pn1 . Hence, all the Schubert varieties PLnj and PL? nj are linear spaces. However, not all linear spaces are Schubert varieties. Points and codimension one linear spaces are always Schubert varieties. Every one-dimensional subspace is isotropic. Similarly, the restriction of Q to a codimension one linear space has a one-dimensional kernel W , hence it is of the form W ? . We conclude that points and codimension one linear spaces are rigid. If 1 < dim.M / < n  1, then M is typically not of the form F or F ? for an isotropic subspace F . Hence, the linear spaces PM are not Schubert varieties and the corresponding Schubert classes are not rigid.

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The following theorem generalizes this example and can be proved by exhibiting explicit restriction varieties that have the same class as the Schubert variety but are not Schubert varieties. Theorem 5.45 ([C4, Theorem 6.2]). Let  I be a Schubert class in the cohomology of S G.k; n/. (1) If s D 0 and j > k  j C 1 for some j , then  I is not rigid. (2) If s  1 and s > max. sC1 ; s1 C 1/, then  I is not rigid. Corollary 5.46 ([C4, Corollary 6.3]). (1) If the Schubert class 1 kC1;2 kC2;:::;k in the cohomology of G.k; n/ can be represented by a smooth subvariety of G.k; n/, then the Schubert class I1 ;:::;k can also be represented by a smooth subvariety of S G.k; n/. (2) If there exists an index i < k such that m  i  1 > i > i C1 C 2 or if there exists an index 1 < i < k such that m  i > i 1 D i C 1 > i C1 C 2, then I1 ;:::;k cannot be represented by a smooth subvariety of S G.k; n/. (3) If the Schubert class 1 ;:::;k in the cohomology of G.k; m/ can be represented by a smooth subvariety of G.k; m/, then the Schubert class 1 ;:::;k I in the cohomology of S G.k; n/ can be represented by a smooth subvariety of S G.k; n/. (4) If there exists an index i < k such that i < i < i C1 C 2 or an index 1 < i < k  1 such that i  1 < i 1 D i  1 < i C1  2, then 1 ;:::;k I cannot be represented by a smooth subvariety of S G.k; n/. Robles and The [RT] have characterized the multi rigid Schubert classes in Lagrangian Grassmannians. Express a./ as .˛1i1 ; : : : ; ˛tit /. They show that a Schubert class I in S G.m; 2m/ is multi rigid if and only if (1) il  2 and ˛l1  ˛l  2 for all 1 < l  t, (2) i1  2 if ˛1 > 0 and s  3 if s > 1. The following problem remains largely open. Problem 5.47. Characterize the rigid, multi rigid and smoothable Schubert classes in flag varieties and isotropic flag varieties.

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[BiC] S. Billey and I. Coskun, Singularities of generalized Richardson varieties. Comm. Alg. 40(2) (2012), 1466–1495. [BiL] S. Billey and V. Lakshmibai, Singularities of Schubert varieties. Springer, 2000. [BH]

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I. Coskun, A Littlewood–Richardson rule for two-step flag varieties. Invent. Math. 176(2) (2009), 325–395.

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I. Coskun and C. Robles, Flexibility of Schubert classes. Differ. Geom. Appl. 31(6) (2013), 759–774.

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I. Coskun and R. Vakil, Geometric positivity in the cohomology of homogeneous varieties and generalized Schubert calculus. Algebraic Geom. Seattle (2005) 1, 77–124.

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W. Fulton and P. Pragacz, Schubert varieties and degeneracy loci, volume 1689 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 1998. Appendix J by the authors in collaboration with I. Ciocan-Fontanine.

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[HRT] R. Hartshorne, E. Rees, and E. Thomas, Nonsmoothing of algebraic cycles on Grassmann varieties. Bull. Amer. Math. Soc. 80(5) (1974), 847–851. [Ho1] J. Hong, Rigidity of smooth Schubert varieties in Hermitian symmetric spaces. Trans. Amer. Math. Soc. 359 (2007), 2361–2381. [Ho2] J. Hong, Rigidity of singular Schubert varieties in Gr.m; n/. J. Diff. Geom. 71(1) (2005), 1–22. [HM] J. Hong and N. Mok, Characterization of smooth Schubert varieties in rational homogeneous manifolds of Picard number 1. J. Algebraic Geom. 22(2) (2013), 333–362. [Kl]

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V. Lakshmibai and C. S. Seshadri, Singular locus of a Schubert variety. Bull. Amer. Math. Soc. 11(2) (1984), 363–366.

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On Plücker equations characterizing Grassmann cones Letterio Gatto and Parham Salehyan1 Contents 1 Preliminaries and notation . . . . . . . . . . . . . . . . 2 Hasse–Schmidt derivations on an exterior algebra 3 Schubert derivations on ZŒX . . . . . . . . . . . . . . 4 Proof of theorem 0.1 . . . . . . . . . . . . . . . . . . . . 5 The Grassmann cone in a polynomial ring . . . . . 6 An example . . . . . . . . . . . . . . . . . . . . . . . . . . 7 The Grassmann cone in infinite exterior power . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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102 104 106 109 110 116 119 124

To Piotr Pragacz, on the occasion of his 60th birthday

Introduction This survey article, which may also serve as background material while reading, for instance, [3, Section 4] and parts of [23, 26, 28], has the purpose to advertise the notion of Schubert derivation on an exterior algebra, introduced in [10] (see also [13]), by showing how it provides another approach to look at the quadratic equations describing the Plücker embedding of Grassmannians – a very classical and widely studied subject. In particular, it allows i) to “discover” the vertex operators generating the fermionic vertex superalgebra (in the sense of [8, Section 5.3]); ii) to compute their bosonic expressions as in [17, Lecture 5]; iii) to interpret them in terms of Schubert derivations and iv) to provide an almost effortless deduction of the celebrated Hirota bilinear form of the KP hierarchy (after Kadomtsev and Petviashvilii) [17]. Let B WD QŒx1 ; x2 ; : : : and denote by B.0/ its quotient field. A formal P pseudo differential operator with coefficients in B.0/ is a formal Laurent series i n ai .x/@i , where @1 denotes a formal inverse of the operator @ WD @=@x1 . The KP hierarchy concerns the evolution of a first order normalized pseudo differential operator, 1 Work was partially supported by PRIN “Geometria sulle Varietà Algebriche”, by GNSAGA-INDAM and Istituto Superiore Mario Boella. The second author was sponsored by FAPESP, Proc. 2015/04513-8.

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L WD @ C u1 .x/@1 C    2 B.0/ ..@1 // obeying the Lax equations dL D Œ.Ln /C ; L; dxn

(0.1)

where .Ln /C denotes the differential part of the n-th power of L – a linear ordinary differential operator of order n. Equations (0.1) arise as compatibility conditions for an isospectral deformation of an ordinary differential operator [23, Section 4]. It is easily seen that the first two non trivial equations of the hierarchy (n D 2; 3) yield the celebrated KP equation   3 1 fyy  ft  fxxx  3ffx D 0; 4 4 x upon identifying x1 D x, x2 D y, x3 D t and f WD u1 [23, Section 4]. It is a fundamental observation due to Sato [26, 27], and widely developed by his Kyoto school [5, 6, 15], that solutions L to the KP hierarchy (0.1) can be identified with the points of a Grassmannian parametrizing infinite dimensional subspaces of an infinite dimensional vector space. Indeed, explicit solutions to the KP equation in Lax form can be constructed by tau functions which, for the limited purposes of this paper, we define just as those polynomials  2 B satisfying    (0.2) Resz  .z/ ˝ .z/ D 0; where z is a formal variable, Resz denotes the coefficient of z 1 of a formal Laurent  series and .z/;  .z/ W B ! B..z// are the “bosonic vertex operators”: 0 0 1 1 X 1 @ X A and .z/ WD exp @ xi z i A exp @ iz i @xi i 1 i 1 0 0 1 1 (0.3) X X 1 @ A  :  .z/ WD exp @ xi z i A exp @ iz i @xi i 1

i 1

For a tau function  2 B, let 0

1 X 1 @ A  .x/ 2 B.0/ Œz 1 : P .z/ D  .x/1 exp @ iz i @xi i 1

Define P .@/ to be the evaluation of P .z/ at z 1 D @1 . Then L D P .@/1  @  P .@/ is a normalized formal pseudo–differential operator that satisfies the Lax equations (0.1) [3, Section 4]. The point we shall focus on is that (0.2), also known as Hirota bilinear form of the KP hierarchy, and which we refer to as the KP hierarchy tout court, encodes

On Plücker equations characterizing Grassmann cones

99

the Plücker equations of the cone of decomposable tensors of a semi-infinite exterior power of an infinite–dimensional vector space. This fact is mentioned and/or explained in a number of different ways, e.g. in [2, 3, 7, 18, 19, 23, 24, 25, 28] and surely in many more references. From our part we shall recover expression (0.2) by considering the limit for r ! 1 of the main formula we prove in the present article, that characterizes Grassmann cones of decomposable tensors in an r-th exterior power (r  1). For this purpose, let M0 be a free abelian group of infinite countable rank with  a basis B0 WD .b0 ; b1 ; : : :/Vand let .ˇj /j 0 be the basis of the restricted dual M0 r such that ˇjV .bi / D ıij . Let M0 be the r-th exterior power of M0 . The Grassmann cone GrVof r M0 is the image of the (non-surjective) multilinear alternating map M0r ! r M0 , given by .m1 ; : : : ; mr / 7! m1 ^    ^ mV r. It is well-known, see e.g. [3, Section 4], that m 2 r M0 belongs to Gr if and only if X .ˇi ym/ ˝ .bi ^ m/ D 0; (0.4) i 0

Vr1

where ˇi ym 2 M0 denotes the contraction of m against ˇi (Section 1.3). Equation (0.4) is equivalent to  X X Resz .ˇi z i 1 ym/ ˝ .bj z j ^ m/ D 0; (0.5) i 0

j 0

a trick we learned in [17, Section 7.3]. We combine P(0.5) with the following Vobservation: there are unique formal power series ˙ .z/ WD i 0 ˙i z ˙i 2 EndZ . M0 /ŒŒz ˙1  such that 8 V 0, hi is a homogeneous polynomial of degree i in .e1 ; : : : ; er / provided that, for all 1  j  r, each ej is given weight j . The Schur determinant associated to the sequence Hr and to  2 Pr is by definition  .Hr / WD det.hj j Ci /1i;j r 2 Br : L Using the well–known fact that Br D 2Pr Z   .Hr /, the map r W Br !

r ^

M0 ;

(0.6)

given by  .Hr / 7! Œbr , defines an isomorphism of abelian groups (the “bosonV fermion correspondence of order r”). It enables to equip r M0 with a V structure of free Br -module of rank 1, generated by Œbr0 , that we shall denote by r Mr . We regard  .z/,   .z/ as maps Br ! Br Œz 1  as well, by defining  .z/ .Hr / D r1 . .z/Œbr / and   .z/ .Hr / D r1 .  .z/Œbr /. Denote by  .z/Hr and   .z/Hr , respectively, the sequences ..z/hj /j 2Z and .  .z/hj /j 2Z in Br Œz 1 . The following equalities (Proposition 5.3)  .z/hn D

n X

hnj z j

and

  .z/hn D hn  hn1 z 1

j D0

hold in Br Œz 1 , for all r  1, and Theorem 0.1 admits the following rephrasing:

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V P Theorem 0.2. The element m WD 2Pr a Œbr 2 r M0 belongs to Gr if and only if the equality below holds in Br1 ˝Z BrC1 Resz

P ;2Pr

a a Er1 .z/ . .z/Hr1 / ˝

1   .z/ .HrC1 / D 0. ErC1 .z/ (0.7)

For example, for r D 2 one may easily recover the equation of the Klein’s quadric cutting out the Grassmannian G.2; 4/ in its Plücker embedding (Section 6). Other examples are discussed in the book [13]. They all indicate that even for detecting the Grassmann cone G2 , computations are quite painful, surely not as easy as checking the simpler condition m ^ m D 0. What makes Theorem 0.2 interesting, however, is  that, on one hand, the maps r .z/ W Br ! BrC1 ..z// and r .z/ W Br ! Br1 ..z//, defined by r .z/ .Hr / WD

1 ErC1 .z/

  .z/ .HrC1 /

and



r .z/ .Hr / WD Er1 .z/ . .z/Hr1 / and occurring in Formula (0.7), are precisely truncated versions of the vertex operators displayed in (0.3) and, on the other hand, the Schubert derivations  .z/ and   .z/ are well defined also for r D 1. More precisely, P let .e1 ; e2 ; : : :/ be a sequence of infinitely many indeterminates and E1 .z/ WD i 0 .1/i ei z i . For all p 2 B1 there is r  0 such that p 2 Bs for all s  r. We have: Theorem 0.3. An element  2 B1 WD ZŒe1 ; e2 ; : : : is solution to the equation   1   .z/p D 0 (0.8) Resz E1 .z/ .z/p ˝Z E1 .z/ if and only if there exists s  0 such that r . / 2 Gr for all r  s. It turns out that Equation (0.8) expresses the KP hierarchy (0.2) over the integers. It has been obtained by using the indeterminates ei and hj (which may be interpreted as elementary and complete symmetric polynomials), often more convenient than the variables .xi / used in (0.2) and (0.3). This is also sanctioned in the couple of important and relatively recent articles [12, 16]. By abuse V of notation, let us write Gr ˝Q for the Grassmann cone of decomposable tensors of r .M0 ˝Z Q/. Let .x1 ; x2 ; : : :/ be the sequence of indeterminates over Q, implicitly defined by the equality: 0 1 X xi z i A WD E1 .z/: exp @ i 1

An immediate check shows that B WD B1 ˝ Q D QŒx1 ; x2 ; : : :.

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Corollary 0.4. An element  2 B is a solution to the KP hierarchy (0.2) if and only if there exists s  0 such that r . / 2 Gr ˝ Q for all r  s. The paper is organized as follows. Section 1 sets the preliminaries and notation used throughout the paper. Section 2 recalls a few facts concerning Hasse–Schmidt (HS ) derivations on exterior algebras as introduced in [10] and treated in more details in [13]. The section proclaims the most powerful tool of the theory which we call (as in [10]) integration by parts formula. That the transpose of an HS -derivation is an HS -derivation as well is also shown in Section 2, a fact heavily used to prove Theorem 0.1. Schubert derivations are studied in Section 3, where a few technical lemmas leading to the approximation Br ! Br ..z// of the vertex operators are discussed. A pivotal aspect of Section 5 is that the Schubert derivations  .z/ and its inverse   .z/ enjoy a stability property enabling to define them as maps Br ! Br Œz 1 . Their limit for r ! 1 give rise to the ring homomorphisms B ! BŒz 1 , which enter in the expression of the vertex operators. The crucial property that  .z/;   .z/ commute with taking Schur determinants, proven in Section 5, is obtained by exploiting a powerful determinantal formula due to Laksov and Thorup [20]. Section 6 is entirely devoted to the standard example of decomposable tensors in a second wedge power, faced via Theorem 0.2. Eventually, Section 7 is concerned with the limit of forV mulaV(0.7) for r ! 1, whereby 1 M0 shall be understood as the projective limit lim r M0 in the category of graded modules. There, instead of (re)showing that taufunctions correspond to decomposable tensors in some infinite use V exterior power, we 1 .m/ them to define the Grassman cone G1 as the locus of m 2 1 M0 such that 1 solves the KP hierarchy, where 1 is the analogous of (0.6) for r D 1. We are grateful to S. G. Chiossi, A. Kasman, A. Ricolfi and I. Scherbak for useful comments and corrections and to C. Araujo, M. Jardim, P. Mulassano and P. Piccione for unreserved support. Thanks are also due to the anonymous referee for his/her patient careful reading and remarks, that helped us to improve the shape of the article. This paper is dedicated to Professor Piotr Pragacz, a living example of mathematical experience, on the occasion of his sixtieth birthday.

1 Preliminaries and notation This section is to fix notation and to list the pre-requisites we shall need in the sequel. 1.1. We denote by P the set of all monotonic non increasing sequence  WD .1  2     / of non negative integers all zero but finitely many, said partitions. The length `./ 2 P is ]fi j i ¤ 0g, the number of its non-zero parts; its weight jj WD P i 1 i . We shall denote by Pr the set of all partitions of length at most r and by Pr;n WD f WD .1 ; : : : ; r / 2 Pr j 1  n  rg the elements of Pr bounded by n  r. An expression .1i1 ; 2i2 ; : : : ; / such that all ij are zero, but finitely many, denotes the partition having ij parts equal to j . The

On Plücker equations characterizing Grassmann cones

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Young diagramme of  WD .1 ; : : : ; r / 2 P is an array Y ./ of left-justified rows of boxes, such that the j-th row has j boxes, for 1  j  r. The set Pr is a monoid with respect to the sum  C  D .i C i /i 1 whose neutral element is the null partition .0/ with all the parts equal to zero. If ;  2 Pr , we shall write   if the Young diagram of  is contained in the Young diagram of . abelian group 1.2. We shall denote by M0 the free V L ZŒX V and by B0 WD .b0 ; b1 ; : : :/ its standard basis .1; X; X 2; : : :/. Let M0 WD r 0 r M0 be the exterior algebra V of M0 . Then 0V M0 WD Z and, for all r  1, the r-th exterior power of M0 is the Z-linear span of r B0 WD .Œbr j  2 Pr /, where Œbr WD br ^ b1Cr1 ^    ^ br1C1 . In particular Œbr0 WD Œbr.0/ D b0 ^ b1 ^    ^ br1 . V  1.3. Let m 2 r M0 , r  1. Its contraction, ˇym, against ˇ 2 M0 is the unique Vr1 element of M0 such that r1 ^



.ˇym/ D .ˇ ^ /.m/; 8 2 M0 : (1.1) V V It turns out that ˇy W . M0 ; ^/ ! . M0 ; ^/ is the unique derivation of degree 1 such that ˇym D ˇ.m/, for all m 2 M0 . 1.4. Let B0 D Z and for r  1 denote by Br the polynomial ring ZŒe1 ; : : : ; er . Accordingly, we let E0 .z/ D 1 and Er .z/ D 1  e1 z C    C .1/r er z r 2 Br Œz for r  1. The equality X X 1 hn z n D .1  Er .z//i ; D E .z/ r n2Z i 0

read in the abelian group of formal Laurent series Br ŒŒz 1 ; z, defines the bilateral sequence Hr WD .hj /j 2Z of elements of Br . Then, by construction, hj D 0 if j < 0, h0 D 1 and for all j > 0, hj is a polynomial in e1 ; : : : ; er of weighted degree j , after declaring that ei is given degree i . Remark 1.5. For the terms of the sequence Hr , the more careful notation hr;n should be preferred, in place of just hn , to keep track of their dependence on r. To make the notation less heavy we decided however to drop the subscript r, hoping for the context being sufficient to avoid confusions. L 1.6. It is well known that (e.g. [22, p. 41]) Br D 2Pr Z   .Hr /; where  .Hr / WD det.hj j Ci /1i;j r : Each partition 0 WD .1i1 : : : r ir / is conjugated to a partition  2 Pr , of weight i1 C 2i2 C    C rir , whose Young diagramme is the transpose of Y .0 /. Define L  e WD e1i1    erir and let .Br /w WD jjDw Z  e . The direct sum decomposition L Br WD w 0 .Br /w , shall be referred to as the weight graduation of Br . Then .Br /w is Lthe submodule of Br of all the polynomials of weight w and it coincides with jjDw Z   .Hr /.

