Classifying Spaces of Degenerating Polarized Hodge Structures. (AM-169) [Course Book ed.] 9781400837113

Table of contents :
Contents
Introduction
Chapter 0. Overview
0.1 Hodge Theory
0.2 Logarithmic Hodge Theory
0.3 Griffiths Domains and Moduli of PH
0.4 Toroidal Partial Compactifications of Г\D and Moduli of PLH
0.5 Fundamental Diagram and Other Enlargements of D
0.6 Plan of This Book
0.7 Notation and Convention
Chapter 1. Spaces of Nilpotent Orbits and Spaces of Nilpotent i-Orbits
1.1 Hodge Structures and Polarized Hodge Structures
1.2 Classifying Spaces of Hodge Structures
1.3 Extended Classifying Spaces
Chapter 2. Logarithmic Hodge Structures
2.1 Logarithmic Structures
2.2 Ringed Spaces (X^log, O^log X )
2.3 Local Systems on X^log
2.4 Polarized Logarithmic Hodge Structures
2.5 Nilpotent Orbits and Period Maps
2.6 Logarithmic Mixed Hodge Structures
Chapter 3. Strong Topology and Logarithmic Manifolds
3.1 Strong Topology
3.2 Generalizations of Analytic Spaces
3.3 Sets Eσ and E^♯σ
3.4 Spaces Eσ, Г\DΣ, E^♯σ, and D^♯Σ
3.5 Infinitesimal Calculus and Logarithmic Manifolds
3.6 Logarithmic Modifications
Chapter 4. Main Results
4.1 Theorem A: The Spaces Eσ, Г\DΣ and Г\DΣ♯
4.2 Theorem B: The Functor PLHФ
4.3 Extensions of Period Maps
4.4 Infinitesimal Period Maps
Chapter 5. Fundamental Diagram
5.1 Borel-Serre Spaces (Review)
5.2 Spaces of SL(2)-Orbits (Review)
5.3 Spaces of Valuative Nilpotent Orbits
5.4 Valuative Nilpotent i-Orbits and SL(2)-Orbits
Chapter 6. The Map ψ
6.1 Review of [CKS] and Some Related Results
6.2 Proof of Theorem 5.4.2
6.3 Proof of Theorem 5.4.3 (i)
6.4 Proofs of Theorem 5.4.3 (ii) and Theorem 5.4.4
Chapter 7. Proof of Theorem A
7.1 Proof of Theorem A (i)
7.2 Action of σC on Eσ
7.3 Proof of Theorem A for Г(σ)^gp\Dσ
7.4 Proof of Theorem A for Г\DΣ
Chapter 8. Proof of Theorem B
8.1 Logarithmic Local Systems
8.2 Proof of Theorem B
8.3 Relationship among Categories of Generalized Analytic Spaces
8.4 Proof of Theorem 0.5.29
Chapter 9. ♭-Spaces
9.1 Definitions and Main Properties
9.2 Proofs of Theorem 9.1.4 for Г\X^♭BS, Г\D^♭BS, and Г\D^♭BS, val
9.3 Proof of Theorem 9.1.4 for Г\D^♭SL(2),≤1
9.4 Extended Period Maps
Chapter 10. Local Structures of DSL(2)
10.1 Local Structures of DSL(2)
10.2 A Special Open Neighborhood U(p)
10.3 Proof of Theorem 10.1.3
10.4 Local Structures of DSL(2),≤1 and Г\D^♭SL(2),≤1
Chapter 11. Moduli of PLH with Coefficients
11.1 Space Г\ D^AΣ
11.2 PLH with Coefficients
11.3 Moduli
Chapter 12. Examples and Problems
12.1 Siegel Upper Half Spaces
12.2 Case GR ≃ O(1, n − 1, R)
12.3 Example of Weight 3 (A)
12.4 Example of Weight 3 (B)
12.5 Relationship with [U2]
12.6 Complete Fans
12.7 Problems
Appendix
A1 Positive Direction of Local Monodromy
A2 Proper Base Change Theorem for Topological Spaces
References
List of Symbols
Index

Citation preview

Classifying Spaces of Degenerating Polarized Hodge Structures

Annals of Mathematics Studies Number 169

Classifying Spaces of Degenerating Polarized Hodge Structures

Kazuya Kato and Sampei Usui

PRINCETON UNIVERSITY PRESS PRINCETON AND OXFORD 2009

Copyright © 2009 by Princeton University Press Published by Princeton University Press 41 William Street, Princeton, New Jersey 08540 In the United Kingdom: Princeton University Press 6 Oxford Street, Woodstock, Oxfordshire 0X20 1TW All Rights Reserved Library of Congress Cataloging-in-Publication Data Kato, K. (Kazuya) Classifying spaces of degenerating polarized Hodge structures / Kazuya Kato and Sampei Usui. p. cm. — (Annals of mathematics studies ; no. 169) Includes bibliographical references and index. ISBN 978-0-691-13821-3 (cloth : acid-free paper) — ISBN 978-0-691-13822-0 (pbk. : acid-free paper) 1. Hodge theory. 2. Logarithms. I. Usui, Sampei. II. Title. QA564.K364 2009 514
.74—dc22 2008039091 British Library Cataloging-in-Publication Data is available This book has been composed in LATEX Printed on acid-free paper. ∞ The publisher would like to thank the authors of this volume for providing the camera-ready copy from which this book was printed press.princeton.edu Printed in the United States of America 10 9 8 7 6 5 4 3 2 1

Suugaku wa mugen enten kou kokoro Koute kogarete harukana tabiji by Kazuya Kato and Sampei Usui, which was translated by Luc Illusie as L'impossible voyage aux points à l'in ni N'a pas fait battre en vain le coeur du géomètre

Contents

Introduction Chapter 0. Overview 0.1 0.2 0.3 0.4 0.5 0.6 0.7

Hodge Theory Logarithmic Hodge Theory Griffiths Domains and Moduli of PH Toroidal Partial Compactifications of
\D and Moduli of PLH Fundamental Diagram and Other Enlargements of D Plan of This Book Notation and Convention

Chapter 1. Spaces of Nilpotent Orbits and Spaces of Nilpotent i-Orbits 1.1 1.2 1.3

Hodge Structures and Polarized Hodge Structures Classifying Spaces of Hodge Structures Extended Classifying Spaces

Chapter 2. Logarithmic Hodge Structures 2.1 2.2 2.3 2.4 2.5 2.6

Logarithmic Structures log Ringed Spaces (X log , OX ) log Local Systems on X Polarized Logarithmic Hodge Structures Nilpotent Orbits and Period Maps Logarithmic Mixed Hodge Structures

Chapter 3. Strong Topology and Logarithmic Manifolds 3.1 3.2 3.3 3.4 3.5 3.6

Strong Topology Generalizations of Analytic Spaces  Sets Eσ and Eσ   Spaces Eσ ,
\D , Eσ , and D Infinitesimal Calculus and Logarithmic Manifolds Logarithmic Modifications

Chapter 4. Main Results 4.1 4.2 4.3 4.4

Theorem A: The Spaces Eσ ,
\D , and
\D  Theorem B: The Functor PLH Extensions of Period Maps Infinitesimal Period Maps

1 7 7 11 24 30 43 66 67 70 70 71 72 75 75 81 88 94 97 105 107 107 115 120 125 127 133 146 146 147 148 153

viii

CONTENTS

Chapter 5. Fundamental Diagram 5.1 5.2 5.3 5.4

157

Borel-Serre Spaces (Review) Spaces of SL(2)-Orbits (Review) Spaces of Valuative Nilpotent Orbits Valuative Nilpotent i-Orbits and SL(2)-Orbits

158 165 170 173



Chapter 6. The Map ψ : Dval → DSL (2) 6.1 6.2 6.3 6.4

175

Review of [CKS] and Some Related Results Proof of Theorem 5.4.2 Proof of Theorem 5.4.3 (i) Proofs of Theorem 5.4.3 (ii) and Theorem 5.4.4