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2 Hasse–Schmidt derivations on an exterior algebra 2.1. Given any module M over a commutative ring A with unit, there is an obvious A-module isomorphism EndA .M /ŒŒz ! HomA .M; M ŒŒz/;

(2.1)

where EndA .M /ŒŒz denotes the formal power series with EndA .M /-coefficients in an indeterminate z. If D.z/ 2 EndA .M /ŒŒz, we denote in the same way its image through the Map (2.1). V Definition 2.2.VA Hasse–Schmidt (HS ) derivation of M is an algebra homomorV phism D.z/ W M ! M ŒŒz, i.e: ^ M: (2.2) D.z/.m1 ^ m2 / D D.z/m1 ^ D.z/m2 ; 8m1 ; m2 2 Define a sequence D WD .D0 ; D1 ; : : :/ of endomorphisms of equality: X Dj m  z j WD D.z/m:

V

M through the

j 0

Then Equation (2.2) holds if and only if the sequence D obeys the higher order Leibniz rules: i X Dj m1 ^ Di j m2 ; i  0: (2.3) Di .m1 ^ m2 / D j D0

and D1 is a (usual) derivation of the In particular V D0 is an algebra homomorphism, V , provided that D D id . Moreover, if D0 is an automorphism of ^-algebra M 0 0 M V V M , it turns out that the formal power series D.z/ is invertible in EndA . M /ŒŒz. Denote by D.z/ its inverse. V V Proposition 2.3 (see V [10]). The set HS. M / of all HS -derivations on M is a subalgebra of EndZ . M /ŒŒz, with respect to the product D.z/E.z/ D

j XX .Di ı Ej i /z j ; j 0 i D0

P

P

where i 0 Di z i WD D.z/ and i 0 Ei z i WD E.z/. In particular if D0 is an autoV morphism of M , then the inverse formal power series D.z/ is an HS -derivation if and only if D.z/ is. Proposition V 2.4. Let V f .z/ 2 EndA .M /ŒŒz. There exists a unique HS -derivation Df .z/ W M ! M ŒŒz such that Df .z/m D f .z/.m/ for all m 2 M .

On Plücker equations characterizing Grassmann cones

105

Proof. Let us first prove the existence. For all r  1, consider the unique A-linear V b.z/.m1 ˝  ˝mr / D extension of the map Df .z/ W M ˝r ! r M ŒŒz defined by f f .z/m1 ^    ^ f .z/mr : This map is clearly alternating and thus it factorizes through V V f f a unique homomorphism Dr .z/ W r M ! Vr M ŒŒz such that Dr .z/.m1 ˝    ˝ mr / D f .z/m1 ^    ^ f .z/mr . Each m 2 M is a finite sum m1 C    C ms of V homogeneous elements, i.e. mi 2 i M for some i  0 (notice that Df .z/a D a for P all a 2 A). Define Df .z/m as siD1 Dif .z/mi . We want to show that for m1 ; m2 2 V M Df .z/.m1 ^ m2 / D Df .z/m1 ^ Df .z/m2 : Without loss of Vgenerality, we may assume they are homogeneous with respect to the graduation of M , i.e. m1 D m11 ^    ^ m1r and m2 WD m21 ^    ^ m2s . Thus Df .z/.m1 ^ m2 / D Df .z/.m11 ^    ^ m1r ^ m21 ^    ^ m2s / D f .z/m11 ^    ^ f .z/m1r ^ f .z/m21 ^    ^ f .z/m2s D D .f .z/m11 ^    ^ f .z/m1r / ^ .f .z/m21 ^    ^ f .z/m2s / D Df .z/m1 ^ Df .z/m2 : V b b To prove unicity, let D.z/ be any HS -derivation on M such that D.z/m D f .z/m V for all m 2 M . Then for all homogeneous element m WD m1 ^    ^ mr 2 r M : b b b b D.z/m D D.z/.m 1 ^    ^ mr / D D.z/m1 ^    ^ D.z/mr D f .z/m1 ^    ^ f .z/mr D Df .z/.m1 ^    ^ mr / D D f .z/m:



The main tool of the paper is the following observation for which we omit the totally obvious proof. It is responsible, in our context, of the emergence of the vertex operators. V Proposition 2.5 (Integration by parts). Assume that D.z/ 2 HS. M / is invertible in the sense of Proposition 2.3. Then the integration by parts formula holds: ^     M: (2.4) 8m1 ; m2 2 D.z/m1 ^ m2 D D.z/ m1 ^ D.z/m2 ;  2.6 (Duality). Let now M0 be a free abelian group with countable basis B0 as in Section 1. Let ˇj 2 M0_ WD HomZ .M0 ; Z/ such that ˇj .bi / D ıij . The restricted L  dual of M0 is M0 WD j 0 Z  ˇj . The equality 1 ^    ^ r .m1 ^    ^ mr / WD det. i .mj //1i;j r ; V V   defines a natural identification of r M0 with . r M0 / . In particular .ˇr ^ ˇ1Cr1 ^    ^ ˇr1C1 /2Pr

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is the basis of . ıi1 j1    ıir jr .

Vr



M0 / dual of

Vr

B0 , i.e. ˇi1 ^    ^ ˇir .bj1 ^    ^ bjr / D

Definition 2.7. The transpose of D.z/ 2 HS. V  V  D.z/T W M0 ! M0 ŒŒz, defined by .D.z/T /.m/ D .D.z/m/; Proposition 2.8. If D.z/ 2 HS.

V

V

M0 / is the module homomorphism

8.; m/ 2

^



M0 

^

M0 :

M0 /, then D.z/T is an HS -derivation of

V



M0 . V   Proof. By definition, D.z/T ˇ.m/ D ˇ.D.z/m/ for all ˇ 2 M0 . As each  2 M0 is a V sum of homogeneous components, without loss of generality we may assume    2 r M0 , i.e.  WD 1 ^    ^ r for some i 2 M0 . Thus   D.z/T .1 ^    ^ r /.m1 ^    ^ mr / D 1 ^    ^ r D.z/.m1 ^    ^ mr / D 1 ^    ^ r .D.z/m1 ^    ^ D.z/mr / D det.i .D.z/mj // D det.D.z/T i .mj // D D D.z/T 1 ^    ^ D.z/T r .m1 ^    ^ mr /: V b b D D.z/T , for all  2 The unique HS -derivation D.z/ on M0 such that D.z/ V   r M0 . Then it must coincide with it M0 , coincides with D.z/T when evaluated on V  T and D.z/ 2 HS. M0 /. 

3 Schubert derivations on ZŒX  3.1. With the same notation as Section 1, Prop. 2.4 guarantees the existence of unique HS -derivations X ^ ^ i z i W M0 ! M0 ŒŒz and C .z/ WD i 0

X ^ ^  C .z/ D .1/i  i z i W M0 ! M0 ŒŒz i 0

P

j such that C .z/bi D j 0 bi Cj z and  C .z/bi D bi  bi C1  z. In particular j bi D bi Cj and  j bi D 0 if j  2. They are one the inverse of the other:

C .z/ C .z/ D  C .z/C .z/ D 1V M0 : We shall call them Schubert derivations, in compliance with the terminology introduced in [10, 11]. The motivation comes from following Pieri-like formula ([10, Theorem 2.4]): X Œbr ; .i  0/ (3.1) i Œbr D 

On Plücker equations characterizing Grassmann cones

107

where the sum is taken over all the partitions  2 Pr such that 1  1      r  r and jj D jj C i . In addition, a Giambelli-like formula holds ([10, Formula (17)] or [20, Theorem 0.1]): for all  2 Pr Œbr D  . C /Œbr0 WD det.j j Ci /1i;j r  Œbr0 ; (3.2) L where by conventions j D 0 if j < 0. If M0;n WD n1 j D0 Zbj , formula (3.1) tells us Vr that M0;n is an irreducible representation of the cohomology ring H  .Gr .Cn /; Z/: the latter is in fact generated as a Z-algebra by the special Schubert cycles ci .Qr /, the i -th Chern classes of the universal quotient bundle over Gr .Cn /, traditionally denoted by i . So the reason we are using the same notation, more than anV abuse, is to emphasize that we are working precisely with the same objects, seeing r M0 as a module over the cohomology (as in [4, p. 303]) of the Grassmannian Gr .C1 /. 3.2. We similarly define a sort of “mirror” of the Schubert derivation C .z/, namely ^  X  .z/ WD i z i 2 HS M0 (3.3) i 0

which, by definition, is the unique HS V -derivation such that j bi D bi j if i  j and 0 otherwise. Its inverse in EndZ . M0 /ŒŒz 1 , X   .z/ D .1/j  j z j ; j 0

is the unique HS -derivation such that   .z/bi WD bi  bi 1 z 1 for all i  0. In particular, for all r  0:   .z/Œbr0 D Œbr0 : (3.4) We note V in passing that for j > 0, both  j and j are locally nilpotent, i.e. for all m 2 M0 there exists N 2 N such that .j /N m D 0 (resp. . j /N m D 0). Lemma 3.3. For all r  1, let .1r / be the partition .1; : : : ; 1/, with r parts equal to one. Then for all  2 Pr we have  ˙r Œbr D Œbr˙.1r / : Proof. Indeed  ˙r Œbr (Cf. the defining formulas at the beginning of 3.1 and 3.2) is the coefficient of the monomial z ˙r in the expansion of  ˙ .z/Œbr D  ˙ .z/.br ^    ^ br1C1 / in powers of z. Since  ˙ .z/ is a HS -derivation, we have  ˙ .z/.br ^    ^ br1C1 / D  ˙ .z/br ^    ^  ˙ .z/br1C1 D .br  br ˙1 z ˙1 / ^    ^ .br1C1  br1C1 ˙1 z ˙1 / and is so apparent that the coefficient of z ˙r is Œbr˙.1r / , as desired.



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Lemma 3.4. The following equality holds in .

Vr

M0 /ŒŒz:

 C .z/m D .1/r z r   .z/ r m: Proof. Without loss of generality, we may assume m D Œbr . In this case r

r

C .z/Œb D C .z/.br ^ b1Cr1 ^    ^ br1C1 /

(definition of Œb ) V ( C .z/ 2 HS. M0 /)

D C .z/br ^    ^ C .z/br1C1 D .br  br C1 z/ ^    ^ .br1C1  brC1 z/ D .1/r z

r

(definition of C .z/bj )

.br C1  br z 1 / ^    ^ .brC1  br1C1 z 1 / (highlights .1/r z r )

D .1/r z r Π .z/br C1 ^    ^  .z/brC1  D .1/r z r  .z/.br C1 ^    ^ brC1 / r

D .1/r z r  .z/ r Œb

(definition of  .z/) V (  .z/ 2 HS. M0 /) (3.3 applied to r ).



An analogue of Lemma 3.4 holds for   .z/ as well, up to an additional hypothesis. Lemma 3.5. For all Œbr 2

Vr

M0 such that `./ D r (i.e. r > 0), then

  .z/Œbr D .1/r z r  C .z/ r Œbr :

(3.5)

Notice that if `./ < r, the right hand side of (3.5) is zero and then (3.5) fails to be true in general. Proof. Applying the definition of   .z/ we eventually arrive to the equality:   .z/Œbr D   .z/.br ^ b1Cr1 ^    ^ br1C1 / D .br  br 1 z 1 / ^    ^ .br1C1  br2C1 z 1 /;

(3.6)

by just imitating the first few steps of the proof of Lemma 3.4. In turn, the left hand side of (3.6) can be written as .1/r z r .br 1  br z/ ^    ^ .br2C1  br1C1 z/ D .1/r z r  C .z/br 1 ^    ^  C .z/b1 Cr2 V i.e., using that  C .z/ 2 HS. M0 /: D .1/r z r  C .z/.br 1 ^    ^ br1C1 / D .1/r z r  C .z/ r Œbr ; where the last equality holds because of the hypothesis r > 0.



T Recall Definition 2.7. The next easy lemma identifies V  the transpose C .z/ of C .z/, relating it with the Schubert derivations on M0 .

On Plücker equations characterizing Grassmann cones

Lemma 3.6. Let ˇi Cj .

P

T j j 0 j z

WD  .z/T 2 EndA .

V

109



T M0 /ŒŒz 1 . Then j ˇi D

Proof. In fact, for all k  0: T ˇi .bk / D ˇi .j bk / D ˇi .bkj / D ıi;kj D ıi Cj;k D ˇi Cj .bk / j



which proves the claim.

4 Proof of theorem 0.1 The starting point is the following well–known criterion ([3, Section 4] or, in an infinite dimensional context, [17, Proposition 7.2]). V Proposition 4.1. An element m 2 r M0 belongs to Gr if and only if the equality X .ˇi ym/ ˝ .bi ^ m/ D 0 (4.1) i 0

holds in

Vr1

M0 ˝

VrC1

M0 .

Proof. See e.g. [13, Theorem 6.1.7]. Recall from Section 3.1 and Lemma 3.6 that X X bi z i D C .z/b0 and ˇj z j 1 D z 1  .z/T ˇ0 : i 0

j 0

Equation (4.1) can be rewritten, imitating [3, 17], in the equivalent form Resz .z 1  .z/T ˇ0 /ym ˝ C .z/b0 ^ m// D 0;

(4.2)

i.e. m 2 Gr if and only it satisfies equation (4.2). We have: Proposition 4.2. The following equality holds in

VrC1

M0 :

C .z/b0 ^ m D .1/r z r C .z/  .z/.b0 ^  r m/:

(4.3)

Proof. First of all   C .z/b0 ^ m D C .z/ b0 ^  C .z/m ;

(4.4)

because of integration by parts (2.4). Lemma 3.4 applied to  C .z/m gives, after simplification: b0 ^  C .z/m D .1/r z r   .z/.b0 ^  r m/: Substituting in the last side of (4.4) gives (4.3), as desired.



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Proposition 4.3. The following equality holds in

Vr1

M0 :

.z 1  .z/T ˇ0 /ym D .1/r1 z r  C .z/ rC1 .ˇ0 y .z/m/: V  Proof. Let 2 r1 M0 be arbitrarily chosen. Then .z 1  .z/T ˇ0 ym/ D z 1 . .z/T ˇ0 ^ /m

(4.5)

(by definition (1.1))

D z 1  .z/T .ˇ0 ^   .z/T /m (integration by parts (2.4)) D z 1 .ˇ0 ^   .z/T / .z/m

(definition of  .z/T )

D z 1   .z/T .ˇ0 y .z/m/

(definition (1.1) of contraction)

D z 1 .  .z/T .ˇ0 y .z/m/

(definition of   .z/T )

whence the equality z 1  .z/T ˇ0 ym D z 1   .z/.ˇ0 y .z/m/: Notice now that ˇ0 y  .z/m is a linear combination of elements Œbr1 associated to  partitions of length exactly r  1. Thus we can apply Lemma 3.5 to get z 1  .z/T ˇ0 ym D z 1   .z/.ˇ0 y .z/m/ D .1/r1z r  C .z/ rC1 .ˇ0 y .z/m/:



4.4. Substitution of expressions (4.5) and (4.4) into (4.2) concludes the proof of Theorem 0.1. 

5 The Grassmann cone in a polynomial ring V 5.1.VNotation and convention as in Section 1. Let r Mr be the Br -module structure on r M0 given by: ei Œbr WD  i Œbr ; which turns ei 2 Br into an eigenvalue of  i . Accordingly, we have  C .z/Œbr D Er .z/Œbr and then

0 C .z/Œbr D C .z/ @Er .z/

X

1 hn z n A Œbr D

n 0

0 1 X X hn z n A Œbr D hn Œbr z n : D C .z/ C .z/ @ n 0

n 0

111

On Plücker equations characterizing Grassmann cones

V Thus hi is the eigenvalue of i seen as endomorphism of r Mr , i.e. i Œbr D r h  : In particular, using (3.2), the homomorphism of abelian groups r W Br ! Vi Œb r M0 given by  .Hr / 7!  .Hr /Œbr0 D  . C .z//Œbr0 D Œbr ;

(5.1)

is an isomorphism, as it maps the Z-basisV. .Hr // of Br to the Z-basis Œbr of V r M0 . By abuse of notation, for all m 2 r M0 we shall write m WD r1 .m/; Œbr0 i.e. for the unique element of Br carrying the Br -basis element Œbr0 of

Vr

Mr to m.