175 186 190 195

Chapter 7. Proof of Theorem A

205

7.1 Proof of Theorem A (i) 7.2 Action of σC on Eσ 7.3 Proof of Theorem A for
(σ )gp \Dσ 7.4 Proof of Theorem A for
\D

205 209 215 220

Chapter 8. Proof of Theorem B 8.1 8.2 8.3 8.4

226

Logarithmic Local Systems Proof of Theorem B Relationship among Categories of Generalized Analytic Spaces Proof of Theorem 0.5.29

226 229 235 241

Chapter 9. -Spaces

244

9.1 9.2

Definitions and Main Properties    Proofs of Theorem 9.1.4 for
\XBS ,
\DBS , and
\DBS,val

9.3 9.4

Proof of Theorem 9.1.4 for
\DSL(2),≤1 Extended Period Maps



248 249 

Chapter 10. Local Structures of DSL(2) and
\DSL(2),≤1 10.1 10.2 10.3 10.4

Local Structures of DSL(2) A Special Open Neighborhood U (p) Proof of Theorem 10.1.3  Local Structures of DSL(2),≤1 and
\DSL(2),≤1

Chapter 11. Moduli of PLH with Coefficients A
\D

11.1 Space 11.2 PLH with Coefficients 11.3 Moduli

Chapter 12. Examples and Problems 12.1 12.2 12.3 12.4

Siegel Upper Half Spaces Case GR  O(1, n − 1, R) Example of Weight 3 (A) Example of Weight 3 (B)

244 246

251 251 255 263 269 271 271 274 275 277 277 281 290 295

CONTENTS

12.5 Relationship with [U2] 12.6 Complete Fans 12.7 Problems

Appendix A1 A2

Positive Direction of Local Monodromy Proper Base Change Theorem for Topological Spaces

References List of Symbols Index

ix 299 301 304 307 307 310 315 321 331

Classifying Spaces of Degenerating Polarized Hodge Structures

Introduction This book is the full detailed version of the paper [KU1]. In [G1], Griffiths defined and studied the classifying space D of polarized Hodge structures of fixed weight w and fixed Hodge numbers (hp,q ). In [G5], Griffiths presented a dream of adding points at infinity to D. This book is an attempt to realize his dream. In the special case w = 1,

h1,0 = h0,1 = g,

other hp,q = 0,

(1)

the classifying space D coincides with Siegel’s upper half space hg of degree g. If g = 1, hg is the Poincaré upper half plane h = {x + iy | x, y ∈ R, y > 0}. For a congruence subgroup
of SL(2, Z), that is, for a subgroup of SL(2, Z) which contains the kernel of SL(2, Z) → SL(2, Z/nZ) for some integer n ≥ 1, the quotient
\h is a modular curve without cusps. We obtain the compactification
\(h ∪ P1 (Q)) of the modular curve
\h by adding points at infinity called cusps (i.e., the elements of the finite set
\P1 (Q)). In the case (1) with g general, for a congruence subgroup
of Sp(g, Z), by adding points at infinity, we have toroidal compactifications of
\hg [AMRT] and the Satake-Baily-Borel compactification of
\hg [Sa1], [BB]. All these compactifications coincide when g = 1, but, when g > 1, there are many toroidal compactifications and the Satake-Baily-Borel compactification is different from them. The theory of these compactifications is included in a general theory of compactifications of quotients of symmetric Hermitian domains by the actions of discrete arithmetic groups. The points at infinity are often more important than the usual points. For example, the Taylor expansion of a modular form at the standard cusp (i.e., the class of ∞ ∈ P1 (Q) modulo
) of the compactified modular curve
\(h ∪ P1 (Q)) is called the q-expansion and is very important in the theory of modular forms. However, the classifying space D in general is rarely a symmetric Hermitian domain, and we cannot use the general theory of symmetric Hermitian domains when we try to add points at infinity to D. In this book, we overcome this difficulty. We discuss two subjects in this book.

SUBJECT I. TOROIDAL PARTIAL COMPACTIFICATIONS AND MODULI OF POLARIZED LOGARITHMIC HODGE STRUCTURES A toroidal compactification of
\hg is defined depending on the choice of a certain fan (cone decomposition). If the fan is not sufficiently big, we have a toroidal

2

INTRODUCTION

partial compactification of
\hg , which need not be compact and which is locally isomorphic to an open set of a toroidal compactification. In this book, for general D, we construct a kind of toroidal partial compactification
\D of
\D associated with a fan  and a discrete subgroup
of Aut(D) satisfying a certain compatibility with . In the case (1), the classes of polarized Hodge structures in
\hg converge to a point at infinity of
\hg when the polarized Hodge structures become degenerate. As in [Sc], nilpotent orbits appear when polarized Hodge structures become degenerate. In our definition of D for general D, a nilpotent orbit itself is viewed as a point at infinity. As is discussed in detail in this book, the theory of nilpotent orbits is regarded as a local aspect of the theory of polarized logarithmic Hodge structures. The “polarized logarithmic Hodge structure” (PLH) is formulated by using the theory of logarithmic structures introduced by Fontaine and Illusie and developed in [Kk1],[KkNc], and it is something like the logarithmic degeneration of the PH (polarized Hodge structure). We give here a rough illustration of the idea of the PLH. Let X be a complex manifold endowed with a divisor Y with normal crossings, and let U = X − Y . Let H be a PLH on X with respect to the “logarithmic structure of X associated with Y .” Then the restriction H |U of H to U is a family of usual PH parametrized by U . At x ∈ U , the fiber H (x) of H is a usual PH. At Y , this family can become degenerate in the classical sense. At each point x ∈ Y , the fiber of H at x corresponds to a nilpotent orbit (in the classical theory) toward x ∈ Y . Schematically, we have the following: (H (x) : a PH)
 a fiber over a point x ∈ U 

(H (x) : a PLH)
 a fiber over a point x ∈ U  extension

(H |U : a family of PH on U ) −−−−→ (H : PLH on X)     degeneration toward x ∈ Y  a fiber over a point x ∈ Y  (a nilpotent orbit toward x)

(2)

(H (x) : a PLH).

(See 0.2.20, 0.4.25.) Our main theorem concerning Subject I is stated roughly as follows (for the precise statement, see Theorem 0.4.27 below). Theorem.
\D is the fine moduli space of “polarized logarithmic Hodge structures” with a “
-level structure” whose “local monodromies are in the directions in .” Roughly speaking,
\D = (polarized Hodge structures with a “
-level structure”) ∩
\D =
\{σ -nilpotent orbit | σ ∈ }   “polarized logarithmic Hodge structures” with a “
-level structure” = whose “local monodromies are in the directions in .”

3

INTRODUCTION

Here, a σ -nilpotent orbit is a nilpotent orbit in the direction of the cone σ . For σ = {0}, a σ -nilpotent orbit is nothing but a point of D; hence we can regard D ⊂ D . In the classical case (1),
\D for a congruence subgroup
and for a sufficiently big  is a toroidal compactification of
\D. Already, in this classical case, this theorem gives moduli-theoretic interpretations of the toroidal compactifications of
\hg . The space
\D has a kind of complex structure, but a delicate point is that, in general, this space can have locally the shape of a “complex analytic space with a slit” (for example, C2 minus {(0, z) | z ∈ C, z  = 0}), and hence it is often not locally compact. However, it is very close to a complex analytic manifold.
\D is a logarithmic manifold in the sense of 0.4.17 below. Infinitesimal calculus can be performed nicely on
\D . These phenomena were first examined in the easiest nontrivial case in [U2]. One motivation of Griffiths for adding points at infinity to D was the hope that the period map ∗ →
\D (∗ = {q ∈ C | 0 < |q| < 1}) associated with a variation of polarized Hodge structure on ∗ could be extended to  →
\(D ∪ (points at infinity)) ( = {q ∈ C | |q| < 1}). By using the above main theorem and the nilpotent orbit theorem of Schmid, we can actually extend the period map to  →
\D for some suitable  (see 0.4.30 and 4.3.1, where a more general result is given).

SUBJECT II. THE EIGHT ENLARGEMENTS OF D AND THE FUNDAMENTAL DIAGRAM In the classical case (1) above, there is another compactification
\DBS of
\D (
is a congruence subgroup of Sp(g, Z)) called the Borel-Serre compactification, where DBS is the Borel-Serre space denoted by D¯ in [BS], which is a real manifold with corners containing D as a dense open set. For general D, by adding to D points at infinity of different kinds, we obtain eight enlargements of D with maps among them which form the following fundamental diagram (3) (see 5.0.1).

Fundamental Diagram DSL(2),val

D,val

←

↓ D



D,val ↓

←



D

→

→

DBS,val

↓

↓

DSL(2)

DBS

(3)

4

INTRODUCTION

Note that the space D that appeared in Subject I sits at the left lower end of this diagram. The left-hand side of this diagram has Hodge-theoretic nature, and the right-hand side has the nature of the theory of algebraic groups. These are related  by the middle map D,val → DSL(2) , which is a geometric interpretation of the SL(2)-orbit theorem of Cattani-Kaplan-Schmid [CKS]. In the case (1) with g = 1, the largest  exists. For this , D = h ∪ P1 (Q), and the above diagram becomes hBS

=

 ←

h ∪ P1 (Q) 

hBS

=

hBS

hBS 

=

(4)

hBS



h ∪ P1 (Q)

←

hBS

The space hBS is described as follows. It is the union of open subsets hBS (a) for a ∈ P1 (Q). hBS (∞) = {x + iy | x ∈ R, 0 < y ≤ ∞} ⊃ h = {x + iy | x ∈ R, y > 0}. The action of SL(2, Q) on h extends to a continuous action of SL(2, Q) on hBS , and we have g(hBS (a)) = hBS (ga) for g ∈ SL(2, Q) and a ∈ P1 (Q). In particular, all hBS (a) are homeomorphic to each other. The map hBS → h = h ∪ P1 (Q) for the biggest  is the identity map on h and sends elements of hBS (a) − h to a for a ∈ P1 (Q). In the case (1) for general g, the fundamental diagram becomes DSL(2),val

=

↓ D,val

←

↓ D



D,val

→

DSL(2)

DBS,val ↓

=

DBS

(5)

↓ ←



D

In this case, for a subgroup
of Sp(g, Z) of finite index and for a suitable ,
\D is a toroidal compactification [AMRT] of
\D,
\D,val is obtained from
\D   by blow-ups, and the maps
\D →
\D and
\D,val →
\D,val are proper surjective maps whose fibers are products of finite copies of S1 . On the other hand,
\DBS is the Borel-Serre compactification [BS] of
\D =
\hg . The spaces DBS  n and D are real manifolds with corners (they are like R m × R≥0 locally). Already, in this classical case, the fundamental diagram (5) gives a relation between toroidal compactifications of
\hg and the Borel-Serre compactification of
\hg , which were not known before (see 0.5.28 below).

5

INTRODUCTION

For general D, these eight spaces are defined as D = (the space of nilpotent orbits) (1.3.8),  D = (the space of nilpotent i-orbits) (1.3.8), DSL(2) = (the space of SL(2)-orbits) (5.2.6), DBS = (the space of Borel-Serre orbits) (5.1.5), D,val = (the space of valuative nilpotent orbits) (5.3.5),  D,val = (the space of valuative nilpotent i-orbits) (5.3.5), DSL(2),val = (the space of valuative SL(2)-orbits) (5.2.7), DBS,val = (the space of valuative Borel-Serre orbits) (5.1.6). The space DBS was constructed in [KU2] and [BJ] independently, by using the work [BS] of Borel-Serre on Borel-Serre compacifications. The spaces DSL(2) , DSL(2),val , and DBS,val are defined in [KU2]. Roughly speaking, these eight spaces appear as follows:
\D is like an analytic manifold with slits,  D and DSL(2) are like real manifolds with corners and slits, DBS is a real manifold with corners,  
\D,val and D,val are the projective limits of “blow-ups” of
\D and D , respectively, associated with rational subdivisions of , DSL(2),val and DBS,val are the projective limits of certain “blow-ups” of DSL(2) and DBS , respectively. 



The maps
\D →
\D and
\D,val →
\D,val are proper surjective maps whose fibers are products of a finite number of copies of S1 , where this number is varying. Like nilpotent orbits, SL(2)-orbits also appear in the theory of degenerations of polarized Hodge structures [Sc], [CKS]. The fundamental diagram (3) shows how nilpotent orbits, SL(2)-orbits, and the theory of Borel and Serre are related. In this book, we study all these eight spaces. To prove the main theorem in Subject I and to prove that
\D has good properties such as the Hausdorff property, nice infinitesimal calculus, etc., we need to consider all other spaces in the diagram (3); we discuss the spaces from the right to the left in the fundamental diagram (3) to deduce the nice properties of
\D , starting from the properties of the BorelSerre compactifications (which were proved in [BS] by using arithmetic theory of algebraic groups). The organization of this book is as follows. In Chapter 0, we give an overview of the book. In Chapters 1–4, we formulate the main theorem in Subject I. In Chapters 5–8, we prove the main theorem, considering all eight enlargements of D in the fundamental diagram (3), and we also prove various properties of the eight enlargements. In Chapters 9–12, we give complementary results. The authors are grateful to Professor Chikara Nakayama for ongoing discussions and constant encouragement. He carefully read various versions of the manuscript and sent us lists of useful comments and advice. In particular, we owe him very heavily for the proof of Theorem 4.3.1. The authors are also grateful to Professor

6

INTRODUCTION

Kazuhiro Fujiwara for stimulating discussions and advice and to Professor Akira Ohbuchi for inputting the figures in the electronic file. Parts of this work were done when the first author was a visitor at Institut Henri Poincaré and when the second author was a visitor at Institute for Advanced Study; the hospitality of each is gratefully appreciated. The first line of the Japanese poem (5-7-5 syllables) in the book’s epigraph was composed by Kato and then, following a Japanese tradition of collaboration, the second line (7-7 syllables) was composed by Usui. We are very grateful to Professor Luc Illusie for his beautiful translation. This work was partly supported by Grants-in-Aid for Scientific Research (B) (2) No. 11440003, (B) No. 14340010, (B) No. 16340005; (A) (1) No. 11304001, (B) No. 19340008, (B) No. 15340009 from Japan Society for the Promotion of Science.

Chapter Zero Overview

In this chapter, we introduce the main ideas and results of this book. In Section 0.1, we review the basic idea of Hodge theory. In Section 0.2, we introduce the basic idea of logarithmic Hodge theory. In Section 0.3, we review classifying spaces D of Griffiths (i.e., Griffiths domains) as the moduli spaces of polarized Hodge structures. In Section 0.4, we describe our toroidal partial compactifications of the classifying spaces of Griffiths and our result that they are the fine moduli spaces of polarized logarithmic Hodge structures. In Section 0.5, we describe the other seven enlargements of D in the fundamental diagram (3) in Introduction and state our results on these spaces. In this chapter, we explain the above subjects by presenting examples. Hodge theory (Section 0.1) and logarithmic Hodge theory (Section 0.2) are explained by using the example of the Hodge structure on H 1 (E, Z) of an elliptic curve E and its degeneration arising from the degeneration of E. This example appears first in 0.1.3 and then continues to appear as an example of each subject. The classifying space D (Section 0.3) and its various enlargements (Sections 0.4 and 0.5) are explained by using the following three examples: (i) D = h, the upper half plane; (ii) D = hg , Siegel’s upper half space ((i) is a special case of (ii). In the case (ii), we mainly consider the case g = 2.); (iii) an example of weight 2 for which D is not a symmetric Hermitian domain. These examples appear first in 0.3.2 and then continue to appear as examples of each subject. In this chapter, we do not generally give proofs.