Definition 5.2. Let  .z/;   .z/ W Br ! Br Œz 1  be defined as:  .z/ .Hr / WD

 .z/Œbr Œbr0

and

  .z/ .Hr / WD

  .z/Œbr : Œbr0

The  .z/-image of hn D .n/ .Hr / could in principle depend on the integer r. However this is not the case. Proposition 5.3. For all r  1, the following equalities hold in the ring Br Œz 1 :  .z/hn D

n X

hni z j

and

  .z/hn D hn  hn1 z 1 :

(5.2)

j D0

Proof. We have: .  .z/hn /Œbr0 D  .z/.hn Œbr0 /

(definition of  .z/hn )

D  .z/.Œb0r1 ^ br1Cn //

(writing Œbr.n/ as Œb0r1 ^ br1Cn )

D  .z/Œb0r1 ^  .z/br1Cn  Pr1Cn  r1 Œb0 ^ br1Cnj z j D j D0

(since  .z/ 2 HS.

D

Pn j D0

V

M0 /)

(apply (2.2) and the definition of  .z/)

 hnj z j Œbr0 ,

which proves the first equality in (5.2). To prove the second equality of (5.2), we can argue either by observing that   .z/ is the inverse of  .z/ or again by direct computation: .  .z/hn /Œbr0 D   .z/.hn Œbr0 / D   .z/Œbrn D   .z/Œbr1 ^   .z/br1Cn 0 D Œbr1 ^ .br1Cn  br1Cn1 z 1 / D .hn  hn1 z 1 /Œbr0 :  0

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Proposition 5.4. We have: b0 ^  r Œbr D ŒbrC1 D  .HrC1 /ŒbrC1 : 0 

(5.3)

Proof. Indeed b0 ^  r Œbr D b0 ^  r .br ^ b1Cr1 ^    ^ b1Cr / D b0 ^ b1Cr ^ b2Cr1 ^    ^ brCr

(definition of Œbr ) (Lemma 3.3)

(definition of ŒbrC1 D ŒbrC1   ) VrC1 Mr we have then equality (5.3) (due to rC1 . .HrC1 // D In the BrC1 -module , by (5.1)).  ŒbrC1  In other words the expressions of  .z/hn and   .z/hn in the ring Br Œz 1  do not depend on the integer r. 5.5. Let us agree to simply write Res.f / for the coefficient of X 1 of a formal Laurent series f 2 Z..X 1//. To prove Theorem 5.7 below, we need a powerful result due to Laksov and Thorup [20, Theorem 0.1.(2)] (see also [21]). Let us introduce a few new pieces of notation. Let   1 D X r  e1 X r1 C    C .1/r er 2 Br ŒX ; pr .X / D X r Er X be the generic polynomial of degree r. If f WD a0 X  C a1 X 1 C    C a 2 Br ŒX  is any polynomial of degree  , then an easy computation shows that     i 1 X i 1 f .X / h1 h2 X f .X / WD Res 1 C C    Res C pr .X / Xr X X2 (5.4)  X aj hi rCj : D j D0

for all i  0. Following [20], the residue of .f0 ; f1 ; : : : ; fr1 / 2 Br ŒX r is, by definition: ˇ ˇ ˇ Res.f0 / Res.f1 /  Res.fr1 / ˇˇ ˇ ˇ Res.Xf0 / Res.Xf1 /  Res.Xfr1 / ˇˇ ˇ Res.f0 ; f1 ; : : : ; fr1 / D ˇ ˇ: :: :: :: :: ˇ ˇ : : : : ˇ ˇ ˇRes.X r1 f0 / Res.X r1 f1 /    Res.X r1 fr1 /ˇ (5.5) We shall use the following

On Plücker equations characterizing Grassmann cones

113

Theorem 5.6 ([20], Theorem 0.1 (2)). Let f0 ; f1 ; : : : ; fr1 2 Br ŒX . Then f0 .1/b0 ^ f1 .1 /b0 ^    ^ fr1 .1 /b0  D Res

 f0 .X / fr1 .X / fr2 .X / b0 ^ b1 ^    ^ br1 : ; ;:::; pr .X / pr .X / pr .X /

   i 1 X frj .X / b0 ^ b1 ^    ^ br1 : D det Res pr .X / 1i;j r

(5.6)

Denote by  .z/Hr and   .z/Hr , respectively, the following Br Œz 1 -valued sequences: ! j X hj i z i and .  .z/hj /j 2Z D .hj  hj 1 z 1 /j 2Z : ..z/hj /j 2Z D i D0

j 2Z

Theorem 5.7. The operators  .z/;   .z/ W Br ! Br Œz 1  commute with taking Schur determinants, i.e.:   .z/ .Hr / D  .  .z/Hr /

and

 .z/ .Hr / D  ..z/Hr /: (5.7)

Proof. To prove the first of equalities (5.7), one observes that .  .z/ .Hr //Œbr0 D   .z/Œbr D   .z/br ^    ^   .z/br1C1 D f0 . 1 /b0 ^ f1 .1 /b0 ^    ^ fr1 .1 /b0 ; where fi . 1 / stands for fi .X / D X ri  X ri 1 z 1 evaluated at X D 1 . By Theorem 5.6 and formula (5.4): f0 .1 /b0 ^ f1 .1 /b0 ^    ^ fr1 .1 /b0   f0 .X / fr1 .X / b0 ^ b1 ^    ^ br1 D Res ;:::; pr .X / pr .X / D det.hj j Ci  hj j Ci 1 z 1 /1i;j r b0 ^ b1 ^    ^ br1 D  . .z/Hr /b0 ^ b1 ^    ^ br1 : To verify the second equality of (5.7) we exploit the first one, just proven, and the fact that  .z/ and   .z/ are one inverse of the other. Then  .z/ .Hr / D  .z/ .  .z/ .z/Hr / D  .z/  .z/ . .z/Hr / D  . .z/Hr /:



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Lemma 5.8. For all  2 Pr , the following equality holds in

Vr1

M0

 rC1 .ˇ0 yŒbr / D  .Hr1 /Œbr1 0 :

(5.8)

Proof. If `./ D r then r > 0 and then ˇ0 yŒbr D 0. So, both sides of (5.8) vanish. If `./ < r instead, one may write Œbr D b0 ^ Œbr1 C.1r1 / ; where  C .1r1 / D .1 C 1; : : : ; r1 C 1/. Thus r1 r1  rC1 .ˇ0 y.b0 ^ Œbr1 D  .Hr1 /Œbr1 0 C.1r1 / // D  rC1 .ŒbC.1r1 / / D Œb



as claimed. Proposition 5.9. The equality below holds in

Vr1

M0 :

 rC1 .ˇ0 y .z/Œbr / D  ..z/Hr1 /Œbr1 0 Proof. If one defines .n/ . .z/H0 / to be z n , the equality holds for r D 1. To check the formula in the remaining cases, let us preliminary observe that  . .z/Hr / is a P linear combination   a .z 1 / .Hr /, whose coefficients a .z 1 / 2 ZŒz 1  do not depend on the chosen r > 1. Thus:      rC1 ˇ0 y .z/Œbr D  rC1 ˇ0 y..z/. .Hr /Œbr0 /   D  rC1 ˇ0 y..z/ .Hr //Œbr0 /   D  rC1 ˇ0 y. . .z/Hr /Œbr0 0 1 X a .z 1 / .Hr /Œbr0 A D  rC1 @ˇ0 y D

X

 

a .z 1 /   rC1 .ˇ0 y .Hr /Œbr0 /

 

D

X

a .z 1 /   .Hr1 /Œbr1 0

 

D  . .z/Hr1 /Œbr1 0 :





5.10. Let r .z/ W Br ! BrC1 ..z// and r .z/ W Br ! Br1 ..z// defined by: r .z/ .Hr / WD and

1

  .z/ .HrC1 /;

(5.9)

r .z/ .Hr / WD Er1 .z/ . .z/Hr1 /:

(5.10)



ErC1 .z/

On Plücker equations characterizing Grassmann cones

115

Expression (5.9) can be also written in the form r .z/ .Hr / D

1  .  .z/HrC1 /; ErC1 .z/

by virtue of Theorem 5.7, according which   .z/ .HrC1 / D  .  .z/HrC1 / for all  2 PrC1 . Similarly, the equality  . .z/Hr1 / D  .z/ .Hr1/ surely holds for all  2 Pr1 . However, if `./ D r, in general  . .z/Hr1 / ¤  .z/ .Hr1 /, because  ..z/H of Pr1 / ¤ 0 in spite of the vanishing P n n  .Hr1 /. For example, if r D 1, n 0 hn z n D .1  e1 z/1 D e z , n 0 1 i.e. hn D hn1 . This implies ˇ ˇ ˇh 1 ˇˇ D h21  h2 D h21  h21 D 0: .1;1/ .H1 / D ˇˇ 1 h2 h1 ˇ On the other hand

ˇ ˇ ˇ ˇ 1 ˇ h1 C 1 ˇˇ ˇ z ˇ ˇ .1;1/ ..z/H1 // D ˇ ˇ ˇ ˇ ˇh C h1 C 1 h C 1 ˇ ˇ 2 ˇ 1 2 z z z  2 1 h1 h1 1 D h1 C  h2   2 D ¤ 0: z z z z P Proof of Theorem 0.2. According to Theorem 0.1, it follows that 2Pr;n a Œbr 2 Gr if and only if X rC1 Resz a a  C .z/ rC1 .ˇ0 y .z/Œbr1  / ˝Z C .z/  .z/.b0 ^  r Œb / ;2Pr

(5.11) V V Vr1 vanishes in . r Mr1 ˝ r MrC1 /..z//. Since  rC1 .ˇ0 y .z/Œbr1 0 / 2 Mr1 , it is an eigenvector of  C .z/ corresponding to the eigenvalue Er1 .z/. SimiV larly,   .z/.b0 ^  r Œbr / belongs to rC1 M0 , which is an eigenmodule of C .z/ corresponding to the eigenvalue 1=ErC1 .z/. Thus, expression (5.11) can be rewritten as X a a Er1 .z/. rC1 .ˇ0 y .z/Œbr / 0 D Resz ;2Pr

1

  .z/.b0 ^  r ŒbrC1  /D ErC1 .z/ X Er1 .z/ . .z/Hr1 /Œbr1 D Resz 0 ˝

;2Pr

˝

1   .z/ .HrC1 /ŒbrC1 0 ErC1 .z/

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0 D @Resz

1

X

Er1 .z/ ..z/Hr1 /Œbr1 ˝ 0

;2Pr

Œbr1 0

1   .z/ .HrC1 /A ErC1 .z/

˝ ŒbrC1 ; 0

where in last equality we used Propositions 5.4 and 5.9. Let us now identify the tensor product Br1 ˝Z BrC1 with the polynomial ring in 2r indeterminates 0 00 Br1 ˝ BrC1 D ZŒe10 ; : : : ; er1 ; e100 ; : : : ; erC1 :

(5.12)

V V Then (0.7) follows because Œbr1 ˝ ŒbrC1 is a basis of r1 Mr1 ˝Z rC1 MrC1 0 0 over Br1 ˝Z BrC1 thought of a ring through the indentification (5.12).  0 0 0 00 5.11. Let Er1 .z/ D 1  e10 z C    C .1/r1 er1 z r 2 Br1 Œz and ErC1 .z/ D 00 rC1 00 rC1 0 1  e1 z C    C .1/ erC1 z 2 BrC1 Œz. Similarly, let 0 .z/ D Hr1

X

h0n z n

00 Hr1 .z/ D

and

n 0

X

h00n z n

n 0

0 00 0 00 be the inverses of Er1 .z/ and ErC1 .z/ in Br1 ŒŒz Š BrC1 ŒŒz and BrC1 ŒŒz respectively. Then formula (0.7) reads

Resz

0 Er1 .z/ X 0 00 a a  . .z/Hr1 /    .z/ .HrC1 / D 0: 00 ErC1 .z/

(5.13)

;

6 An example To show how formula (5.13) works, in this section we proceed to determine all the polynomials p.H2 / D a0 C a1 h1 C a2 h2 C a11 .11/ .H2 / C a21 .21/ .H2 / C a22 .22/ .H2 / 2 B2 ;

(6.1)

V such that p.H2 /Œb20 belongs to the Grassmann cone G2 of 2 M0 . Needless to say, we expect to find the same expression of the Klein quadric hypersurface of P5 . For this purpose we need a few preliminaries to speed up computations, which by the way could be carried out directly. 6.1. Consider the two sub-modules n M0 WD V of r M0 . They fits in the exact sequence

L

tn

i 0 Z bnCi

and M0;n WD

0 ! n M0 ! M0 ! M0;n ! 0;

Ln1 i D0

Zbi

On Plücker equations characterizing Grassmann cones

117

which maps bi to itself where tn is the truncation V L if 0  i  nV 1 and to zero otherwise. Hence r M0;n is the submodule 2Pr;n ZŒbr of r M0 . It can be V V seen itself as the epimorphic image of the truncation tnr W r M0 ! r M0;n which maps Œbr to itself if  2 Pr;n and to 0 otherwise. Again, we have an exact sequence 0 ! n M0 ^

r1 ^

M0 !

r ^

M0 !

r ^

M0;n ! 0;

Vr1 V where M0 is the submodule of r M0 generated by all bi1 ^  ^bir 2 Vr n M0 ^ M0 such that ij  n for at least one 1  j  r. Vr r Let Ir;n be M0;n , where Vrthe kernel of the composition tn ı r W Br ! M0 is the isomorphism (5.1). It is a simple exercise to check that r W Br ! hnrC1Cj 2 Ir;n for all j  0. Indeed, a simple argument (see [13, Chapter 5] or [11] for details) shows that Ir;n D .hnCr1 ; : : : ; hn /. Let ZŒe1 ; : : : ; er  .hnrC1 ; : : : ; hn / L be the canonical projection. Then Br;n D 2Pr;n Z .Hr;n /, where by Hr;n we have denoted the sequence .hj C Ir;n /j 0 of Br;n and then  .Hr;n/ D  .Hr / C Ir;n. Clearly,  .Hr;n/Œbr0 D  .Hr /Œbr0 if  2 Pr;n. Define i .hj C Ir;n / D j hn CIr;n (resp.  i .hj CIr;n / D  j hn CIr;n). Then Theorem 0.2 has the following n W Br ! Br;n WD

Corollary 6.2. A tensor Resz

X

P

a Œbr is decomposable if and only if

2Pr;n

a a Er1 .z/ . .z/Hr1;n / ˝

;2Pr;n

1  .  .z/HrC1;n / D 0 ErC1 .z/

in the tensor product .Br1;n ˝Z BrC1;n /..z//. Proof. In fact by Theorem 0.2, 0 D Resz

P 2Pr;n

X

a Œbr 2 Gr if and only if

Er1 .z/ ..z/Hr1 /Œbr1 0

;2Pr

1   .z/ .HrC1 /ŒbrC1 0 ErC1 .z/ X D Resz Er1 .z/ ..z/Hr1;n /Œbr1 0 ˝

;2Pr

˝

1 ErC1 .z/

  .z/ .HrC1;n /ŒbrC1 : 0

Moreover formula (5.13) and Corollary 6.2 easily imply:



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Corollary 6.3. The tensor Resz

0 Er1 .z/ 00 ErC1 .z/

X

P 2Pr;n

a Œbr 2

Vr

M0 is decomposable if and only if

0 00 a a  ..z/Hr1;n /    .z/ .HrC1;n / D 0:



;2Pr;n

6.4. Going back to our original purpose of finding all p.H2 / 2 B2 such that p.H /Œb20 2 G2 , where p.H2 / is like in (6.1), we use the B2 -module structure of V V V2 2 M WD B2 ˝ 2M0 . Since p.H2 /Œb20 2 2M2;4 , the B2 -module structure of V2 2 M2 factorizes through that of B2;4 , i.e. p.H2 /Œb20 D p.H2;4 /Œb20 . We have B1;4 WD

B1 ZŒx Š 4 ; .h4 / .x /

and

B3;4 WD

B3 ZŒy Š 4 ; .h2 ; h3 ; h4 / .y /

where we have set x D e1 C.h4 / and y D e1 C.h2 ; h3 ; h4 /. In particular, hi C.h4 / D x i and h1 C .h2 ; h3 ; h4 / D y. Thus p. .z/H1;4 / D a0 C a1  .z/h1 C a2  .z/h2 C a11 .11/ . .z/H1 / C a21 .21/ . .z/H1 / C .22/ . .z/H1 / C I1;4 D a0 C a1 .x C z 1 / C a2 .x 2 C xz 1 C z 2 / from which X



2 .z/p.H2;4 / D E1 .z/

a  . .z/H1;4 // C I1;4

2P2;4

     1 x 1 2 C a2 x C C 2 C a0 C a1 x C z z z  2 2 x x x x C a11 C a21 C a22 2 : C z z2 z z

1  xz D z2

On the other hand p.  .z/H3;4 / D

X

a  .  .z/H3;4 /

2P2;4

D

X

a.1 ;2 / det.hj j Ci 1  hj j Ci z 1 / C I3;4

2P2;4

and so the equality 2 .z/p.H2;4 / D

X z2 a.1 ;2 / det.hj j Ci 1 hj j Ci z 1 /CI3;4 (6.2) E3 .z/ 2P2;4

119

On Plücker equations characterizing Grassmann cones

holds in B3;4 ..z//. Since 1 C I3;4 D 1 C h1 z C I3;4 D 1 C yz; E3 .z/ formula (6.2) becomes:

    1 y 1 y 2  a2 C a11 y  C 2 2 .z/p.H2;4 / D z .1 C yz/ a0 C a1 y  z z z z   y2 y2 y C a22 2 :  C a21 z2 z z 2

A few computations, carried out by means of the CoCoA software [1], eventually tell 

Resz .2 .z/p.H2;4 //  .2 .z/p.H2;4 // D .a11 a2 C a1 a21  a0 a22 /x 3 C .a11 a2  a1 a21 C a0 a22 /x 2 y  .a11 a2  a1 a21 C a0 a22 /xy 2 C .a11 a2  a1 a21 C a0 a22 /y 3 D .a11a2  a1 a21 C a0 a22 /.y 3  y 2 x C x 2 y  x 3 /: So, the latter expression is identically zero if and only if the following Plücker equation holds: (6.3) a11 a2  a1 a21 C a0 a22 D 0:

7 The Grassmann cone in infinite exterior power Proposition 7.1. Let r  1 be fixed. Then for all .i1 ; : : : ; ir / 2 Nr , the equalities below hold in Br :  .z/.hi1    hir / D  .z/hi1     .z/hir and   .z/.hi1    hir / D   .z/hi1      .z/hir :

(7.1)

Proof. Let us begin by checking the first of (7.1). Since . .HrP / j  2 Pr / is a basis of Br , each product of the form hi1    hir is a linear combination 2Pr;n a  .Hr /. Therefore X X a  .Hr // D a  .z/ .Hr /  .z/.hi1    hir / D  .z/. D

X

2Pr;n

2Pr;n

a  ..z/Hr /;

2Pr;n

where last equality is due to Theorem 5.7. In other words: X a det. .z/hj j Ci / D  .z/hi1     .z/hir :  .z/.hi1    hir / D 2Pr;n

The proof for   .z/ is evidently analogous.



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7.2. Let B1 be the polynomial ring ZŒe1 ; e2 ; : : : in infinitely many indeterminates .e1 ; e2 ; : : :/. It is the projective limit of the rings Br in the category of graded Zalgebras, in the following sense. Recall notation and convention of Section 1.6. For all s > r there are obvious projection maps .Bs /w ! .Br /w . They are defined by  .Hs / 7!  .Hr / if  2 Pr and by  .Hs / 7! 0 otherwise. Let .B1 /w WD lim.Br /w . Clearly, for all w 2 N there exists r > 0 such that .B1 /w D .Br /w , the module of polynomials in .e1 ; : L : : ; er / of weighted degree w (See 1.6). The ring B1 is by definition the direct sum Lw 0 .B1 /w . In particular, for all w  0 there exists s  0 such L that .B1 /w WD 0i w .B1 /i is isomorphic, as abelian group, to .Br /w WD 0i w .Br /i , for all r  s. P Let E1 .z/ D 1  e1 z C e2 z 2 C    and n 0 hn z n D .E1 .z//1 2 B1 ŒŒz. In this case the non-zero terms of the sequence H1 D .hj /j 2Z are algebraically independent and so B1 is the free polynomial algebra ZŒh1 ; h2 ; : : :. Moreover B1 D L  .H1 /.  2Pr Corollary 7.3. The maps  .z/;   .z/ W B1 ! B1 Œz 1  are ring homomorphisms and are each other’s inverses, when regarded as elements of .EndZ .B1 //Œz 1 . Proof. Consider an arbitrary product in hi1    his 2 B1 and let w D i1 C    C is . There exists a sufficiently large r > maxfw; sg such that .B1 /w D .Br /w . For such a choice of r, we have  .z/.hi1    his / D  .z/hi1     .z/his and   .z/.hi1    his / D   .z/hi1      .z/his 

by virtue of Proposition 7.1.

7.4. Proof of Corollary 0.3. Each p 2 B1 is a finite linear combination of , where P is the set of all the partitions. Then there exist 0  r1  n . .H1 //2PP such that p WD 2Pr ;n a  .H1 /. Notice that the maximum weight of the parti1 tions possibly occurring in the sum is w WD r1 .n  r/. Suppose that there is r2  0 such that r .p/ 2 Gr for all r  r2 . Then by Theorem 0.2 Resz

X ;2Pr1 ;n

a a Er1 .z/ . .z/Hr1 / ˝

1 ErC1 .z/

  .z/ .HrC1 / D 0;

(7.2) in .Br1 /w ˝ .BrC1 /w (notation as in 7.2). Fix r  max.r1 ; r2 ; w/ big enough such that .B1 /w is isomorphic to .Br /w . Then the left hand side of (7.2) is equal to 0 1 X X a a Er1 .z/ . .z/Hr1 / ˝ @ hi z i A    .z/ .HrC1 /; Resz ;2P

0i r

(7.3)

On Plücker equations characterizing Grassmann cones

121

P because, to compute the residue, the formal power series i 0 hi z i D .ErC1 .z//1 contributes with only finitely many summands. Therefore, invoking the isomorphism .B1 /w Š .Br /w , (7.3) is equivalent to X X 0 D Resz a a Er1 .z/ . .z/H1 / ˝ hi z i    .z/ .H1 /; (7.4) ;2P

0i r

P in .B1 ˝ B1 /..z//. We may now replace Er1 .z/ with E1 .z/ and 0i r hi z i P with i 0 hi z i D .E1 .z//1 in formula (7.4), because adding powers of z does not alterate the residue. We have so proven that if r .p/ 2 Gr for a big enough r, then it satisfies equation (0.8). Conversely, if p satisfies (0.8), then (7.4) holds, and since the weight of the partitions involved to express p as linear combination of Schur polynomials is bounded by a given positive integer w, there exists a big enough s such that .Br1 /w and .BrC1 /w are both isomorphic to .B1 /w , for all r  s. Thus equation (7.4) is equivalent to (7.3) and then to (7.2), i.e. r .p/ 2 Gr for all r  s. As Corollary 7.3 implies the commutation  . .z/H1 / D  .z/ .H1 /, for all  2 P, equation (7.2) can be rewritten as X X 1 0 D Resz E1 .z/a  .z/ .H1 / ˝ a   .z/ .H1 / E1 .z/ 2P 2P

D Resz E1 .z/ .z/p ˝

1   .z/p: E1 .z/



7.5. Let B WD B1 ˝Z Q and define the sequence X WD .x1 ; x2 ; : : :/ through the equality X X hn z n D exp. xi z i /; n 0

i 1

holding in BŒŒz, in such a way that each hn can be regarded as a function of .x1 ; x2 ; : : :/. Standard calculations show that hn is a polynomial expression of .x1 ; : : : ; xn /, homogeneous of degree n with respect to the weight graduation of B1 (for which hn and xn have degree n). Lemma 7.6. The following equalities hold in the ring B for all j  1 and n  0: @j hn j @x1

D

@hn D hnj : @xj

Proof. For all j  1:

0 1 X X @ @ zn D hn z n D exp @ xj z j A @x @x @x j j j n 0 n 0 j 1 0 1 X X D z j exp @ xj z j A D hn z nCj : X @hn

j 1

n 0

(7.5)

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Comparing the coefficient of z n in the first and last side gives @hn D hnj : @xj In particular @hn =@x1 D hn1 . Iterating j times the operator @=@x1 gives (7.5), as desired.  Let 1 .z/ .H1 / WD and

1   .z/ .H1 / 2 B1 ..z// E1 .z/



1 .z/ .H1 / WD E1 .z/ .z/ .H1 / 2 B1 ..z//: 



Define .z/;  .z/ to be, respectively, 1 .z/ ˝ 1Q W B ! B..z// and 1 .z/ ˝ 1Q W B ! B..z//. Corollary 0.4 follows immediately from Corollary 0.3 and the following: Theorem 7.7 (Cf. [17], Theorem 5.1). We have: 0 0 1 1 X 1 @ X A xi z i A exp @ .z/ D exp @ iz i @xi i 1

0

and 

 .z/ D exp @

X i 1

Proof. Since X

(7.6)

i 1

1

0

1 X 1 @ A: xi z i A exp @ iz i @xi

(7.7)

i 1

0

X

1

1 hn z n D exp @ xi z i A ; D E1 .z/ n 0 i 1  P  i it follows that E1 .z/ D exp  i 1 xi z and the first factors involved on the left hand side of (7.6) and (7.7) are explained. Let us now observe that:   hn1 1 @ hn :   .z/hn D hn  D 1 z z @x1 1 0 X tn A at t D z 1 @ , and Evaluating the well–known identity 1  t D exp @ n @x1 n 1 using (7.5), we have 0

0 1 1 X 1 @i X 1 @ A hn D exp @ A hn :   .z/hn D exp @ iz i @x1i iz i @xi i 1 i 1

(7.8)

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On Plücker equations characterizing Grassmann cones

0

1 X 1 @ A W B ! BŒz 1  is a ring homomorphism, Now, we observe that exp @ iz i @xi i 1 X 1 @ : because it is the exponential of the first order differential operator  iz i @xi i 1

0

Thus

1

  .z/ D exp @

X 1 @ A; iz i @xi i 1

because both sides are ring homomorphisms and by (7.8) they coincide on hn , for all n  0, which generate B as a Q-algebra. The proof that 0 1 X 1 @ A  .z/ D exp @ iz i @xi i 1

is similar, but arguing that  .z/ is the inverse of   .z/ in EndQ .B/Œz 1  turns it easier.  Remark 7.8. As pointed out in the introduction, for each  -function, the normalized first order formal pseudo-differential operator L WD P .@/@P .@/1 , where P .z/ D

  .z/ 2 B.0/ Œz 1  

is a solution of the KP hierarchy in Lax form (0.1). See e.g. [17, Section 7.5]. V L V 7.9. Let . r M0 /w WD jjDw Z  Œbr . The isomorphism Br ! r M0 does reVr . M0 /w and V the map .Br1 /w ! .Br2 /w , for all strict to an isomorphism .Br /w ! V r1  r2 , induce the epimorphism . r1 M0 /w ! . r2 M0 /w mappingVŒbr1 7! Œbr2 1 r (where M0 /w WD Vr by convention Œb D 0 if `./  r). The projective limit . M0 /w , with respect to the above projection maps, may be identified with the lim. free abelian group generated symbols Œb1  ranges over all the parti , whereL V1 V1 by theL M0 D w 0 . M0 /w D 2P ZŒb1 tions of weight w. Let  and 1 W B1 !

1 ^

M0

be the Z-isomorphism defined by  .H1 / 7! Œb1  (the boson-fermion correspondence). ThenV Corollaries 0.3 and 0.4 show thatL one may safely define the Grassmann 1 cone G1 1 M0 as the locus of all m 2 2P ZŒb1 1 .m/ satis such that V fies (0.8). Similarly, the Grassmann cone G1 ˝ Q is the locus of m 2 1 .M0 ˝ Q/ such that .1 ˝ 1/1.m/ 2 B is a tau function for the KP hierarchy (0.2).

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[19] M. E. Kazarian and S. K. Lando, An algebro-geometric proof of Witten’s conjecture. J. Amer. Math. Soc. 20(4) (2007), 1079–1089. [20] D. Laksov and A. Thorup, A Determinantal Formula for the Exterior Powers of the Polynomial Ring. Indiana Univ. Math. J. 56(2) (2007), 825–845. [21] D. Laksov and A. Thorup, Schubert Calculus on Grassmannians and Exterior Products. Indiana Univ. Math. J. 58(1) (2009), 283–300. [22] I. G. Macdonald, Symmetric Functions and Hall Polynomials, Second Edition. Clarendon Press, Oxford, 1995. [23] M. Mulase, Algebraic theory of the KP equations. Perspectives in Mathematical Physics, Conf. Proc. Lecture Notes Math. Phys., III, Int. Press, Cambridge, MA, 151–217, 1994. [24] M. Mulase, Matrix integrals and integrable systems. Topology, geometry and field theory, 111–127, World Sci. Publ., River Edge, NJ, 1994. [25] A. Pressley and G. Segal, Loop groups. Oxford Mathematical Monographs, Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1986. [26] M. Sato, Soliton Equations as Dynamical Systems on infinite dimensional Grassmann Manifolds. RIMS Kokioroku 439 (1981), 30–46. [27] M. Sato, The KP Hierarchy and Infinite-Dimensional Grassmann Manifolds. In Theta Functions—Bowdoin 1987, Part 1 (L. Ehrenpreis, R. C. Gunning, eds.), Proc. of Symposia in Pure Math. 49, Amer. Math. Soc., Providence, RI, 1989, 51–66. [28] G. Segal and G. Wilson, Loop Groups and Equations of KdV Type. Inst. Hautes Études Sci. Publ. Math. 61 (1985), 5–65.

Kempf–Laksov Schubert classes for even infinitesimal cohomology theories Thomas Hudson1 and Tomoo Matsumura2 Contents 1 2

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Algebraic cobordism and infinitesimal theories . . . . . . . . . . . . . . . . . . . . . 2.1 Oriented cohomology theories . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Chern classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 The construction of the Lazard ring and its multiplicative structure . . . 2.4 Infinitesimal theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Segre classes for infinitesimal theories . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Determinantal formula for Grassmann bundles . . . . . . . . . . . . . . . . . . . . . 4.1 Degeneracy loci and Kempf–Laksov resolutions . . . . . . . . . . . . . . . 4.2 Computing  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Pfaffian formula for Lagrangian Grassmannian . . . . . . . . . . . . . . . . . . . . . 5.1 Lagrangian degeneracy loci and their Kempf–Laksov resolutions . . . . 5.2 Computing the class C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Appendix: Symmetric functions and their evaluation on virtual vector bundles References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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1 Introduction The purpose of this paper is to present an extension of the classical Kempf–Laksov formula for the Chow ring to an elementary family of oriented cohomology theories known as infinitesimal theories. In 1902, Giambelli [7] described the fundamental classes of the Schubert varieties of the Grassmannian in a closed, determinantal expression involving the Chern classes of the tautological vector bundle. Later, in 1974,

1 A considerable part of this work took place while I was affiliated to GAIA at POSTECH, which I would like to thank for the excellent working conditions. This work was supported by the National Research Foundation of Korea (NRF) grants funded by the Korea government (MSIP) (GAIA, No.2011-0030044) and (MSIP) (ASARC, NRF-2007-0056093). Thanks are also due to Marc Levine and Jerzy Weyman, who introduced me to infinitesimal theories. 2 This work was supported by Grant-in-Aid for Young Scientists (B) 16K17584. Together with my coauthor I would like to thank the anonymous referee, whose comments significantly improved the readability of the paper.

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Kempf and Laksov [14] extended Giambelli’s formula to the Grassmann bundles associated to a vector bundle. The proof essentially consists in the construction, through a tower of projective bundles, of a resolution of singularities and in a Gysin computation, which produces the formula for the fundamental class. Recently, in [11], together with T. Ikeda and H. Naruse, we generalised such computation to K-theory, obtaining a determinantal formula for the Schubert classes. This was achieved by combining the geometric input given by Kempf–Laksov’s resolution with an algorithmic procedure modelled after the one used by Kazarian in [13]. Given this state of things it seemed reasonable to try and see whether or not this procedure could be further generalised to other, more comprehensive, oriented cohomology theories and in [12] we succeeded in extending this approach to algebraic cobordism. In algebraic geometry the notion of oriented cohomology theories was introduced in [16] by Levine and Morel, who were inspired by the work of Quillen in [18] on the category of differential manifolds. Among such theories, algebraic cobordism  represents the universal one and can be used to construct other theories with prescribed formal group laws. The main object of study of this paper, the infinitesimal theories In belong to such class of functors. In general, every oriented cohomology theory is characterised by a formal group law, which encodes the expansion of the first Chern class of the tensor product of two line bundles. More specifically, as one of the features of the universal cohomology theory, Levine–Morel identified the coefficient ring of  with the Lazard ring L, a polynomial ring with integer coefficients in infinitely many variables equipped with the universal formal group law. If on the one hand this is a testimony of the amount of information encoded inside of  , a clear drawback is that explicit computations become far more difficult to perform. However, it should be noticed that this is no longer the case for the In ’s: their formal group laws are given by polynomial expressions in which only one of the variables of L appears. To a certain extent they can be considered as the simplest oriented cohomology theories beyond K-theory. Before we say more about infinitesimal theories, let us mention one last important difference with K-theory. For a general theory not all Schubert varieties have a well defined notion of fundamental class and as a consequence the computation of the Schubert classes depends on the choice of a resolution, which may not be unique. In our context the classes associated to Kempf–Laksov’s resolutions seemed to be a natural choice since they happen to be stable along the natural inclusions connecting the Grassmann bundles associated to bundles of increasing rank, a fact that allowed us to define a generalisation of Schur/Grothendieck polynomials. Another option, developed in [17] by Nakagawa and Naruse for complex cobordism, is to consider Hall–Littlewood resolutions. It is worth pointing out that for flag varieties and flag bundles another generalisation of Schubert classes was studied in [3, 4, 8, 9, 10, 15], by taking into consideration Bott–Samelson resolutions. In order to be able to appreciate how infinitesimal theories differ from  in terms of complexity, it can be worth to have a look at their precise definition. Let ˛n be an indeterminant and set Qn to be quotient of ZŒ˛n  by the relation ˛n2 D 0, with grading given by deg ˛n D n. Following [1, Part II, §5 and §7], in Section 2.3 we introduce the surjective ring homomorphism L ! Qn and set In WD  ˝L Qn . To further

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simplify things, in this paper we will restrict our attention to the case n D 2m. In  the formal group law has the following special fact, as proven in Lemma 2.6, for I2m form " !# 2m1 X .2m/ i 2mi u  v D .u C v/ 1 C ˛2m : (1.1) i uv i D1

Here one sets i.2m/

D

1 d2m

"

! # 2m i  .1/ ; i

with dj being equal to a prime p, in case j D pe  1 for some integer e, and 1 otherwise. One important consequence of (1.1) is that the formal inverse, the power series expressing the first Chern class of a dual line bundle, becomes particularly simple. Actually, since u  v is divisible by u C v, one obtains ˇu D u or, in other words, c1 .L_ / D c1 .L/ precisely as in the Chow ring. This is yet further evidence of how dealing with infinitesimal theories only involves a mild increase in complexity with respect to the classical setting. As in both [11] and [12], a key role in our computations is played by the Segre classes of vector bundles Sl .E/. Although in all three cases the definition we use is the exact analogue of that given by Fulton in [6], it is worth mentioning that the derivation of the main properties presented in this paper is independent of and slightly different from the one in [12]. In fact, in view of the simplicity of the formal group law, it is possible to avoid some steps in the construction and the resulting expressions become more explicit. In particular, a direct usage of Quillen’s formula (which for  was established by Vishik in [20]) allows us to describe each Segre class as a symmetric function in the Chern roots. We go on to obtain a generating function for the whole collection of Segre classes, which is then used to define the relative Segre classes Sl .E  F /, so to extend the realm of definition to the Grothendieck group of vector bundles. Finally, we recover the following geometric interpretation which, as in [12], constitutes the main ingredient of our derivation of the Kempf–Laksov formula. Proposition A (Propositions 3.2, 3.6 and 3.8). Let E and F be vector bundles over a smooth scheme X , respectively of rank e and f . Consider the dual projective bundle