0.1 HODGE THEORY 0.1.1 First we recall the basic idea of Hodge theory. For a topological space X, the homology groups Hm (X, Z) and the cohomology groups H m (X, Z) are important invariants of X. If X is a projective complex analytic manifold, the cohomology groups H m (X, Z) have finer structures: C ⊗Z H m (X, Z) = H m (X, C) is endowed with a decreasing filtration F = (F p )p∈Z , called Hodge filtration. The cohomology group H m (X, Z) remembers X merely as a topological space, but, with this Hodge filtration, the pair (H m (X, Z), F ) becomes a finer invariant of X which remembers the analytic structure of X (not just the topological structure of X) often very well.

8

CHAPTER 0

0.1.2 For example, in the case m = 1, the Hodge filtration F on H 1 (X, C) is given by F p = H 1 (X, C) for p ≤ 0, F p = 0 for p ≥ 2, and F 1 is the image of the injective map H 0 (X, 1X ) → H 1 (X, C).

(1)

Here H 0 (X, 1X ) is the space of holomorphic differential forms on X, and (1) is the map that sends a differential form ω ∈ H 0 (X, 1X ) to its cohomology 1 class in H 1 (X, C): Under the identification  H (X, C) = Hom(H1 (X, Z), C), the cohomology class of ω is given by γ  → γ ω (γ ∈ H1 (X, Z)). For the definition of F p of H m (X, C) for general p, m, see 0.1.7. 0.1.3 Elliptic curves. An elliptic curve X over C is isomorphic to C/(Zτ + Z) for some τ ∈ h, where h is the upper half plane. For X = C/(Zτ + Z) with τ ∈ h, H1 (X, Z) is identified with Zτ + Z, H 1 (X, Z) is identified with Hom(Zτ + Z, Z), and the Hodge filtration on H 1 (X, C) = Hom(Zτ + Z, C) is described as follows. The space H 0 (X, 1X ) is a one-dimensional C-vector space with the basis dz, where z is the coordinate function of C, and where we regard dz as a differential form on the quotient space X = C/(Zτ + Z) of C. Let (γj )j =1,2 be the Z-basis of H1 (X, Z) that is identified with the Z-basis (τ, 1) ofZτ + Z, and let (ej )j =1,2 be the dual  Z-basis of H 1 (X, Z). Since γ1 dz = τ and γ2 dz = 1, the cohomology class of dz coincides with τ e1 + e2 , and hence F 1 H 1 (X, C) is the C-subspace of H 1 (X, C) generated by τ e1 + e2 . 0.1.4 Elliptic curves (continued). If X is an elliptic curve, we cannot recover X merely from H 1 (X, Z). In fact, H 1 (X, Z)  Z2 for any elliptic curve X over C, and we cannot distinguish different elliptic curves from this information. However, if we consider the Hodge filtration, we can recover X from (H 1 (X, Z), F ) as X  HomC (F 1 , C)/H1 (X, Z) = HomC (F 1 , C)/ Hom(H 1 (X, Z), Z). (1)  Here H1 (X, Z) is embedded in HomC (F 1 , C) via the map γ  → (ω  → γ ω) (γ ∈ H1 (X, Z), ω ∈
(X, 1X )), and the isomorphism X  HomC (F 1 , C)/H1 (X, Z) sends x ∈ X to the class of the homomorphism F 1 → C, ω  → γ ω, where γ is a path in X from the origin 0 of X to x (the choice of γ is not unique, but the  class of the map ω  → γ ω modulo H1 (X, Z) is independent of the choice of γ ). In (1), the middle group is identified with the right one in which Hom(H 1 (X, Z), Z) is embedded in HomC (F 1 , C) via the composition Hom(H 1 (X, Z), Z) → HomC (H 1 (X, C), C) → HomC (F 1 , C), which is injective. If X = C/(Zτ + Z) with τ ∈ h, this isomorphism X  HomC (F 1 , C)/H1 (X, Z) is nothing but the original presentation X = C/(Zτ + Z) where HomC (F 1 , C) is identified with C by the evaluation at dz.

9

OVERVIEW

0.1.5 Now we discuss Hodge structures. A Hodge structure of weight w is a pair (HZ , F ) consisting of a free Z-module HZ of finite rank and of a decreasing filtration F on HC := C ⊗Z HZ (that is, a family (F p )p∈Z of C-subspaces of HC such that F p ⊃ F p+1 for all p), which satisfies the following condition (1):  (1) H p,q , where H p,q = F p ∩ F¯ q . HC = p+q=w

Here F¯ q denotes the image of F q under the complex conjugation HC → HC , a ⊗ x  → a¯ ⊗ x (a ∈ C, x ∈ HZ ). We have 

Fp = (2) H p ,w−p , F p /F p+1  H p,w−p . p
≥p

We say (HZ , F ) is of Hodge type (hp,q )p,q∈Z , where hp,q = dimC H p,q if p + q = w, and hp,q = 0 otherwise (these numbers hp,q are called the Hodge numbers). 0.1.6 For a projective analytic manifold X and for m ∈ Z, the pair (HZ , F ) with HZ = H m (X, Z)/(torsion) and F the Hodge filtration becomes a Hodge structure of weight m. For example, if X is the elliptic curve C/(Zτ + Z) with τ ∈ h, then H 1,0 = C(τ e1 + e2 ), H 0,1 = C(τ¯ e1 + e2 ), and H 1 (X, C) = H 1,0 ⊕ H 0,1 since τ  = τ¯ . The theory of homology groups and cohomology groups is important in the study of topological spaces. Similarly, Hodge theory (the theory of Hodge structures) is important for the study of analytic spaces. 0.1.7 For a projective analytic manifold X, the Hodge filtration F p on H m (X, C) is defined as follows. Let d

d

d

•X = (OX − → 1X − → 2X − → ···) p p be the de Rham complex of X where X = OX 1X is the sheaf of holomorphic ≥p p-forms on X (OX is set in degree 0). Let X be the degree ≥ p part of •X . Then p F is defined as ≥p

F p := H m (X, X ) → H m (X, •X )  H m (X, C). ≥p

Here H m (X, X ) and H m (X, •X ) denote the mth hypercohomology groups of ≥p complexes of sheaves. The canonical homomorphism H m (X, X ) → H m (X, •X ) is known to be injective. The isomorphism H m (X, •X )  H m (X, C) comes from the exact sequence of Dolbeault d

d

d

→ 1X − → 2X − → ··· . 0 → C → OX −

10

CHAPTER 0

We have isomorphisms p

H p,m−p  F p /F p+1  H m−p (X, X ), where the second isomorphism is obtained by applying H m (X, ) to the exact ≥p+1 ≥p p sequence of complexes of sheaves 0 → X → X → X [−p] → 0. 0.1.8 It is often important to consider a polarized Hodge structure, that is, a Hodge structure endowed with a polarization. A polarization on a Hodge structure (HZ , F ) of weight w is a nondegenerate bilinear form  ,  : HQ × HQ → Q (HQ := Q ⊗Z HZ ) which is symmetric if w is even and is antisymmetric if w is odd, satisfying the following conditions (1) and (2). (1) F p , F q  = 0 for p + q > w. (2) Let CF : HC → HC be the C-linear map defined by CF (x) = i p−q x for x ∈ H p,q . Then the Hermitian form ( , )F : HC × HC → C, defined by(x, y)F = CF (x), y, ¯ is positive definite. Here, in (2),  ,  is regarded as the natural extension to the C-bilinear form. The Hermitian form ( , )F in (2) is called the Hodge metric associated with F . The condition (1) (resp. (2)) is called the Riemann-Hodge first (resp. second) bilinear relation. 0.1.9 For a projective analytic manifold X, we have the intersection form  ,  : H m (X, Q) × H m (X, Q) → Q induced by an ample line bundle on X (see [G1], [GH]). The triple (H m (X, Z),  , , F ) becomes a polarized Hodge structure. Elliptic curves (continued) For the elliptic curve X = C/(Zτ + Z) (τ ∈ h), the standard polarization of X gives the antisymmetric pairing  ,  : H 1 (X, Q) × H 1 (X, Q) → Q characterized by e2 , e1  = 1. This pairing is nothing but the cup product H 1 (X, Q) × H 1 (X, Q) → H 2 (X, Q)  Q. It satisfies τ e1 + e2 , τ e1 + e2  = 0, (τ e1 + e2 , τ e1 + e2 )F = i 1−0 τ e1 + e2 , τ¯ e1 + e2  = i(τ¯ − τ ) = 2 Im(τ ) > 0. Hence (H 1 (X, Z),  , , F ) is indeed a polarized Hodge structure. 0.1.10 It is often very useful to consider analytic families of Hodge structures and of polarized Hodge structures.

11

OVERVIEW

Let X be an analytic manifold. After Griffiths ([G3], also [D2], [Sc]), a variation of Hodge structure (VH) on X of weight w is a pair (HZ , F ) consisting of a locally constant sheaf HZ of free Z-modules of finite rank on X and of a decreasing filtration F of HO := OX ⊗Z HZ by OX -submodules which satisfy the following three conditions: p

(1) F p = HO for p  0, F p = 0 for p  0, and gr F = F p /F p+1 is a locally free OX -module for any p. (2) For any x ∈ X, the fiber (HZ,x , F (x)) is a Hodge structure of weight w. (3) (d ⊗ 1HZ )(F p ) ⊂ 1X ⊗OX F p−1 for all p. Here (3) is called the Griffiths transversality. A polarization of a variation of Hodge structure (HZ , F ) of weight w on X is a bilinear form  ,  : HQ × HQ → Q which yields for each x ∈ X a polarization  , x on the fiber (HZ,x , F (x)). In this case, the triple (HZ ,  , , F ) is called a variation of polarized Hodge structure (VPH).

0.1.11 Let X and Y be analytic manifolds and f : Y → X be a projective, smooth morphism. Then for each m ∈ Z, we obtain a VH (variation of Hodge structure) of weight m on X: HZ = R m f∗ Z/(torsion), ≥p

F p = R m f∗ ( Y /X ) → R m f∗ ( •Y /X )  OX ⊗Z HZ . Indeed, on each fiber Yx := f −1 (x) (x ∈ X), HZ,x = H m (Yx , Z)/(torsion) and ≥p F p (x) = H m (Yx , Yx ) form the Hodge structure. If we fix a polarization of Y over X, this VH becomes a VPH (variation of polarized Hodge structure) on X ([G3]; cf. also [Sc], [GH]).

0.2 LOGARITHMIC HODGE THEORY From now on, we discuss degeneration of Hodge structures by the method of logarithmic Hodge theory. The logarithmic Hodge theory uses the magic of the theory of logarithmic structures introduced by Fontaine-Illusie. It has a strong connection with the theory of nilpotent orbits as discussed in Section 0.4. In the story of Beauty and the Beast, the Beast becomes a nice man because of the love of the heroine. Similarly, a degenerate object becomes a nice object because of the magic of LOG. (Beast) (degenerate object)

Love Of Girl

−−−−−−→ LOG

−−→

(a nice man), (a nice object).

The authors learned this mysterious coincidence of letters from Takeshi Saito.

12

CHAPTER 0

0.2.1 Elliptic curves (continued). We observe what happens for the Hodge structure on H 1 of an elliptic curve when the elliptic curve degenerates. In this section, the idea of logarithmic Hodge theory is explained by use of this example. Let  = {q ∈ C | |q| < 1} be the unit disc. Then we have a standard family of degenerating elliptic curve f : E → , which is a morphism of analytic manifolds, having the following property. (1) For q ∈  with q  = 0, f −1 (q) = C× /q Z . This is an elliptic curve. In fact, taking τ ∈ C with q = exp(2π iτ ), we have τ ∈ h (since |q| < 1), and ∼

→ C× /q Z , C/(Zτ + Z) −

(z mod (Zτ + Z))  → (exp(2π iz) mod q Z ).

(2) f −1 (0) = P1 (C)/(0 ∼ ∞). The definition of E will be given in 0.2.10 below. E is a two-dimensional analytic manifold and f −1 (0) is a divisor with normal crossings on E. These look like the left-hand side of Figure 1. This family degenerates at q = 0 as is described on the left-hand side of Figure 1. All the fibers f −1 (q) for q ∈ ∗ are homeomorphic to the surface of a doughnut, whereas the central fiber f −1 (0) has a degenerate shape. However, as we will see below, the central fiber recovers its lost body as in the right-hand side of Figure 1 by the magic of its logarithmic structure. We will explain this magical process in the following.

Figure 1

13

OVERVIEW

0.2.2 Elliptic curves (continued). Let ∗ =  − {0}, let E ∗ = f −1 (∗ ) ⊂ E, and let f
: E ∗ → ∗ be the restriction of f to E ∗ . Since f
: E ∗ → ∗ is projective and smooth, the polarized Hodge structures on H 1 of the elliptic curves C× /q Z (q ∈ ∗ ) form a variation of polarized Hodge structure (HZ
, F
). Here, HZ
= R 1 f∗
Z is a locally constant sheaf on ∗ of Z-modules of rank 2, and the filtration F
on O∗ ⊗Z d

HZ
= R 1 f∗
( •E ∗ /∗ ) = R 1 f∗
(OE ∗ − → 1E ∗ /∗ ) is given by (F
)p = O∗ ⊗Z HZ
for p ≤ 0, (F
)p = 0 for p ≥ 2, and (F
)1 = f∗
( 1E ∗ /∗ ) ⊂ O∗ ⊗Z HZ
. 0.2.3 Elliptic curves (continued). This variation of Hodge structure on ∗ does not extend to a VH on . First of all, the local system HZ
on ∗ does not extend to a local system on . HZ
extends to the sheaf R 1 f∗ Z on , but this sheaf is not locally constant. The stalk (R 1 f∗ Z)0 = H 1 (f −1 (0), Z) of this sheaf at 0 ∈  is of rank 1, not 2. In fact, e1 ∈ H 1 (C/(Zτ + Z), Z)  H 1 (C× /q Z , Z) for q ∈ ∗ (0.1.3) extends to a global section of R 1 f∗ Z on , but e2 is defined only locally on ∗ , depending on the choice of τ with q = exp(2π iτ ). There is no element of (R 1 f∗ Z)0 that gives e2 in H 1 (C/(Zτ + Z), Z)  H 1 (C× /q Z , Z) for q ∈ ∗ near to 0. We show that by a magic of the theory of logarithmic structure, HZ
does extend over the origin as a local system in the logarithmic world (0.2.4–0.2.10), and the variation of polarized Hodge structure (HZ
, F
) also extends over the origin as a logarithmic variation of polarized Hodge structure (0.2.15–0.2.20). 0.2.4 By a monoid, we mean a commutative semigroup with a neutral element 1. A homomorphism of monoids is assumed to preserve 1. A logarithmic structure on a local ringed space (X, OX ) is a sheaf of monoids MX on X endowed with a homomorphism α : MX → OX , where OX is regarded as × × a sheaf of monoids with respect to the multiplication, such that α : α −1 (OX ) → OX is an isomorphism. × We regard OX as a subsheaf of MX via α −1 . 0.2.5 Example. A standard example of a logarithmic structure is given as follows. Let X be an analytic manifold, let D be a divisor on X with normal crossings, and let U = X − D. (That is, locally, X = n a polydisc with coordinates q1 , . . . , qn , D = {q1 · · · qr = 0} (0 ≤ r ≤ n), and U = (∗ )r × n−r .) Then α

MX = {f ∈ OX | f is invertible on U } → OX is a logarithmic structure on X. This is called the logarithmic structure on X associated with D.