P .E/ ! X with the tautological quotient line bundle O.1/. Then we have     cf O.1/ ˝ F _ D Sf eC1 .E  F /; where the relative Segre classes are given by the following expression # " 2m1 X .2m/ St .E  F / D ct .F _ /St .E/ 1 C ˛2m 2mi p2mi .F _ /t i i D1

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involving the power sum symmetric functions pj as well as the Segre classes

Sk .E/ D hk .E/  ˛2m

2m1 X

.1/l l.2m/  pl .E/  h2mCkl .E/ ;

lD0

where 0.2m/ WD

2mC1 . d2m

We now explain our main result, comparing it with the original Kempf–Laksov formula. Let E be a vector bundle of rank n over a smooth quasi-projective variety X and consider the Grassmann bundle Grd .E/ ! X parametrising rank d subbundles, with S being its tautological bundle. In order to define the degeneracy loci of Grd .E/, we assume to be given a complete flag 0 D F n    F 1 E. Then, to every partition  D .1 ; : : : ; r / of length r such that 1  n  d we associate the e KL ! X , its Kempf–Laksov resolution (see degeneracy locus X Grd .E/ and X  Section 4.1). The original Kempf–Laksov formula for the Chow ring reads:   : ŒX CH D det ci Cj i .E=F i i Cd / 1i;j r

 I2m .Grd .E//

On the other hand, in we need a little more notation in order to be e KL ! Grd .E/. Let A .k/ WD Sl .S _  able to express the class  associated to X  l .E=F k /_ / for each l 2 Z and k D 0; 1; : : : ; n and recall that the multi-Schur determinant is defined as follows: for each .l1 ; : : : ; lr / 2 Zr and k1 ; : : : ; kr 2 f0; 1; : : : ; ng, we denote .ki / r/ 1/    Al.k  D det.Al Cj / : DetŒAl.k i 1i;j r r 1 i

Our explicit closed formula is given by the following theorem. Theorem B (Theorem 4.6). Let  D .1 ; : : : ; r / be a partition of length r such that  .Grd .E// is 1  n  d . Let ki WD i  i C d . The Kempf–Laksov class  in I2m given by

 D Det A.k1 1 /    A.kr r / C ˛2m 0 @

m1 X

.2m/ .1/mCl mCl

lDmC1

X

1 .k1 /

.kb / a/ Det A1    A.ka CmCl    Ab Cml    A.kr r / A :

1a 0 and let ' W S ! B be a quasi-elliptic surface defined over k. Throughout this paper, we assume that any exceptional curve of the first kind is not contained in fibers. Such a surface exists if and only if p is equal to 2 or 3, and the multiplicities of multiple fibers are all equal to p (cf. Bombieri–Mumford [1]). We denote by g the genus of the curve B. In this section, we recall some facts on quasi-elliptic surfaces. We denote by pFi .i D 1; : : : ; / the multiple fibers. Let T be the torsion part of R1 ' OS . Then, there exists a Cartier divisor f on B such that R1 ' OS =T Š OB .f/. The canonical divisor formula of S is given by KS  '  .KB  f/ C

 X

a i Fi ;

i D1

where  deg f D .OS / C t with t D the length of the torsion part of R1 ' OS and 0  ai  p  1. Here,  means linear equivalence. If pFi is a tame multiple fiber, then we have ai D p  1. For details, see Bombieri–Mumford [1]. Lemma 2.1. The Albanese variety Alb.S / of S is isomorphic to the Jacobian variety J.B/ of B. Proof. Let W S ! Alb.S / be the Albanese mapping. If Alb.S / is a point, then by the universality of Albanese variety we see that the Jacobian variety J.B/ of B is also a point. Now, assume Alb.S / is not a point. Since the general fiber of ' is a rational curve with one cusp, the fibers are contracted by . Therefore, .S / is a curve. We have, by the universality of Albanese variety, a commutative diagram: 'W

S ! B # # Alb.S / ! J.B/ [ [ .S / B:

By this diagram, we have a morphism .S / ! B. Therefore, by the Stein factorization theorem, we see that .S / is isomorphic to B. Therefore, by the universality of Jacobian variety, we conclude Alb.S / Š J.B/ (see also Katsura–Ueno [4], Lemma 3.4). We find the following lemma and corollary in Lang [5] and Raynaud [6]. We give here an easy proof for the lemma. Lemma 2.2. Let ' W S ! B be a quasi-elliptic surface over a non-singular complete curve B with genus g. Then, we have the inequality .OS /  .1  g/=3. Proof. By Noether’s formula and the self-intersection number c1 .S /2 D 0 of the first Chern class of S , we have 12.OS / D c1 .S /2 C c2 .S / D 2  4q.S / C b2 .S /:

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By Lemma 2.1, we have q.S / D g. Denoting by .S / the Picard number of S , we have also b2 .S /  .S /  2. Hence, we have .OS /  .1  g/=3. Corollary 2.3. (i) If g D 1, then .OS /  0. (ii) If g D 0, then .OS /  1.

3 Multicanonical systems In this section, let k be an algebraically closed field of characteristic 3. Let ' W S ! B be a quasi-elliptic surface defined over k. Example 3.1. In characteristic 3, we consider the quasi-elliptic surface ' W S ! P1 which is given by a non-singular complete model of the surface defined by t 2 .t  1/2 z C t 2 .t  1/2 C x 3 C tz 3 D 0 Here, t is a parameter of the base curve P1 . By Lang [5] p.485, this surface has two tame multiple fibers at t D 0; 1, and we have .OS / D 1. We denote the two tame multiple fibers by 3F0 and 3F1 . The canonical divisor KS is given by KS

 '  .KP1  f/ C 2F0 C 2F1  F C 2F0 C 2F1 :

Here, f is a Cartier divisor on P1 with  deg f D .OS / D 1 such that OP1 .f/ Š R1 ' OS , and F is a general fiber of ' W S ! P1 . Since we have F  3F0  3F1 , we see 4KS  2F0 C 2F1 . Therefore, we have dim H0 .S; OS .4KS // D 1, and j4KS j does not give the structure of quasi-elliptic surface. If m  5, then we have dim H0 .S; OS .mKS //  2, and jmKS j gives the structure of quasi-elliptic surface. We have the following theorem. Theorem 3.1. Assume that the characteristic p D 3. Then, for any quasi-elliptic surface f W S ! B with .S / D 1 over k and for any m  5, the multicanonical system jmKS j gives the unique structure of quasi-elliptic surface, and the number 5 is best possible. Proof. The method of the proof is similar to the one in Iitaka [2], Katsura–Ueno [4] and Katsura [3]. Since the Kodaira dimension is equal to 1, the structure of quasielliptic surface is unique. The Kodaira dimension of S is equal to 1 if and only if . /

2g  2 C .OS / C t C

 X i D1

.ai =3/ > 0:

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Therefore, we need to find the least integer m such that . / m.2g  2 C .OS / C t/ C

 X Œmai =3  2g C 1 i D1

holds under the condition . /. Here, Œr means the integral part of a real number r. We have the following 6 cases: Case (I) g  2 Case (II-1) g D 1; .OS / C t  1 Case (II-2) g D 1; .OS / D 0; t D 0 Case (III-1) g D 0; .OS / C t  3 Case (III-2) g D 0; .OS / C t D 2 Case (III-3) g D 0; .OS / D 1; t D 0 We check . / under the condition . / for each case. In Case (I), by Lemma 2.2, we have 2g  2 C .OS /  5.g  1/=3. Hence, if m  3, . / holds. In Case (II-1), if m  3, . / holds. In Case (II-2), all multiple fibers are tame in this case, and we have at least one multiple fiber by . /. Since ai D 2, . / holds for m  5. In Case (III-1), . / holds for m  1. In Case (III-2), since .OS /  1 by Corollary 2.3, we have t  1. Therefore, the number of wild fibers is less than or equal to 1. If there exists at least one tame multiple fiber, then . / holds for m  2. If there exist no tame fibers and only one wild fiber, then by . / we have a1  1. Therefore, . / holds for m  3. In Case (III-3), all multiple fibers are tame, and we have   2 by . /. Therefore, . / holds for m  5. The result on the best possible number in characteristic 3 follows from Example 3.1. In characteristic 2, we can also consider a similar question to the one in characteristic 3. We have still difficulties to decide the best possible number. For example we need to solve the following question. Question 3.1. Does there exist a quasi-elliptic surface over an elliptic curve with only one tame multiple fiber and with .OS / D 0 in characteristic 2? If there don’t exist such quasi-elliptic surfaces, then we can show that in characteristic 2, . / holds for m  4 and that the best possible number is equal to 4. Namely, we have M D 4 in characteristic 2.

References [1] E. Bombieri and D. Mumford, Enriques’ classification of surfaces in char. p, II, In Complex Analysis and Algebraic Geometry (W. L. Baily Jr. and T. Shioda, eds.), Iwanami Shoten, Tokyo, and Princeton Univ. Press, Princeton, NJ, 1977, 22–42. [2] S. Iitaka, Deformations of compact complex surfaces, II. J. Math. Soc. Japan 22 (1970), 247– 261.

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[3] T. Katsura, Multicanonical systems of elliptic surfaces in small characteristics. Compositio Math. 97 (1995), 119–134. [4] T. Katsura and K. Ueno, On elliptic surfaces in characteristic p. Math. Ann. 272 (1985), 291– 330. [5] W. Lang, Quasi-elliptic surfaces in characteristic three. Ann. Scient. Ec. Norm. Sup. 12 (1979), 473–500. [6] M. Raynaud, Surfaces elliptiques et quasi-elliptiques, manuscript.

Characteristic classes of mixed Hodge modules and applications Lauren¸tiu Maxim and Jörg Schürmann1 Contents 1 2

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Calculus of mixed hodge modules . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Mixed Hodge modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Grothendieck groups of algebraic mixed Hodge modules. . . . . . . 3 Hodge–Chern and Hirzebruch classes of singular varieties . . . . . . . . . . 3.1 Construction. Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Application: characteristic classes of toric varieties . . . . . . . . . . 4 Hirzebruch–Milnor classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 V -filtration. Nearby and vanishing cycles . . . . . . . . . . . . . . . . . 4.3 Specialization of Hodge–Chern and Hirzebruch classes . . . . . . . 4.4 Application: Hirzebruch–Milnor classes of singular hypersurfaces 5 Equivariant characteristic classes . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Motivation. Construction. Properties . . . . . . . . . . . . . . . . . . . . 5.2 Symmetric products of mixed Hodge modules . . . . . . . . . . . . . . 5.3 Application: characteristic classes of symmetric products . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Dedicated to Piotr Pragacz on the occasion of his 60th birthday

1 We thank J.-P. Brasselet, S. Cappell, A. Libgober, M. Saito, J. Shaneson, S. Yokura for many useful discussions and for collaborating with us on various parts of the research described in this note. These notes are based on the authors’ lectures at the 2013 CMI Workshop “Mixed Hodge Modules and Their Applications”, as well as the first author’s lecture at the IMPANGA 15 conference. L. Maxim was partially supported by grants from NSF, NSA, and by a fellowship from the Max-Planck-Institut für Mathematik, Bonn. J. Schürmann was supported by the SFB 878 “groups, geometry and actions”.

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1 Introduction We give an overview of recent developments on the interaction between characteristic class theories for singular spaces and Saito’s theory of mixed Hodge modules in the complex algebraic context, updating the existing survey [43]. The emphasis here is on applications. There are two versions of characteristic classes associated to mixed Hodge modules, cf. [5, 43]. The K-theoretical classes, called Hodge–Chern classes, capture information about the graded pieces of the filtered de Rham complex associated to the filtered D-module underlying a mixed Hodge module. Application of the Todd class transformation td of Baum–Fulton–MacPherson then gives classes in (Borel– BM Moore) homology H WD H2 , which we call un-normalized Hirzebruch classes. Both versions are defined by natural transformations DRy W K0 .MHM.X // ! K0 .X / ˝ ZŒy ˙1  and resp.

Ty WD td ı DRy W K0 .MHM.X // ! H .X /Œy; y 1 on the Grothendieck group of mixed Hodge modules on a variety X . For the norby we renormalize the classes Ty by multiplying by malized Hirzebruch classes T i .1 C y/ on Hi ./. These transformations are functorial for proper pushdown and external products, and satisfy some other properties which one would expect for a theory of characteristic classes for singular spaces. For “good” variations of mixed Hodge structures (and their extensions along a normal crossing divisor) the corresponding Hodge–Chern and resp. Hirzebruch classes have an explicit description in terms of (logarithmic) de Rham complexes. On a point space, these classes coincide with the Hodge polynomial y of a mixed Hodge structure. From this point of view, the Hirzebruch characteristic classes of mixed Hodge modules can be regarded as higher homology class versions of the Hodge polynomials defined in terms of the Hodge filtration on the cohomology of the given mixed Hodge module. This will be explained in Section 3, after some relevant background about mixed Hodge modules is presented in Section 2. As a first application, these characteristic classes are computed in the case of toric varieties, see Section 3.2. In Section 4, we discuss the specialization property of Hodge–Chern and resp. Hirzebruch classes, which can be viewed as a Hodge-theoric counterpart of the Verdier specialization result for the MacPherson Chern class. In more detail, for a globally defined hypersurface X D ff D 0g given by the zero-fiber of a complex algebraic function f W M ! C defined on an algebraic manifold M , it is proved in [42] that the Hodge–Chern and resp. normalized Hirzebruch class transformations commute with specialization defined in terms of the nearby cycles fH with rat. fH / D p f , in the sense that .1 C y/  DRy ı . fH Œ1/ D i Š ı DRy and by ı . H Œ1/ D i Š ı T by ; T f

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for i W X ! M the inclusion map and i Š the corresponding Gysin homomorphism. The proof of this result, discussed in Section 4.3, uses the algebraic theory of nearby and vanishing cycles given by the V -filtration of Malgrange–Kashiwara (as recalled in Section 4.2) in the D-module context, together with a specialization result about the filtered de Rham complex of the filtered D-module underlying a mixed Hodge module. As an application, we give in Section 4.4 a description of the Hirzebruch–Milnor classes of (globally defined) complex hypersurfaces (i.e., the difference between the corresponding virtual and functorial Hirzebruch characteristic classes) in terms of vanishing cycles related to the singularities of the hypersurface. These Hirzebruch– Milnor classes are a Hodge-theoretic version of the so-called Milnor(–Chern) class of a hypersurface, so our results from Section 4.4 can be seen as a generalization of a Milnor class formula obtained by Parusi´nski–Pragacz [37]. See also [32, 33, 29] for calculations of Hirzebruch–Milnor classes in the case of projective hypersurfaces (and even projective complete intersections). In Section 5, we discuss equivariant characteristic class theories for singular varieties. More precisely, following [13], for varieties with finite algebraic group actions we define for a “weak” equivariant complex of mixed Hodge modules localized Atiyah–Singer classes in the (Borel–Moore) homology of the fixed point sets. These are equivariant versions of the above Hirzebruch classes, and enjoy similar functorial properties. They can be used for computing explicitly the Hirzebruch classes of global quotients, e.g., of symmetric products of quasi-projective varieties. In Section 5.2, we recall the definition of symmetric products of mixed Hodge module complexes. For a complex quasi-projective variety X , let X .n/ WD X n =†n denote its n-th symmetric product, with n W X n ! X .n/ the natural projection map. In [31], we define an action of the symmetric group †n on the n-fold external n self-product M of an arbitrary bounded complex of mixed Hodge modules M 2 D b MHM.X /. By construction, this action is compatible with the natural action on the underlying Q-complexes. There are, however, certain technical difficulties associated with this construction, since the difference in the t-structures of the underlying Dmodules and Q-complexes gives certain differences of signs. In [31] we solve this problem by showing that there is a sign cancellation. For a complex of mixed Hodge modules M 2 D b MHM.X /, we can therefore define its n-th symmetric product (in a way compatible with rat) by: M.n/ WD .n Mn /†n 2 D b MHM.X .n/ /: In Section 5.3 we discuss a generating series formula for the un-normalized Hirzebruch classes of symmetric products of a mixed Hodge module complex, which generalizes to this singular setting many of the generating series formulae in the literature. The key technical point is a localization formula for the Atiyah–Singer classes, which relies on understanding how Saito’s functors GrpF DR behave with respect to external products. For more recent developments see [29, 30], where we obtain refined generating series formulae for equivariant characteristic classes valued in the delocalized Borel–Moore homology of external and symmetric products of varieties, and

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resp. for characters of (virtual) cohomology representations of external products of suitable coefficients on (possibly singular) complex quasi-projective varieties (e.g., (complexes of) constructible or coherent sheaves, or (complexes of) mixed Hodge modules). These formulae generalize our previous results for symmetric (and alternating) powers of such coefficients, and apply also to other Schur functors.

2 Calculus of mixed hodge modules For the sake of completeness and coherence of exposition, in this section we give a brief overview of the theory of mixed Hodge modules and their Grothendieck calculus. 2.1 Mixed Hodge modules. To any complex algebraic variety X , Saito associated a category MHM.X / of algebraic mixed Hodge modules on X (cf. [38, 39]). If X is smooth, an object of this category consists of an algebraic (regular) holonomic Dmodule .M; F / with a good filtration F , together with a perverse sheaf K of rational vector spaces, both endowed a finite increasing filtration W such that the isomorphism ˛ W DR.M/an ' K ˝QX CX is compatible with W under the Riemann–Hilbert correspondence (with ˛ a chosen isomorphism). Here we use left D-modules. The sheaf D X of algebraic differential operators on X has the increasing filtration F with Fi DX given by the differential operators of degree  i (i 2 Z). A good filtration F of the algebraic holonomic D-module M is then given by a bounded from below, increasing, and exhaustive filtration Fp M by coherent algebraic OX -modules such that   Fi D X Fp M FpCi M (2.1) for all i; p, and this is an equality for i big enough. In general, for a singular variety X one works with suitable local embeddings into manifolds and corresponding filtered D-modules supported on X . In addition, these objects are required to satisfy a long list of properties (which are not needed here). The forgetful functor rat is defined as rat W MHM.X / ! Perv.QX / ;

..M; F /; K; W / 7! K ;

with Perv.QX / the abelian category of perverse sheaves on X . For the following result, see [39][Thm. 0.1 and Sect. 4] for more details: Theorem 2.1 (M. Saito). The category MHM.X / is abelian, and the forgetful functor rat W MHM.X / ! Perv.QX / is exact and faithful. It extends to a functor rat W D b MHM.X / ! Dcb .QX /

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to the derived category of complexes of Q-sheaves with algebraically constructible cohomology. There are functors f ; fŠ ; f  ; f Š ; ˝;  on D b MHM.X / which are “lifts” via rat of the similar (derived) functors defined on Dcb .QX /, and with .f  ; f / and .fŠ ; f Š / pairs of adjoint functors. One has a natural map fŠ ! f , which is an isomorphism for f proper. p

The usual truncation  on D b MHM.X / corresponds to the perverse truncation   on Dcb .X /, so that rat ı H D p H ı rat ;

where H stands for the cohomological functor in D b MHM.X / and perverse cohomology (with respect to the middle perversity).

p

H denotes the

Example 2.2. Let X be a complex algebraic manifold of pure complex dimension n, with V WD .L; F; W / a good (i.e., admissible, with quasi-unipotent monodromy at infinity) variation of mixed Hodge structures on X . Then L WD L ˝QX OX with its integrable connection r is a holonomic (left) D-module with ˛ W DR.L/an ' LŒn, where we use the shifted de Rham complex r

r

DR.L/ WD ŒL !    ! L ˝OX nX  with L in degree n, so that DR.L/an ' LŒn is a perverse sheaf on X . The filtration F on L induces by Griffiths’ transversality a good filtration Fp .L/ WD F p L on L as a filtered D-module. Moreover, ˛ is compatible with the induced filtration W defined by W i .LŒn/ WD W i n LŒn and W i .L/ WD .W i n L/ ˝QX OX : This data defines a mixed Hodge module VH Œn on X , with rat.VH Œn/ ' LŒn. Hence rat.VH Œn/Œn is a local system on X . In what follows, we will often use the same symbol V to denote both the variation and the corresponding (shifted) mixed Hodge module. Definition 2.3. A mixed Hodge module M on a pure n-dimensional complex algebraic manifold X is called smooth if rat.M/Œn is a local system on X . We have the following result (see [39][Thm. 3.27, Rem. on p. 313]): Theorem 2.4 (M. Saito). Let X be a pure n-dimensional complex algebraic manifold. Associating to a good variation of mixed Hodge structures V D .L; F; W / on X the mixed Hodge module VH Œn as in Example (2.2) defines an equivalence of categories MHM.X /sm ' VMHSg .X /

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between the categories of smooth mixed Hodge modules MHM.X /sm and good variations of mixed Hodge structures on X . This equivalence commutes with external product . For X D pt a point, one obtains in particular an equivalence MHM.pt/ ' MHSp

(2.2)

between mixed Hodge modules on a point space and the abelian category of graded polarizable mixed Hodge structures. By the identification in (2.2), there exists a unique Tate object QH .k/ 2 MHM.pt/ such that rat.QH .k// D Q.k/, with Q.k/ of type .k; k/. For a complex variety X with constant map k W X ! pt, define  H b QH X .k/ WD k Q .k/ 2 D MHM.X /;

with

rat.QH X .k// D QX .k/:

So tensoring with QH X .k/ defines the Tate twist operation .k/ on mixed Hodge modH ules. To simplify the notations, we let QH X WD QX .0/. If X is smooth of complex H dimension n then QX Œn is perverse on X , and QX Œn 2 MHM.X / is a single mixed Hodge module, explicitly described by QH X Œn D ..OX ; F /; QX Œn; W /; with griF D 0 D griWCn for all i ¤ 0. Let us also mention here the following result about the existence of a mixed i .X I M/ for M 2 Hodge structure on the cohomology (with compact support) H.c/ D b MHM.X /. Corollary 2.5. Let X be a complex algebraic variety with constant map k W X ! pt. i Then the cohomology (with compact support) H.c/ .X I M/ of M 2 D b MHM.X / gets an induced graded polarizable mixed Hodge structure by: i H.c/ .X; M/ D H i .k.Š/ M/ 2 MHM.pt/ ' MHSp :

In particular: i 1. The rational cohomology (with compact support) H.c/ .X I Q/ of X gets an induced graded polarizable mixed Hodge structure by: H i .X I Q/ D rat.H i .k k  QH // and Hci .X I Q/ D rat.H i .kŠ k  QH // : 2. Let V be a good variation of mixed Hodge structures on a smooth (pure dimensional) complex manifold X . Then H i .X I V/ gets a mixed Hodge structure by H i .X I V/ ' rat.H i .k VH // ; and, similarly, Hci .X I V/ gets a mixed Hodge structure by Hci .X I V/ ' rat.H i .kŠ VH // :

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Remark 2.6. By a deep theorem of Saito ([40][Thm. 0.2, Cor. 4.3]), the mixed Hodge i .X I Q/ defined as above coincides with the classical mixed Hodge structure on H.c/ structure constructed by Deligne ([16, 17]). We conclude this section with a short explanation of the rigidity property for good variations of mixed Hodge structures. Assume X is smooth, connected and of dimension n, with M 2 MHM.X / a smooth mixed Hodge module, so that the underlying local system L WD rat.M/Œn has the property that the restriction map r W H 0 .X I L/ ! Lx is an isomorphism for all x 2 X . Then the (admissible) variation of mixed Hodge structures V WD .L; F; W / is a constant variation since r underlies the morphism of mixed Hodge structures (induced by the adjunction id ! i i  ): H 0 .k MŒn/ ! H 0 .k i i  MŒn/ with k W X ! pt the constant map, and i W fxg ,! X the inclusion of the point. This implies b MŒn D VH ' k  i  VH D k  VH x 2 D MHM.X /: 2.2 Grothendieck groups of algebraic mixed Hodge modules. In this section, we describe the functorial calculus of Grothendieck groups of algebraic mixed Hodge modules. Let X be a complex algebraic variety. By associating to (the class of) a complex the alternating sum of (the classes of) its cohomology objects, we obtain the following identification K0 .D b MHM.X // D K0 .MHM.X //

(2.3)

between the corresponding Grothendieck groups. In particular, if X is a point, then K0 .D b MHM.pt// D K0 .MHSp /;

(2.4)

and the latter is a commutative ring with respect to the tensor product, with unit ŒQH . For any complex M 2 D b MHM.X /, we have the identification X ŒM D .1/i ŒH i .M/ 2 K0 .D b MHM.X // Š K0 .MHM.X //: (2.5) i 2Z

In particular, if for any M 2 MHM.X / and k 2 Z we regard MŒk as a complex concentrated in degree k, then ŒMŒk D .1/k ŒM 2 K0 .MHM.X //:

(2.6)

All functors f , fŠ , f  , f Š , ˝,  induce corresponding functors on K0 .MHM.//. Moreover, K0 .MHM.X // becomes a K0 .MHM.pt//-module, with the multiplication induced by the exact external product with a point space:  W MHM.X /  MHM.pt/ ! MHM.X  fptg/ ' MHM.X /:

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Also note that

H M ˝ QH X ' M  Qpt ' M

for all M 2 MHM.X /. Therefore, K0 .MHM.X // is a unitary K0 .MHM.pt//module. The functors f , fŠ , f  , f Š commute with external products (and f  also commutes with the tensor product ˝), so that the induced maps at the level of Grothendieck groups K0 .MHM.// are K0 .MHM.pt//-linear. Moreover, by using the forgetful functor rat W K0 .MHM.X // ! K0 .Dcb .QX // ' K0 .Perv.QX //; all these transformations lift the corresponding transformations from the (topological) level of Grothendieck groups of constructible (or perverse) sheaves. Remark 2.7. The Grothendieck group K0 .MHM.X // is generated by the classes f Œj V (or, alternatively, by the classes f ŒjŠ V), with f W Y ! X a proper morphism from a complex algebraic manifold Y , j W U ,! Y the inclusion of a Zariski open and dense subset U , with complement D a normal crossing divisor with smooth irreducible components, and V a good variation of mixed (or pure) Hodge structures on U . This follows by induction from resolution of singularities and from the existence of a standard distinguished triangle associated to a closed inclusion. Let i W Y ,! Z be a closed inclusion of complex algebraic varieties with open complement j W U D ZnY ,! Z. Then one has by [39][(4.4.1)] the following functorial distinguished triangle in D b MHM.Z/: adj

adi

Œ1

(2.7)

jŠ j  ! id ! i i  ! ;

where the maps ad are the adjunction maps, and i D iŠ since i is proper. In particular, we get the following additivity relation at the level of Grothendieck groups: H H b ŒQH Z  D ŒjŠ QU  C ŒiŠ QY  2 K0 .D MHM.Z// D K0 .MHM.Z//:

(2.8)

As a consequence, if S D fS g is a complex algebraic stratification of Z such that for any S 2 S, S is smooth and SN n S is a union of strata, then with jS W S ,! Z denoting the corresponding inclusion map, we get: X ŒQH Œ.jS /Š QH (2.9) Z D S : S2S

If f W Z ! X is a complex algebraic morphism, then we can apply fŠ to (2.7) to get another distinguished triangle adj

adi

Œ1

H  H fŠ jŠ j  QH Z ! fŠ QZ ! fŠ iŠ i QZ ! :

(2.10)

So, on the level of Grothendieck groups, we get the following additivity relation: H H b ŒfŠ QH Z  D Œ.f ı j /Š QU  C Œ.f ı i /Š QY  2 K0 .D MHM.X // D K0 .MHM.X // : (2.11)

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Let K0 .var=X / be the motivic relative Grothendieck group of complex algebraic varieties over X , i.e., the free abelian group generated by isomorphism classes Œf  WD Œf W Z ! X  of morphisms f to X , divided out be the additivity relation Œf  D Œf ı i  C Œf ı j  for a closed inclusion i W Y ,! Z with open complement j W U D ZnY ,! Z. The pushdown fŠ , external product  and pullback g  for these relative Grothendieck groups are defined by composition, exterior product and resp. pullback of arrows. Then we get by (2.11) the following result: Corollary 2.8. There is a natural group homomorphism Hdg W K0 .var=X / ! K0 .MHM.X //; Œf W Z ! X  7! ŒfŠ QH Z; which commutes with pushdown fŠ , exterior product  and pullback g  . The fact that Hdg commutes with external product  (or pullback g  ) follows from the corresponding Künneth (or base change) theorem for the functor fŠ W D b MHM.Z/ ! D b MHM.X / (see [39][(4.4.3)]).

3 Hodge–Chern and Hirzebruch classes of singular varieties 3.1 Construction. Properties. The construction of K-theoretical and resp. homology characteristic classes of mixed Hodge modules is based on the following result of Saito (see [38][Sect. 2.3] and [40][Sect. 1] for the first part, and [39][Sect. 3.10, Prop. 3.11]) for part two): Theorem 3.1 (M. Saito). Let X be a complex algebraic variety. Then there is a functor of triangulated categories b .X / GrpF DR W D b MHM.X / ! Dcoh

(3.1)

commuting with proper pushforward, and with GrpF DR.M/ D 0 for almost all p and b M fixed, where Dcoh .X / is the bounded derived category of sheaves of algebraic OX modules with coherent cohomology sheaves. If X is a (pure n-dimensional) complex algebraic manifold, then one has in addition the following: 1. Let M 2 MHM.X / be a single mixed Hodge module. Then GrpF DR.M/ is the corresponding complex associated to the de Rham complex of the underlying algebraic left D-module M with its integrable connection r: r

r

DR.M/ D ŒM !    ! M ˝OX nX 

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with M in degree n, filtered by r

r

Fp DR.M/ D ŒFp M !    ! FpCn M ˝OX nX  : 2. Let XN be a smooth partial compactification of the complex algebraic manifold X with complement D a normal crossing divisor with smooth irreducible components, and with j W X ! XN the open inclusion. Let V D .L; F; W / be a good variation of mixed Hodge structures on X . Then the filtered de Rham comN

plex .DR.j V/; F / of the shifted mixed Hodge module j V 2 MHM.X/Œn N is filtered quasi-isomorphic to the logarithmic de Rham complex D b MHM.X/ r

r

DRlog .L/ WD ŒL !    ! L ˝OXN nXN .log.D// with the increasing filtration Fp WD F p (p 2 Z) associated to the decreasing F -filtration r

r

F p DRlog .L/ D ŒF p L !    ! F pn L ˝OXN nXN .log.D// ; where L is the canonical Deligne extension of L WD L ˝QX OX . In particular, F Grp DR.j V/ (p 2 Z) is quasi-isomorphic to Gr r

Gr r

GrFp DRlog .L/ D ŒGrFp L !    ! GrFpn L ˝OXN nXN .log.D// : Similar considerations apply to the filtered de Rham complex .DR.jŠ V/; F / of the N N by considershifted mixed Hodge module jŠ V 2 MHM.X/Œn

D b MHM.X/, ing instead the logarithmic de Rham complex associated to the Deligne extension L ˝ O.D/ of L. p

Note that the maps Grr and Gr r in the complexes GrF DR.M/ and respectively GrFp DRlog .L/ are O-linear. The transformations GrpF DR (p 2 Z) induce functors on Grothendieck groups. b Therefore, if K0 .X / ' K0 .Dcoh .X // denotes the Grothendieck group of coherent algebraic OX -sheaves on X , we get group homomorphisms b .X // ' K0 .X / : GrpF DR W K0 .MHM.X // D K0 .D b MHM.X // ! K0 .Dcoh F Notation. In the following we will use the notation GrFp in place of Grp corresponding to the identification F p D Fp , which makes the transition between the increasing filtration Fp appearing in the D-module language and the classical situation of the decreasing filtration F p coming from a good variation of mixed Hodge structures.

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169

Definition 3.2. For a complex algebraic variety X , the Hodge–Chern class transformation DRy W K0 .MHM.X // ! K0 .X / ˝ ZŒy ˙1  is defined by ŒM 7! DRy .ŒM/ WD

X

.1/i Hi GrpF DR.M/  .y/p

(3.2)

i;p

(where the sign of the variable y is chosen to fit with Hirzebruch’s convention in (3.8) below). The un-normalized homology Hirzebruch class transformation is defined as the composition

(3.3) Ty WD td ı DRy W K0 .MHM.X // ! H .X / ˝ Q y; y 1 where td W K0 .X / ! H .X / ˝ Q is the Baum–Fulton–MacPherson Todd class

transformation [3], which is linearly extended over Z y; y 1 . Here, we denote by BM H .X / the even degree Borel–Moore homology H2 .X / of X . The normalized homology Hirzebruch class transformation is defined as the composition 1 b (3.4) T y WD td.1Cy/ ı DRy W K0 .MHM.X // ! H .X / ˝ Q y ˙1 ; yC1 where



1 td.1Cy/ W K0 .X /Œy; y 1 ! H .X / ˝ Q y; y.yC1/

is the scalar extension of the Todd class transformation td together with the multiplication by .1 C y/k on the degree k component. Remark 3.3. By precomposing with the transformation Hdg of Corollary 2.8, we get similar motivic transformations, Chern and resp. Hirzebruch, defined on the relative Grothendieck group of complex algebraic varieties. It is the (normalized) motivic Hirzebruch class transformation which unifies (in the sense of [5]) the previously known characteristic class theories for singular varieties, namely, the (rational) Chern class transformation of MacPherson [24] for y D 1, the Todd class transformation of Baum–Fulton–MacPherson [3] for y D 0, and the L-class transformation of Cappell–Shaneson [10] for y D 1, thus answering positively an old question of MacPherson about the existence of such a unifying theory (cf. [25, 47]). For the rest of the paper, we mostly neglect the more topological L-class transformation related to self-dual constructible sheaf complexes (e.g., intersection cohomology complexes), because the above-mentioned relation to Hirzebruch classes can only be defined at the motivic level, but not on the Grothendieck group of mixed Hodge modules. Remark 3.4. By [43][Prop. 5.21], we have that by .ŒM/ 2 H .X / ˝ QŒy; y 1 ; T

(3.5)

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and, moreover, the specialization at y D 1, b1 .ŒM/ D c .Œrat.M// 2 H .X / ˝ Q T

(3.6)

equals the MacPherson–Chern class of the underlying constructible sheaf complex rat.M/ (i.e., the MacPherson–Chern class of the constructible function defined by taking stalkwise the Euler characteristic). The homology Hirzebruch characteristic classes of a complex algebraic variety by .X /, are obtained by evaluating the Hirzebruch X , denoted by Ty .X / and resp. T transformations of Definition 3.2 on the constant Hodge module M D QH X . Moreover, we have that by .X / 2 H .X / ˝ QŒy: (3.7) Ty .X /; T If X is smooth, then

 DRy .ŒQH X / D ƒy ŒTX ;

(3.8)

where for a vector bundle V on X we set: X ƒy ŒV  WD Œƒp V  y p : p

Indeed, we have that  DR.QH X / D DR.OX /Œn D X ;

with n WD dim X;

and the Hodge filtration F p on X is given by the truncation  p . In particular, the following normalization property holds for the homology Hirzebruch classes: if X is smooth, then by .X / D T by .TX / \ ŒX  ; Ty .X / D Ty .TX / \ ŒX  ; T

(3.9)

by .TX / the un-normalized and resp. normalized versions of the with Ty .TX / and T cohomology Hirzebruch class of X appearing in the generalized Riemann–Roch theorem, see [22]. More precisely, we have un-normalized and resp. normalized power series Qy .˛/ WD

˛.1 C ye˛ / b ˛.1 C ye ˛.1Cy/ / ; Q .˛/ WD 2 QŒyŒŒ˛ y 1  e˛ 1  e˛.1Cy/

with initial terms Qy .0/ D 1 C y;

b y .0/ D 1 Q

(3.10)

(3.11)

so that for X smooth we have Ty .TX / D

dim YX i D1

Qy .˛i / 2 H  .X / ˝ QŒy

(3.12)