14

CHAPTER 0

0.2.6 Elliptic curves (continued). Let f : E →  be as in 0.2.1. Define the logarithmic structure M on  by taking X =  and D = {0} in 0.2.5, and define a logarithmic structure ME on E by taking X = E and D = f −1 (0). × if q ∈ ∗ , and M,0 = Then the stalk of M is given by M,q = O,q × n n≥0 O,0 · q where the last q denotes the coordinate function of . 0.2.7 For a complex analytic space X endowed with a logarithmic structure MX , let Xlog = {(x, h) | x ∈ X; h is a homomorphism MX,x → S1 satisfying (1) below}. Here S1 = {z ∈ C× | |z| = 1} is regarded as a multiplicative group. h(u) =

u(x) × for any u ∈ OX,x . |u(x)|

(1)

We have a canonical map τ : Xlog → X,

(x, h)  → x.

log

The space X has a natural topology, the weakest topology for which the map τ and the maps τ −1 (U ) → S1 , (x, h)  → h(f ), given for each open set U of X and for each f ∈
(U, MX ), are continuous. 0.2.8 Elliptic curves (continued). For  with the logarithmic structure M , the shape of log is as in Figure 2. Here, in log , {0} ⊂  is replaced by S1 since h : M,0 → S1 can send q ∈ M,0 to any element of S1 . Thus, roughly speaking, log has a shape like ∗ (log is an extension of ∗ over the origin without a change of shape).

Figure 2

15

OVERVIEW

Precisely speaking, the inclusion map ∗ → log is a homotopy equivalence. From this, we see that HZ
on ∗ extends to a local system on log of rank 2. We will see in 0.2.10 below, for the continuous map f log : E log → log induced by f : E → , all the fibers (f log )−1 (p) of any p ∈ log are homeomorphic to the surface of a doughnut even if p lies over 0 ∈ , as described on the right-hand side log of Figure 1. The above local system of rank 2 on log is in fact the sheaf R 1 f∗ (Z). 0.2.9 Example. In 0.2.5, consider the case X = n , D = {q1 · · · qr = 0}. Then Xlog = (|| × S1 )r × n−r , where || = {t ∈ R | 0 ≤ t < 1}, which has the natural topology. The map τ : Xlog → X is given by ((rj , uj )1≤j ≤r , (xj )r0 .

16

CHAPTER 0

For example, if p = (0, 1) ∈ || × S1 = log , we have a homeomorphism (f log )−1 (p)  [0, ∞]/(0 ∼ ∞) × S1 which sends the image of (c, 0, u1 , u2 ) ∈ Xlog in E log to (c, u1 ), and the image of (0, c, u1 , u2 ) to (c−1 , u1 ). Hence we have a homeomorphism (f log )−1 (p)  S1 × S1 . We show that, for each p ∈ log , there are an open neighborhood V of p and a homeomorphism (f log )−1 (V )  V × S1 × S1 over V . This shows that any fiber of f log : E log → log is homeomorphic to S1 × S1 . Let B := (the complement of (1, 1) in [0, 1] × [0, 1]) ⊂ (R≥0 ) × (R≥0 ), and let A := B × S1 × S1 ⊂ Xlog . Then the projection A → E log is surjective, and we have a homeomorphism ∼

A/ ∼ − → E log , where for a, a
∈ A, a ∼ a
if and only if either a = a
or {a, a
} = {(1, r, u1 , u2 ), (r, 1, u1 (u1 u2 ), u2 (u1 u2 )−1 )} for some r ∈ || and for some u1 , u2 ∈ S1 . Note that this equivalence relation ∼ in A comes from the equivalence relation associated with the local homeomorphism Xlog → E log . Take a continuous map s : B → [0, 1] such that we have a homeomorphism ∼

B− → || × [0, 1],

(r1 , r2 )  → (r1 r2 , s(r1 , r2 )),

and such that s(r1 , r2 ) = 0 ⇔ r2 = 1,

s(r1 , r2 ) = 1 ⇔ r1 = 1.

For example, the function s(r1 , r2 ) = (1 − r2 )/((1 − r1 ) + (1 − r2 )) has this property. Let p = (r0 , u0 ) ∈ log (r0 ∈ ||, u0 ∈ S1 ) and take an open neighborhood V
of u0 in S1 and a continuous map k : V
× [0, 1] → S1 such that k(u, 1) = k(u, 0)u for any u ∈ V
. For example, if u0 = eiθ with θ ∈ R, we can take V
= {eiλ | a < λ < b} for any fixed a, b ∈ R such that a < θ < b and b − a < 2π and k(eiλ , t) = eiλt (a < λ < b, 0 ≤ t ≤ 1). Let V

= {(u1 , u2 ) | u1 u2 ∈ V
} ⊂ S1 × S1 .

V = [0, 1) × V
⊂ log , Then we have a homeomorphism ∼

B × V

− → V × [0, 1] × S1 , (r1 , r2 , u1 , u2 )  → (r1 r2 , u1 u2 , s(r1 , r2 ), u1 k(u1 u2 , s(r1 , r2 ))). The subset B × V

of A is stable for the relation ∼, and this homeomorphism induces a homeomorphism ∼

(B × V

)/ ∼ − → V × ([0, 1]/(0 ∼ 1)) × S1 .

17

OVERVIEW ∼

→ (f log )−1 (V ), we have a homeomorphism Since (B × V

)/ ∼ − ∼

→ V × ([0, 1]/(0 ∼ 1)) × S1 (f log )−1 (V ) − over V . Note that [0, 1]/(0 ∼ 1)  S1 . See [U3] for a generalization of this local topological triviality of the family E log → log over the base. Thus Figure 1 is completely explained. 0.2.11 The above magic for f : E →  is generalized in logarithmic complex analytic geometry as follows. In logarithmic complex analytic geometry, we consider mainly logarithmic structures called fs logarithmic structures. We say a monoid S is integral if ab = ac implies b = c in S.An integral monoid S is embedded in the group S gp = {ab−1 | a, b ∈ S}. We say a monoid S is saturated if it is integral and if a ∈ S gp and a n ∈ S for some integer n ≥ 1 imply a ∈ S. We say a monoid is fs if it is finitely generated and saturated. For an fs monoid S, S gp is a finitely generated abelian group, and S gp is torsion free if S is torsion free. Let X be a local ringed space. Let S be an fs monoid which is considered as a constant sheaf on X, and let h : S → OX be a homomorphism of sheaves of monoids. The associated logarithmic structure on X is defined as the push-out S˜ of the diagram × h−1 (OX ) −−−−→ S    × OX

in the category of sheaves of monoids, which is endowed with the induced homomorphism α : S˜ → OX (for an explicit description of the push-out, see 2.1.1). A logarithmic structure on X is fs if it is locally isomorphic to the one above. The logarithmic structure in 0.2.5 associated with a divisor with normal crossings on an analytic manifold is an fs logarithmic structure. This can be checked locally by the fact that the logarithmic structure on X = n associated with the r divisor q1 · · · q

r = 0 (0 ≤ r ≤ n) is induced from the homomorphism N → OX , nj r r (nj )1≤j ≤r  → j =1 qj . Note that N (the semigroup law is the addition) is an fs monoid. An fs logarithmic structure MX on X is integral, and hence MX is embedded in gp the sheaf of commutative groups MX . For an fs logarithmic structure MX on X, × the stalk (MX /OX )x at x ∈ X is a sharp fs monoid, and, in particular, torsion-free. Here we say that a monoid S is sharp if S × = {1}, where S × denotes the set of all gp × × )x )gp = (MX /OX )x is a free Z-module invertible elements of S. Hence ((MX /OX of finite rank. For instance, in the above example X = n , if x = (xj )1≤j ≤n ∈ X and if the × number of those j satisfying 1 ≤ j ≤ r and xj = 0 is m, then (MX /OX )x  N m

18

CHAPTER 0

× × ) for such j . (Note that qj ∈ OX,x and this monoid is generated by (qj mod OX,x gp × m for the other j ’s.) We have (MX /OX )x  Z . An analytic space with an fs logarithmic structure is called an fs logarithmic analytic space. For an fs logarithmic analytic space X, the canonical map τ : Xlog → X is proper, gp × )x . and τ −1 (x)  (S1 )m for x ∈ X, where m is the rank of (MX /OX Here “proper” means “proper in the sense of Bourbaki [Bn] and separated” (see 0.7.5). We keep this terminology throughout this book.

0.2.12 Example: Toric varieties. Let S be an fs monoid, and X := Spec(C[S])an = Hom(S, Cmult ) (here Cmult denotes the set C regarded as a multiplicative monoid) be the analytic toric variety. Then S ⊂ C[S] → OX induces a canonical fs logarithmic structure. We have mult X log = Hom(S, R≥0 × S1 ) mult (here R≥0 denotes the set R≥0 regarded as a multiplicative monoid).

Using this, we have a local presentation of X log for any fs logarithmic analytic space X. Let X be an analytic space, let S be an fs monoid, let S → OX be a homomorphism, and endow X with the induced logarithmic structure. Then mult Xlog = X ×Hom(S,Cmult ) Hom(S, R≥0 × S1 )

(1)

(the fiber product as a topological space). For a morphism f : Y → X of local ringed spaces and for a logarithmic structure M on X, the inverse image f ∗ M of M, which is a logarithmic structure on Y , is defined as in 2.1.3. If M is an fs logarithmic structure associated with a homomorphism S → OX with S an fs monoid, the inverse image f ∗ M is the fs logarithmic structure associated with the homomorphism S → OY induced by f . Hence the inverse image of an fs logarithmic structure is an fs logarithmic structure. If f : Y → X is a morphism of analytic spaces, for an fs logarithmic structure M on X and for f ∗ M on Y , we have Y log = Y ×X X log as a topological space. The description of X log in (1) is explained by this since, in that case, the logarithmic structure of X is the inverse image of the canonical logarithmic structure of Spec(C[S])an . Example. Let x = 0 ∈ , and define the logarithmic structure Mx of x as the inverse image of M . This logarithmic structure is induced from the homomorphism S = N → Ox = C, n  → (the image of q n in Ox ) = 0n (note 00 = 1). Hence Mx = n≥0 (C× · q n )  C× × N with α : Mx → Ox = C, c · q n  → c · 0n (c ∈ C× , n ∈ N). Thus a one-point set can have a nontrivial logarithmic structure. We have x log = S1 for this logarithmic structure Mx . 0.2.13 The morphism f : E →  in 0.2.1 is an easiest nontrivial example of logarithmically smooth morphisms (see 2.1.11 for the definition) of fs logarithmic analytic

OVERVIEW

19

spaces. A logarithmically smooth morphism can have degeneration in the sense of classical complex analytic geometry, but, with the magic of the logarithmic structure, it can behave in logarithmic complex analytic geometry like a smooth morphism in classical complex analytic geometry. A wider example of a logarithmically smooth morphism is a morphism with semistable degeneration f : Y →  ( is endowed with

M ), that is, a morphism which, locally on Y , has the form n → , (qj )1≤j ≤n  → rj =1 qj (1 ≤ r ≤ n) with the logarithmic structure of n in 0.2.9. Indeed, it is shown in [U3] that, if f is proper (0.7.5), the associated continuous map f log : Y log → log is topologically trivial locally over the base. Kajiwara and Nakayama [KjNc] proved the following: Let f : Y → X be a proper logarithmically smooth morphism of fs logarithmic log analytic spaces. Then, for any m ≥ 0, the higher direct image functor R m f∗ log sends locally constant sheaves of abelian groups on Y to locally constant sheaves on X log . An fs logarithmic analytic space X is said to be logarithmically smooth if the structural morphism X → Spec(C) is logarithmically smooth. Here Spec(C) is endowed with the trivial logarithmic structure C× . An fs logarithmic analytic space X is logarithmically smooth if and only if, locally on X, there is an open immersion of analytic spaces i : X → Z = Spec(C[S])an with S an fs monoid such that the logarithmic structure of X is the inverse image of the canonical logarithmic structure of Z (i.e., the logarithmic structure of X induced by the homomorphism S → OX defined by i). 0.2.14 Logarithmic differential forms. The name “logarithmic structure” comes from its relation to differential forms with logarithmic poles. For an fs logarithmic analytic space X, the sheaves of logarithmic differential q q-forms ωX on X (q ∈ N) are defined as in 2.1.7. If X is an analytic manifold and D is a divisor on X with normal crossings, and if X is endowed with the logarithmic q q structure associated with D, ωX coincides with the sheaf X (log(D)), the sheaf of q differential q-forms on X which may have logarithmic poles along D. In general, ωX q q is an OX -module, there is a canonical homomorphism of OX -modules X → ωX , q 1 1 ωX is the qth exterior power of ωX , and ωX is generated over OX by 1X and the image gp 1 of a homomorphism d log : MX → ωX . For a morphism Y → X of fs logarithmic q q analytic spaces, the logarithmic version ωY /X of Y /X is also defined (2.1.7). Example. Let X = n with the logarithmic structure as in 0.2.9. Then for x = 1 at x is a free OX,x -module with basis (ωj )1≤j ≤n (xj )1≤j ≤n ∈ X, the stalk of ωX where ωj = d log(qj ) if 1 ≤ j ≤ r and xj = 0, and ωj = dqj otherwise. For a logarithmically smooth morphism Y → X of fs logarithmic analytic spaces, 1 is locally free although 1Y /X may not be locally free if degeneration the sheaf ωY/X occurs in Y → X. Consider the case n = r = 2 of the last example, and consider the logarithmically smooth morphism X → , (x1 , x2 )  → x1 x2 . The sheaf 1X/ is generated by dq1 and dq2 which satisfy the relation q1 dq2 + q2 dq1 = d(q1 q2 ) = 0.

20

CHAPTER 0

1 This indicates that 1X/ is not locally free. However, ωX/ is generated by d log(q1 ) and d log(q2 ), which satisfy d log(q1 ) + d log(q2 ) = d log(q1 q2 ) = 0. This shows 1 that ωX/ is a free OX -module of rank 1 generated by d log(q1 ).

Example. Let S be an fs monoid and let X = Spec(C[S])an endowed with the canonical logarithmic structure. Then we have an isomorphism ∼

1 OX ⊗Z S gp − → ωX , f ⊗ g  → fd log(g).

Example. Let x = 0 ∈  and endow x with the inverse image of M . Then ωx1 is the one-dimensional C-vector space generated by the image of d log(q) ∈ 1
(, ω ). Thus a one point set can have a nontrivial logarithmic differential form. Now we talk about Hodge filtration and logarithmic Hodge theory. The key point log is that, for an fs logarithmic analytic space X, we define a sheaf of rings OX on X log . log Roughly speaking, OX is the ring generated over OX by log(M gp ). First, by considering the example f : E →  of 0.2.1, we will see why such a ring is necessary. 0.2.15 Elliptic curves (continued). Let f
: E ∗ → ∗ be as in 0.2.2. Let HZ
= R 1 f∗
(Z) and let M
= R 1 f∗
( •E ∗ /∗ ). We have seen that HZ
extends to the local system HZ = R 1 f∗ (Z) on log (0.2.8). On the other hand, the O∗ -module M
with a decreasing filtration F
(0.2.2) log

d

• 1 extends to a locally free O -module M := R 1 f∗ (ωE/ ) = R 1 f∗ (OE − → ωE/ ) of 1 1 0 rank 2 with the decreasing filtration defined by M = M, M = f∗ (ωE/ ) → M, M2 = 0. We have

M = O e1 ⊕ O ω ⊃ M1 = O ω,

(1)

where we denote by the same letter e1 the image of the global section e1 of • R 1 f∗ Z under R 1 f∗ Z → R 1 f∗ (ωE/ ), and we denote by ω the differential form −1 (2πi) dt1 /t1 on E with logarithmic poles along f −1 (0) (where t1 is the coordinate function of X in 0.2.10). Although the relation between HZ
and M
is simply M
= O∗ ⊗Z HZ
, the relation between HZ and M is not so direct. They are related only after being tensored log by a sheaf of rings O defined below. Let q be the coordinate function of . Then on ∗ , via the identification M
= O∗ ⊗Z HZ
, we have ω = (2π i)−1 log(q)e1 + e2

(2)

where e2 is taken by fixing a branch of the multivalued function (2π i)−1 log(q), and the same branch of this function is used in the formula (2). This formula (2) follows from the fact that the restriction of ω to each fiber C× /q Z  C/(Zτ + Z) (q ∈ ∗ , τ = (2πi)−1 log(q)) is dz = τ e1 + e2 on C/(Zτ + Z). Locally on log , e2 extends to a local section of HZ , and we have HZ = Ze1 ⊕ Ze2 .