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(and similarly for b T y .TX /), with f˛i g the Chern roots of TX . These two power series are related by the following relation b y .˛/ D .1 C y/1  Qy .˛.1 C y//; Q

(3.13)

which explains the use of the normalized Todd class transformation td.1Cy/ in the by .X /. definition of T Note that for the values y D 1, 0, 1 of the parameter, the class b T y specializes to    the total Chern class c , Todd class td , and L-polynomial L , respectively. Indeed, b y .˛/ 2 QŒyŒŒ˛ becomes respectively the power series Q ˛=.1  e˛ /;

1 C ˛;

˛= tanh ˛:

Moreover, by (3.7) we are also allowed to specialize the parameter y in the homology by .X / to the distinguished values y D 1; 0; 1. For example, classes Ty .X / and T for y D 1, we have the identification (cf. [5]) b1 .X / D c .X / ˝ Q T

(3.14)

with the total (rational) Chern class c .X / of MacPherson [24]. Also, for a variety X with at most Du Bois singularities (e.g., rational singularities, such as toric varieties), we have by [5] that b0 .X / D td .ŒOX / DW td .X / ; T0 .X / D T

(3.15)

for td the Baum–Fulton–MacPherson Todd class transformation [3]. Indeed, in the language of mixed Hodge modules, X has at most Du Bois singularities if, and only if, the canonical map

b OX ! GrF0 DR.QH X / 2 Dcoh .X /

is a quasi-isomorphism (see [40][Cor. 0.3]). It is still only conjectured that if X is a compact complex algebraic variety which is also a rational homology manifold, then b T 1 .X / is the Milnor–Thom homology L-class of X (see [5]). This conjecture is proved in [13][Cor. 1.2] for projective varieties X of the form Y =G, with Y a projective manifold and G a finite group of algebraic automorphisms of Y , see Remark 5.6. By the Riemann–Roch theorem of [3], the Todd class transformation td commutes with the pushforward under proper morphisms, so the same is true for the by of Definition 3.2. If we apply this observaHirzebruch transformations Ty and T tion to the constant map k W X ! pt with X compact, then the pushforward for H is identified with the degree map, and we have K0 .pt/ D Z;

H .pt/ D Q;

MHM.pt/ D MHSp ;

where MHSp is, as before, Deligne’s category of graded-polarizable mixed Q-Hodge structures. By definition, we have for H  2 D b MHSp that X p j T y .H  / D y .H  / WD .1/j dimC GrF HC .y/p : (3.16) Ty .H  / D b j;p

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by .X / is Hence, for X compact and connected, the degree-zero part of Ty .X / or T identified with the Hodge polynomial   (3.17) y .X / WD y H  .X / ; Z

i.e.,

Z

y .X / D

Ty .X / D X

by .X /: T

(3.18)

X

Remark 3.5. The Hodge polynomial of (3.16) is just a special case of the Hodge– Deligne polynomial defined by taking into consideration both the Hodge and the weight filtration. However, simple examples show that there is no characteristic class theory incorporating the weight filtration of mixed Hodge modules (e.g., see [5][Ex. 5.1] and [43][p. 428]). The Hodge–Chern and resp. Hirzebruch characteristic classes have good functorial properties, but are very difficult to compute in general. In the following, we use Theorem 3.1 to compute these characteristic classes in some simple examples. Example 3.6. Let XN be a smooth partial compactification of the complex algebraic manifold X with complement D a normal crossing divisor with smooth irreducible components, with j W X ,! XN the open inclusion. Let V D .L; F; W / be a good variation of mixed Hodge structures on X . Then the filtered de Rham comN N is by Theorem 3.1(2)

D b MHM.X/ plex .DR.j V/; F / of j V 2 MHM.X/Œn filtered quasi-isomorphic to the logarithmic de Rham complex DRlog .L/ with the increasing filtration Fp WD F p (p 2 Z) associated to the decreasing F -filtration induced by Griffiths’ transversality. Then X .1/i ŒHi .GrFp DRlog .L//  .y/p DRy .Œj V/ D i;p

D

X

ŒGrFp DRlog .L/  .y/p

(3.19)

p

D

X

pi

.1/i ŒGrF

.L/ ˝OXN iXN .log.D//  .y/p ;

i;p p

where the last equality uses the fact that the complex GrF DRlog .L/ has coherent (locally free) objects, with OXN -linear maps between them. Let us now define X N ˙1 ; Gr y .Rj L/ WD ŒGrFp .L/  .y/p 2 K 0 .X/Œy p

N denoting the Grothendieck group of algebraic vector bundles on X. N with K 0 .X/ Therefore, (3.19) is equivalent to the formula:     DRy .Œj V/ D Gr y .Rj L/ \ ƒy 1XN .log.D// \ ŒOXN  : (3.20)

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Similarly, by setting Gr y .jŠ L/ WD

X N ˙1 ; ŒOXN .D/ ˝ GrFp .L/  .y/p 2 K 0 .X/Œy p

we obtain the identity:     DRy .ŒjŠ V/ D Gr y .jŠ L/ \ ƒy 1XN .log.D// \ ŒOXN  :

(3.21)

Here, the pairing N  K0 .X/ N ! K0 .X/ N \ WD ˝OXN W K 0 .X/ is defined by taking the tensor product. In particular, for j D id W X ! X we get the following Atiyah–Meyer type formula (compare [7, 27]):   (3.22) DRy .ŒV/ D Gr y .L/ \ ƒy .TX / \ ŒOX  : Let us now discuss the following multiplicativity property of the Hodge–Chern and resp. homology Hirzebruch class transformations: Proposition 3.7. The Hodge–Chern class transformation DRy commutes with external products, i.e.: DRy .ŒM  M0 / D DRy .ŒM  ŒM0 / D DRy .ŒM/  DRy .ŒM0 /

(3.23)

for M 2 D b MHM.Z/ and M0 2 D b MHM.Z 0 /. A similar property holds for the (un-) normalized Hirzebruch class tranformations. First note that since the Todd class transformation td commutes with external products, it suffices to justify the first part of the above proposition (refering to the Hodge– Chern class). For X and X 0 and their partial compactifications as in Example 3.6, we have that:     0 0 1 1 0 .log.D  X [ X  D // '  .log.D//   .log.D // : 1X N XN 0 XN XN 0 Therefore,   ƒy 1XN XN 0 .log.D  X 0 [ X  D 0 // \ ŒOXN XN 0  D         ƒy 1XN .log.D// \ ŒOXN   ƒy 1XN 0 .log.D 0 // \ ŒOXN 0  : Recall that the Grothendieck group K0 .MHM.Z// of mixed Hodge modules on the complex variety Z is generated by classes of the form f .j ŒV/, with f W XN ! Z N V as before. Finally one also has the multiplicativity proper and X; X; .f  f 0 / D f  f0

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for the pushforward for proper maps f W XN ! Z and f 0 W XN 0 ! Z 0 on the level of Grothendieck groups K0 .MHM.//, as well as for K0 ./ ˝ ZŒy ˙1 . So the claim follows. We conclude this section with a discussion of the following additivity property for Hodge–Chern classes and resp. homology Hirzebruch classes (see [32][Prop. 5.1.2]): Proposition 3.8. Let X be a complex algebraic variety, and fix M 2 D b MHM.X / with underlying Q-complex K. Let S D fS g be a complex algebraic stratification of X such that for any S 2 S, S is smooth, S n S is a union of strata, and the sheaves Hi KjS are local systems on S for any i . Let jS W S ,! X denote the inclusion map. Then X X Œ.jS /Š .jS / M D .1/i Œ.jS /Š H i .jS / M ŒM D S

D

X

i



S;i

.1/ .jS /Š H

i Cdim.S/

(3.24)

.jS / MΠdim.S / 2 K0 .MHM.X //;

S;i

where H i Cdim.S/ .jS / M is a smooth mixed Hodge module on the stratum S so that Hi KjS ' p Hi Cdim.S/ .KjS /Πdim.S / underlies a good variation of mixed Hodge structures. In particular, X

.1/i DRy .jS /Š H i Cdim.S/ .jS / MŒ dim.S / ; (3.25) DRy .ŒM/ D S;i

and Ty .M/ D

X

  .1/i Ty .jS /Š H i Cdim.S/ .jS / MŒ dim.S / :

(3.26)

S;i

by . A similar formula holds for the normalized Hirzebruch classes T Moreover, the summands on the right-hand side of formulae (3.25) and resp. (3.26) can be computed as in [32][Sect. 5.2] as follows. For any sets A B, denote the inclusion map by iA;B . Let V be a good variation of mixed Hodge structure on a stratum S . Take a smooth partial compactification iS;Z W S ,! Z such that D WD ZnS is a divisor with simple normal crossings and, moreover, iS;S D Z ıiS;Z for a proper morphism Z W Z ! S . Then in the notations of Example 3.6, we get by functoriality the following result: Proposition 3.9. In the above notations, we have: 



DRy . .iS;S /Š V / D .Z / Gr y ..iS;Z /Š V/ \ ƒy 1Z .log.D// :

(3.27)

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175

In particular, if L denotes the canonical Deligne extension on Z associated to V, we obtain:   X

p q Ty .iS;S /Š V D .1/q .Z / td OZ .D/ ˝ GrF L ˝ Z .log D/ .y/pCq : p;q

(3.28) Let us now consider the situation of Proposition 3.8 in the special case when M D QH X is the constant Hodge module. Then the following additivity holds: X

X

.iS;X / .iS;S /Š QH (3.29) .jS /Š QH ŒQH X D S D S 2 K0 .MHM.X //; S

S

where we use the factorization jS D iS ;X ıiS;S . Hence the formulae (3.25) and (3.26) yield by functoriality the following: X

DRy .ŒQH .iS;X / DRy .iS;S /Š QH (3.30) X / D S ; S

and Ty .X / D

X

  .iS;X / Ty .iS;S /Š QH S :

(3.31)

S

Moreover, in the notations of Proposition 3.9, we get by (3.27):



1 N DRy . .iS;S /Š QH S / D .Z / OZ .D/ ˝ ƒy Z .log D/ 2 K0 .S /Œy;

(3.32)

and

  X Ty .iS;S /Š QH .Z / td OZ .D/ ˝ qZ .log D/ y q S D q 0

D .Z / td OZ .D/ ˝ ƒy 1Z .log D/ 2 H .SN / ˝ QŒy; (3.33) P where for a vector bundle V we set as before: ƒy ŒV  WD p Œƒp V  y p . 3.2 Application: characteristic classes of toric varieties. Let X† be a toric variety of dimension n corresponding to the fan †, and with torus T WD .C /n ; see [11] for more details on toric varieties. Then X† is stratified by the orbits of the torus action. More precisely, by the orbit-cone correspondence, to a k-dimensional cone  2 † there corresponds an .n  k/-dimensional torus-orbit O Š .C /nk . The closure V of O (which is the same in both classical and Zariski topology) is itself a toric variety, and a T -invariant subvariety of X† . In particular, we can apply the results of the previous section in the setting of toric varieties. By using toric geometry, formula (3.32) adapted to the notations of this section yields the following (see [28][Prop. 3.2]):

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Proposition 3.10. For any cone  2 †, with corresponding orbit O and inclusion i W O ,! O D V , we have:

dim.O / DRy . .i /Š QH  Œ!V ; (3.34) O / D .1 C y/ where !V is the canonical sheaf on the toric variety V . The main result of this section is now a direct consequence of Propositions 3.10 and additivity: Theorem 3.11. Let X† be the toric variety defined by the fan †. For each cone  2 † with corresponding torus-orbit O , denote by k W V ,! X† the inclusion of the orbit closure. Then the Hodge–Chern class of X† is computed by the formula: X .1 C y/dim.O /  .k / Œ!V : (3.35) DRy .X† / D 2†

Similarly, the un-normalized homology Hirzebruch class Ty .X† / is computed by: X .1 C y/dim.O /  .k / td .Œ!V /: (3.36) Ty .X† / D 2†

by .X† / is And after normalizing, the corresponding homology Hirzebruch class T computed by the following formula: X by .X† / D .1 C y/dim.O /k  .k / tdk .Œ!V /: (3.37) T ;k

by one gets Recall that by making y D 1 in the normalized Hirzebruch class T the (rational) MacPherson Chern class c . Moreover, since toric varieties have only rational (hence Du Bois) singularities, making y D 0 in the Hirzebruch classes yields the Todd class td . So we get as a corollary of Theorem 3.11 the following result: Corollary 3.12. The (rational) MacPherson–Chern class c .X† / of a toric variety X† is computed by Ehler’s formula: X X .k / tddim.O / .V / D .k / .ŒV /: (3.38) c .X† / D 2†

2†

The Todd class td .X† / is computed by the additive formula: X .k / td .Œ!V /: td .X† / D

(3.39)

2†

Remark 3.13. The results of this section hold more generally for torus-invariant closed algebraic subsets of X† which are known to also have Du Bois singularities. For more applications and examples, e.g., generalized Pick-type formulae for fulldimensional lattice polytopes, see [28].

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177

4 Hirzebruch–Milnor classes In this section, we explain how to compute the homology Hirzebruch classes of globally defined hypersurfaces in an algebraic manifold (but see also [32] for the global complete intersection case). The key technical result needed for this calculation is the specialization property of the Hirzebruch class transformation, obtained by the second author in [42]. We first explain this result in Section 4.3 after some relevant background is introduced in Section 4.2. We complete our calculation of Hirzebruch classes of hypersurfaces in Section 4.4. i

4.1 Motivation. Let X ,! M be the inclusion of an algebraic hypersurface X in a complex algebraic manifold M (or more generally the inclusion of a local complete intersection). Then the normal cone NX M is a complex algebraic vector bundle NX M ! X over X , called the normal bundle of X in M . The virtual tangent bundle of X , that is, TXvir WD Œi  TM  NX M  2 K 0 .X /; (4.1) is independent of the embedding in M , so it is a well-defined element in the Grothendieck group K 0 .X / of algebraic vector bundles on X . Of course TXvir D ŒTX  2 K 0 .X /; in case X is a smooth algebraic submanifold. Let cl  denote a multiplicative characteristic class theory of complex algebraic vector bundles, i.e., a natural transformation (with R a commutative ring with unit)     cl  W K 0 .X /; ˚ ! H  .X / ˝ R; [ ; from the Grothendieck group K 0 .X / of complex algebraic vector bundles to a suitable cohomology theory H  .X / with a cup-product [, e.g., H 2 .X I Z/. Then one can associate to X an intrinsic homology class (i.e., independent of the embedding X ,! M ) defined as follows: clvir .X / WD cl  .TXvir / \ ŒX  2 H .X / ˝ R :

(4.2)

Here ŒX  2 H .X / is the fundamental class of X in a suitable homology theory BM H .X / (e.g., the Borel–Moore homology H2 .X /). Assume, moreover, that there is a homology characteristic class theory cl ./ for complex algebraic varieties, functorial for proper morphisms, obeying the normalization condition that for X smooth cl .X / is the Poincaré dual of cl  .TX / (justifying the notation cl ). If X is smooth, then clearly we have that clvir .X / D cl  .TX / \ ŒX  D cl .X / : However, if X is singular, the difference between the homology classes clvir .X / and cl .X / depends in general on the singularities of X . This motivates the following problem:

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Describe the difference class clvir .X /  cl .X / in terms of the geometry of the singular locus of X . This problem is usually studied in order to understand the complicated homology classes cl .X / in terms of the simpler virtual classes clvir .X / and these difference terms measuring the complexity of singularities of X . The strata of the singular locus have a rich geometry, beginning with generalizations of knots which describe their local link pairs. This “normal data", encoded in algebraic geometric terms via, e.g., the mixed Hodge structures on the (cohomology of the) corresponding Milnor fibers, will play a fundamental role in our study of characteristic classes of hypersurfaces. There are a few instances in the literature where, for the appropriate choice of cl  and cl , this problem has been solved. The first example was for the Todd classes td  , and td .X / WD td .ŒOX /, respectively, with td W K0 .X / ! H .X / ˝ Q the Todd class transformation in the singular Riemann–Roch theorem of Baum– Fulton–MacPherson [3]. Here ŒOX  2 K0 .X / the class of the structure sheaf. By a result of Verdier [45, 19], td commutes with the corresponding Gysin homomorphisms for the regular embedding i W X ,! M . This can be used to show that tdvir .X / WD td  .TXvir / \ ŒX  D td .X / equals the Baum–Fulton–MacPherson Todd class td .X / of X ([45, 19]). If cl  D c  is the total Chern class in cohomology, the problem amounts to comparing the Fulton–Johnson class cF J .X / WD cvi r .X / (e.g., see [19, 20]) with the homology Chern class c .X / of MacPherson [24]. Here c .X / WD c .1X / D c .QX /, with c W K0 .Dcb .X // ! F .X / ! H .X / the functorial Chern class transformation of MacPherson [24], defined on the group F .X / of complex algebraically constructible functions. To emphasize the analogy with the Grothendieck group of mixed Hodge modules (as used in the subsequent sections), we work here with the Grothendieck group of constructible (resp. perverse) sheaf complexes. The difference between the two classes cvi r .X / and c .X / is measured by the so-called Milnor class, M .X /. This is a homology class supported on the singular locus of X , and in the case of a global hypersurface X D ff D 0g it was computed in [37] as a weighted sum in the Chern–MacPherson classes of closures of singular strata of X , the weights depending only on the normal information to the strata. For example, if X has only isolated singularities, the Milnor class equals (up to a sign) the sum of the Milnor numbers attached to the singular points, which also explains the terminology: X    HQ  .Fx I Q/ ; (4.3) M .X / D x2Xsing

where Fx is the local Milnor fiber of the isolated hypersurface singularity .X; x/. More generally, Verdier [46] proved the following specialization result for MacPher-

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179

son’s Chern class transformation (which holds even more generally for every constructible sheaf complex on M ): c .

f

QM / D i Š c .QM /;

(4.4)

where i Š W H .M / ! H1 .X / is the homological Gysin map. This result was used in [37, 41] for computing the (localized) Milnor class M .X / of a global hypersurface X D ff D 0g in terms of the vanishing cycles of f W M ! C: M .X / D c .'f .QM // 2 H .Xsing/ ;

(4.5)

with the support of the (shifted) perverse complex 'f .QM / being contained in the singular locus Xsing of X . Remark 4.1. For a more topological example concerning the Goresky–MacPherson L-class L .X / ([21]) for X a compact complex hypersurface, see [9, 10]. T y is the (total) A main goal here is to explain the (unifying) case when cl  D b normalized cohomology Hirzebruch class of the generalized Hirzebruch–Riemann– Roch theorem [22]. The aim is to show that the results stated above are part of a more general philosophy, derived from comparing the intrinsic homology class (with polynomial coefficients) vir b by .T vir / \ ŒX  2 H .X / ˝ QŒy T y .X / WD T X

(4.6)

by .X / of [5]. This approach is motivated by the with the homology Hirzebruch class T fact that, as already mentioned, the L-class L , the Todd class td  and the Chern class c  , respectively, are all suitable specializations (for y D 1; 0; 1, respectively) of the by ; see [22]. In order to achieve our goal, we need to adapt Verdier’s Hirzebruch class T specialization result (4.4) to the normalized Hirzebruch class transformation. For this, we first need to recall Saito’s definition of nearby and vanishing cycles of mixed Hodge modules in terms of the V -filtration for the underlying filtered D-modules. 4.2 V -filtration. Nearby and vanishing cycles. Let f W M ! C be an algebraic function defined on a complex algebraic manifold M , with X WD ff D 0g a hypersurface in M . Consider the graph embedding i 0 W M ! M 0 WD M  C, with t D pr2 W M 0 ! C the projection onto the second factor. Note that t is a smooth morphism, with f D t ı i 0 . Let I OM 0 be the ideal sheaf defining the smooth hypersurface ft D 0g ' M , i.e., the sheaf of functions vanishing along M . Then the increasing V -filtration of Malgrange–Kashiwara on the algebraic coherent sheaf DM 0 with respect to the smooth hypersurface M M 0 is defined for k 2 Z by Vk DM 0 WD fP 2 DM 0 j P .I j Ck / I j

for all j 2 Zg :

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Here I j WD OM 0 for j < 0. Note that \ Vk DM 0 D f0g and k2Z

[

Vk DM 0 D DM 0 :

k2Z

Moreover, Vk DM 0 jft ¤0g D DM 0 jft ¤0g for all k 2 Z, so that GrkV DM 0 is supported on M . By definition, one has t 2 V1 DM 0 ; @t 2 V1 DM 0

and @t t D 1 C t@t 2 V0 DM 0 :

Also, for the sheaf DM 0 =C of relative differential operators along the fibers of t we have: DM 0 =C V0 DM 0 and Gr0V DM 0 jM D DM Œ@t t : Let M 2 MHM.M / be a given mixed Hodge module, where as before we use the same symbol for the underlying (filtered) holonomic (left) D-module. Then the pushforward D M 0 -module M0 WD i0 M on M 0 admits a unique increasing canonical V -filtration satisfying the following properties: 1. it a discrete, rationally indexed filtration. S 2. ˛ V˛ M0 D M0 , and each V˛ M0 is a coherent V0 DM 0 -module. 3. .Vk DM 0 /.V˛ M0 / V˛Ck M for all ˛ 2 Q; k 2 Z, and t.V˛ M0 / D V˛1 M0 for all ˛ < 0. S 4. @t tC˛ is nilpotent on Gr˛V M0 WD V˛ M0 =V 0 : X

R

(Here X denotes the degree of zero-cycles.) Note that under the identification of ci with the i th elementary symmetric function in x1 ; x2 ; : : : , any weighted homogeneous P polynomial P .c1 ; c2 ; : : : / of degree d is a Z-combination of Schur polynomials jjDd a s . In their study of numerically positive polynomials for ample vector bundles, the authors of [11] showed

221

On a certain family of U.b/-modules

P Theorem 7.1 (Fulton–Lazarsfeld). Such a nonzero polynomial jjDd a s is numerically positive for ample vector bundles if and only if all the coefficients a are nonnegative. Schur functors give rise to Schur bundles V .E/ associated to vector bundles E. In [28, Cor. 7.2], the following result was shown P Corollary 7.2 (Pragacz). In a Z-combination s .V .E// D  a s .E/ of Schur polynomials, all the coefficients a are nonnegative. Indeed, assuming that E is ample, combining the theorem and the fact that the Schur bundle of an ample bundle is ample (see [12]), the assertion follows. To proceed further, we shall need a variant of functors SI ./ associated with sequences of surjections of modules. Suppose that F0 D F  F1  F2  : : :

(7.1)

is a sequence of surjections of R-modules, ring. Let I D P1where R is a commutative P1 e Œik;l  be a shape (see Sect. 4), ik WD lD1 ik;l , and i l WD kD1 ik;l . We define SI0 .F / as the image of the following composition: ‰I .F / W

O

^

^ik .Fk / !

k

OO k

mS

^ik;l .Fk / !

l

O

SQil .Fl / ;

(7.2)

l

where ^ is the diagonalization in the exterior algebra and mS is the multiplication in the symmetric algebra. For w 2 †1 , we define Sw0 .F / D SI0 w .F / : Suppose that R is K, and Fi is spanned by f1 ; f2 ; : : : ; fni . Consider the maximal torus (2.2) T B consisting of diagonal matrices with x1 ; x2 ; : : : on the diagonal, with respect to the basis ffi W i D 1; 2; : : : g. If w 2 †n , then the character of Sw0 , i.e., the trace of the action of T on Sw0 .F /, is equal to Sw0 ww0 . We now pass to filtered vector bundles. By a filtered bundle, we shall mean a vector bundle E of rank n, equipped with a flag of subbundles 0 D En En1 : : : E0 D E ; where rank.Ei / D n  i for i D 0; 1; : : : ; n. For a polynomial P of degree d , and a filtered vector bundle E on a variety X , by substituting c1 .E0 =E1 / for x1 , c1 .E1 =E2 / for x2 ,. . . , c1 .Ei 1 =Ei / for xi ; : : : , we get a codimension d class denoted P .E /. Such a polynomial P will be called numerically positive for filtered ample vector bundles, if for any filtered ample vector bundle E on any d -dimensional variety X , Z P .E / > 0 : X

222

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Write P as a Z-combination of Schubert polynomials P D coefficients aw . We then have (see [10])

P

aw Sw with the unique

P Theorem 7.3 (Fulton). Such a nonzero polynomial P D aw Sw is numerically positive for filtered ample bundles if and only if all the coefficients aw are nonnegative. In [10], in fact, a more general result for r-filtered ample vector bundles is proved. Suppose that (7.1) is a sequence of vector bundles of ranks n; n  1; : : : ; 1. We record Lemma 7.4. If F is an ample vector bundle, then the bundle Sw0 .F / is ample. Proof. The bundle Sw0 .F / is the image of the map (7.2), thus it is a quotient of a tensor product of exterior powers of the bundles Fk which are quotients of F . The assertion follows from the facts (see [12]) that a quotient of an ample bundle is ample, an exterior power of an ample vector bundle is ample, and the tensor product of ample bundles is ample.  Let xi D c1 .Ker.Fni ! Fni C1 //. Write s .Sw0 .F // as a Z-combination X av Sv .x1 ; x2 ; : : : / : (7.3) s .Sw0 .F // D v

Then assuming that F is ample, and observing that F admits a filtration by Ker.F  Fi /, i D 0; : : : ; n, the following result holds true by the theorem and the lemma. Corollary 7.5 (Fulton). The coefficients av in (7.3) are nonnegative. Let E be a filtered vector bundle with Chern roots x1 ; x2 ; : : : and w be a permutation. Then using the notation from the introduction, the Chern roots of the vector bundle Sw .E / are expressions of the form l.x ˛ /, where x ˛ are monomials of Sw . This follows from the character formula for KP modules (cf. Theorem 4.5). Therefore we can restate the corollary in the following way. Corollary 7.6. A Schur function specialized with the expressions l.x ˛ / associated to the monomials x ˛ of a Schubert polynomial is a nonnegative combination of Schubert polynomials. Note 7.7. In [17, Sect. 6], the reader can find a discussion of some developments related to KP modules. It would be interesting to find further applications of KP modules and construct analogues of KP modules for other types.

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Equivariant Chern–Schwartz–MacPherson classes in partial flag varieties: interpolation and formulae Richárd Rimányi1 and Alexander Varchenko2 Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Weight functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Combinatorial description of weight function . . . . . . . . . . . . . . 4 Partial flag manifold, equivariant cohomology, Schubert varieties . 5 Unique classes defined by interpolation . . . . . . . . . . . . . . . . . . 6 Chern–Schwartz–MacPherson classes . . . . . . . . . . . . . . . . . . . 7 CSM classes coincide with classes . . . . . . . . . . . . . . . . . . . . 8 Orthogonality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 CSM classes of Schubert cells in generalized flag manifolds . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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1 Introduction An interesting chapter of enumerative geometry is the theory of characteristic classes of singular varieties. One of the fundamental results about Chern–Schwartz–MacPherson (CSM) characteristic classes is their calculation for degeneracy loci in [10] by Pragacz and Parusi´nski. In this short note—dedicated to the 60th birthday of P. Pragacz—we prove a result about CSM classes of Schubert cells in partial flag varieties. Namely, we present a set of interpolation properties that uniquely determine the sought CSM class. Such interpolation characterisation was known before for the leading term of the CSM class, the fundamental class [4]. A solution of these interpolation conditions is a weight function in the terminology of our earlier works. Weight

1 R.R.

was supported by NSF grant DMS-1200685. was supported in part by NSF grant DMS-1362924, Simons Foundation grant #336826, and the Max Planck Institute in Bonn. 2 A.V.

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functions were considered by Tarasov and Varchenko [15, 16] in the context of qhypergeometric solutions of qKZ differential equations, and turn up in our recent works (with Tarasov, Gorbounov) in the context of geometric interpretations of Bethe algebras. In fact, our results on the interpolation conditions and their solution are present in [11, 12, 13] for certain cohomology classes I —see also one of the main motivations [7, Section 3.3.4]. In the present note we only prove that I is equal to the CSM class of a Schubert cell I . A corollary of this equality is an explicit combinatorial formula to localize CSM classes at any torus fixed point, an analogue for CSM classes of Billey’s formulas [3] to localize Schubert classes in flag manifolds. By giving a proof of the fact that I agrees with the CSM class of a Schubert cell, and by presenting an accessible description of the interpolation conditions and the weight functions (with appropriate convention changes) we hope to bring the attention of researchers in Schubert calculus and characteristic classes to the quantum group aspects of CSM classes. In Section 9 we extend the result to the case of Chern–Schwartz–MacPherson classes of Schubert cells in G=P where P is a parabolic subgroup of a semisimple group G. The authors thank P. Aluffi, L. M. Fehér, and T. Ohmoto for useful discussions on the topic.

2 Weight functions Let us fix non-negative integers n and N , as well as  D .1 ; : : : ; N / 2 f0; 1; 2; : : :gN PN with jj D kD1 k D n. Consider N -tuples I D .I1 ; : : : ; IN /, where Ik

f1; : : : ; ng, jIk j D k , and Ik \ Il D ; for k 6D l. The set of such N -tuples will be denoted by I . For I 2 I we will use the notations I .k/ D

k [

n o .k/ .k/ .k/ Il D i1 < i2 < : : : < i.k/ ;

lD1

.k/ D jI .k/ j D

k X

l

lD1

and the variables o n t D ta.k/ W k D 1; : : : ; N  1; a D 1; : : : ; .k/ ;

z D fz1 ; : : : ; zn g:

For I 2 I , k D 1; : : : ; N  1, a D 1; : : : ; .k/ , b D 1; : : : ; .kC1/ define 8 .kC1/ ˆ  ta.k/ if ib.kC1/ < ia.k/ ia.k/ I .a; b/ D ˆ : 1 if ib.kC1/ D ia.k/ ; .N /

with the convention that tb

D zb .

(2.1)

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Let the group S ./ D S.1/  : : :  S.N 1/ act on the set of variables t in such a way that elements of S.k/ permute the lower indexes of ta.k/ ’s. Let Sym f .t/ D P 2S ./ f . .t//: Q 1 Q.k/ Q.k/ .k/ .k/ Denote e .t/ D N aD1 bD1 .1 C tb  ta /. kD1 Definition 2.1. Define the weight function 2 0 .k/ .kC1/ 13 .k/ .k/ N 1   Y Y  Y .k/ Y Y 1 C t .k/  ta.k/ b @ A5 ; `I .a; b/  WI .t; z/ D Sym 4 .k/ .k/ t  t a aD1 bDaC1 kD1 aD1 bD1 b and the modified weight function WQ I .t; z/ D WI .t; z/=e .t/: .k/

.k/

The weight functions WI are polynomials (despite the appearance of tb  ta factors in the denominators), while the modified weight functions WQ I are rational functions. Example 2.2. For N D 2,  D .1; n  1/ we have W.fkg;f1;:::;ngfkg/ D

k1 Y i D1

1 C zi 

t1.1/



n   Y  zi  t1.1/ : i DkC1

3 Combinatorial description of weight function In this section we show a diagrammatic interpretation of the terms of the weight function. Let I 2 I . Consider a table with n rows and N columns. Number the rows from top to bottom and number the columns from left to right. Certain boxes of this table will be distinguished, as follows. In the k’th column distinguish boxes in the i ’th row if i 2 I .k/ . This way all the boxes in the last column will be distinguished since I .N / D f1; : : : ; ng. Now we will define fillings of the tables by putting various variables in the distinguished boxes. First, put the variables z1 ; : : : ; zn into the last column from top to bottom. Now choose permutations 1 2 S.1/ ; : : : ; N 1 2 S.N 1/ . Put the variables ; : : : ; t .k/..k/ / in the distinguished boxes of the k’th column from top to bottom. t .k/ k .1/ k Each such filled table will define a rational function as follows. Let u be a variable in the filled table in one of the columns 1; : : : ; N  1. If v is a variable in the next column, but above the position of u then consider the factor 1Cvu (‘type-1 factor’). If v is a variable in the next column, but below the position of u then consider the factor v  u (‘type-2 factor’). If v is a variable in the same column, but below the

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position of u then consider the factor .1 C v  u/=.v  u/ (‘type-3 factor’). The rule is illustrated in the following figure. v

u

u

u v

1Cvu

vu

type-1

type-2

v 1Cvu vu type-3

For each variable u in the table consider all these factors and multiply them together. This is “the term associated with the filled table”. One sees that WI is the sum of terms associated with the filled tables corresponding to all choices 1 ; : : : ; N 1 . For example, Wf2g;f1g;f3g is the sum of two terms associated with the filled tables t1.2/ t1.1/

t2.2/

z1 t1.1/

z2 ;

t2.2/

z1

t1.2/

z2 :

z3

z3

The term corresponding to the first filled table is 

      1 C t1.2/  t1.1/ 1 C z1  t2.2/ z2  t1.2/ z3  t1.2/ z3  t2.2/ ƒ‚ … „ ƒ‚ … „ type-1

type-2



 1 C t2.2/  t1.2/   ; t2.2/  t1.2/ ƒ‚ … „ type-3

and the term corresponding to the second filled table is        1 C t2.2/  t1.1/ 1 C z1  t1.2/ z2  t2.2/ z3  t2.2/ z3  t1.2/ ƒ‚ … „ ƒ‚ … „ type-1

type-2

  1 C t1.2/  t2.2/   : t1.2/  t2.2/ ƒ‚ … „ type-3

4 Partial flag manifold, equivariant cohomology, Schubert varieties Let 1 ; : : : ; n be the standard basis in Cn . Consider the partial flag manifold F parameterizing chains 0 D F0 F1 : : : FN D Cn ;

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where dim Fi =Fi 1 D i . The standard action of the torus T D .C /n on Cn induces an action of T on F . The fixed points of this action are the points xI D .Fi / with Fk D spanf i W i 2 I .k/ g. Consider variables i D f i;1 ; : : : ; i;i g for i D 1; : : : ; N , and let  D f1 ; : : : ; N g. The group S D i Si acts on  by permuting the variables with the same first index. The complex coefficient, T equivariant cohomology ring of F is presented as HT .F / D CŒS ˝ CŒz = hf ./ D f .z/ for any f 2 CŒzSn i:

(4.1)

Here i;j for j D 1; : : : ; i are the Chern roots of the bundle whose fiber is Fi =Fi 1 , and z are the Chern roots of the torus. Let Vi D span. 1 ; : : : ; i /. For I 2 I define the Schubert cell I D fF 2 F j dim.Fp \ Vq / D #fi 2 I .p/ ji  qg 8p  N 8q  ng: The Schubert cell I is an affine space of dimension #f.i; j / 2 f1; : : : ; ng2 j i 2 Ia ; j 2 Ib ; a < b; i > j g: The point xI is a smooth point of the Schubert cell I . The weights of the torus action on the tangent space TxI I are zb  za

for

a > b; a 2 Ik ; b 2 Il ; k < l:

The weights of the torus action on a T invariant normal space NxI I to TxI I in TxI F are zb  za for a < b; a 2 Ik ; b 2 Il ; k < l: Hence we have Y Y Y c.TxI I / D .1 C zb  za /;

e.NxI I / D

kb

Y Y Y

.zb  za /;

ka

for the tangent total Chern class and the normal Euler class of I at xI .

5 Unique classes defined by interpolation The restriction of a cohomology class ! 2 HT .F / to the fixed point xJ will be denoted !jxJ . In terms of variables this means the substitution f k;i j i D 1; : : : ; k g 7! fza j a 2 Jk g:

(5.1)

For an S ./ symmetric function f .t; z/ in variables t; z as in (2.1) let f .; z/ denote the substitution fta.k/ j a D 1 : : : ; .k/ g 7! f1 ; : : : ; k g:

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Theorem 5.1. There are unique classes I 2 HT .F / satisfying (I) I jxJ is divisible by c.TxJ J /; (II) I jxI D c.TxI I /e.NxI I /; (III) I jxJ has degree less than dim F D Moreover,

P k