(3)

21

OVERVIEW

When we compare (1), (2), and (3), since log(q) in (2) does not extend over the origin in the classical sense, we do not find a simple relation between HZ and M. However, log log(q) exists on log locally as a local section of j∗ (O∗ ) where j log is the inclusion ∗ log iθ log map  →  . In fact, if y = (0, e ) ∈  = || × S1 (θ ∈ R), where || is as in 0.2.9, if we take a, b ∈ R such that a < θ < b and b − a < 2π , and if we denote by U the open neighborhood {(r, eix ) | r ∈ ||, a < x < b} of y in log , the holomorphic map reiθ  → log(r) + iθ defined on ∗ ∩ U = {(r, eiθ ) | 0 < r < 1, log a < θ < b} is an element of
(∗ ∩ U, O∗ ) =
(U, j∗ (O∗ )), which is a branch log of log(q). All branches of log(q) in j∗ (O∗ ) are congruent modulo 2π iZ. Let log log log O ⊂ j∗ (O∗ ) be the sheaf of subrings of j∗ (O∗ ) on log generated over −1 −1 τ (O ) by log(q). Here τ ( ) is the inverse image of a sheaf. Then from (1), (2), and (3), we have O ⊗τ −1 (O ) τ −1 (M) = O ⊗Z HZ . log

log

0.2.16 log

The sheaf of rings OX . For an fs logarithmic analytic space X, we have a sheaf of log log rings OX , which generalizes the above O . First we consider the case X = Spec(C[S])an for an fs monoid S. Let U be the open subspace Spec(C[S gp ])an of X, and let j log : U → Xlog be the canonical map. log log Then we define OX as a sheaf of subrings of j∗ (OU ) generated over τ −1 (OX ) by gp log the logarithms of local sections of MX (these logarithms exist in j∗ (OU ) and are determined mod 2πiZ just as in the case of ). log The definition of OX for a general fs logarithmic analytic space X is given in 2.2.4. If the logarithmic structure of X is induced from a homomorphism S → OX with S an fs monoid, we have log

log

OX = OX ⊗OZ OZ

with

Z = Spec(C[S])an

log (note that OZ

is explained just above), where we denote the inverse images on X log log log of the sheaves OX , OZ , and OZ , by OX , OZ , and OZ , respectively, for simplicity. gp log log We have a homomorphism log : MX → OX /2π iZ, and OX is generated over gp τ −1 (OX ) by log(MX ). For an fs logarithmic analytic space X and x ∈ X, if the free gp gp × × Z-module (MX /OX )x is of rank r with basis (fj mod OX,x )1≤j ≤r (fj ∈ MX,x ), log log then for any point y of Xlog lying over x, the stalk OX,y of OX is isomorphic to the polynomial ring in r variables over OX,x by ∼

log

OX,x [T1 , . . . , Tr ] − → OX,y ,

Tj  → log(fj )

(log(fj ) is defined only modulo 2π iZ but we choose a branch (a representative) for each j ). Let q,log

ωX

= OX ⊗τ −1 (OX ) τ −1 (ωX ). log

•,log

Then we have the de Rham complex ωX q+1,log defined as in 2.2.6. ωX

q

q,log

on X log with the differential d : ωX

→

22

CHAPTER 0

Example. Let X = n endowed with the logarithmic structure in 0.2.9. Let x ∈ n and let y be a point of X log lying over x. Let m be the number of j such that 1 ≤ j ≤ r log and xj = 0. Then the stalk OX,y is a polynomial ring over OX,x in m variables log(qj ) for such j . Example. Let x = Spec(C) endowed with an fs logarithmic structure (we will call such x an fs logarithmic point). Then Mx = C× × S for some fs monoid S with no invertible element other than 1, where α : Mx → C sends (c, t) ∈ Mx (c ∈ C× , t ∈ S) to 0 if t  = 1, and to c if t = 1. Let r be the rank of S gp which is a free Z-module of finite rank. We have x log  (S1 )r ,

ωx1  Cr ,

log Ox,y  C[T1 , . . . , Tr ]

(y ∈ x log ).

0.2.17 Let X be an fs logarithmic analytic space, let x ∈ X, and let y be a point of Xlog log lying over x. The stalk OX,y is not necessarily a local ring, and has a global ringlog

theoretic nature. Let sp(y) be the set of all ring homomorphisms s : OX,y → C such that s(f ) = f (x) for any f ∈ OX,x . If we fix s0 ∈ sp(y), we have a bijection ∼

× sp(y) − → Hom((MX /OX )x , Cadd ), gp

s  → (f  → s(log(f )) − s0 (log(f )))

(1) for

f ∈

gp MX,x .

Here Cadd is C regarded as an additive group. Let (HZ , F ) be a pair of a local system HZ of free Z-modules of finite rank on log log Xlog and of a decreasing filtration F on the OX -module OX ⊗Z HZ such that F p log log and (OX ⊗Z HZ )/F p are locally free as OX -modules for all p. Then, for each s ∈ sp(y), we have a decreasing filtration F (s) on HC,y = C ⊗Z HZ,y (called the log p specialization of F at s) defined by F p (s) = C ⊗Olog Fy . Here OX,y → C is s. X,y

We will see later that a nilpotent orbit can be regarded as the family (F (s))s∈sp(y) associated with such (HZ , F ). The reason why an orbit of Hodge filtrations, called a nilpotent orbit (not a single Hodge filtration), appears in the degeneration of Hodge log structures is, from the point of view of logarithmic Hodge theory, that the stalk OX,y is still a global ring and we have many specializations at a point y. 0.2.18 Elliptic curves (continued). Let f : E →  be as in 0.2.1, and consider HZ = log 1 ), and the filtration (Mp )p as in 0.2.15. Define R 1 f∗ (Z), M = R 1 f∗ (OE → ωE/ log

a filtration F on O ⊗Z HZ by F p = O ⊗τ −1 (O ) τ −1 (Mp ) ⊂ O ⊗τ −1 (O ) τ −1 (M) = O ⊗Z HZ . log

log

log

Take y ∈ log lying over 0 ∈ . We consider the specializations F (s) for s ∈ sp(y). Take a branch of e2 ∈ HZ at y and take the corresponding branch of log(q) log at y. Then Fy1 is a free O,y -module of rank 1 generated by (2π i)−1 log(q)e1 + e2

23

OVERVIEW log

(0.2.15 (2)). Since O,y is a polynomial ring in one variable log(q) over O,0 , an element s ∈ sp(y) is determined by s(log(q)) ∈ C. The filtration F (s) is described as: F 0 (s) = HC,y , F 2 (s) = 0, and F 1 (s) is the one-dimensional C-subspace of HC,y generated by s((2πi)−1 log(q))e1 + e2 . If the imaginary part of s((2π i)−1 log(q)) is > 0 (that is, if | exp(s(log(q)))| < 1), then (HZ,y , F (s)) is a Hodge structure of weight 1, and for the antisymmetric pairing  ,  : HQ × HQ → Q defined as e2 , e1  = 1, (HZ,y ,  , y , F (s)) becomes a polarized Hodge structure. 0.2.19 The observation in 0.2.18 leads us to the notion of “logarithmic variation of polarized Hodge structure.” Let X be a logarithmically smooth fs logarithmic analytic space. A logarithmic variation of polarized Hodge structure (LVPH) on X of weight w is a triple (HZ ,  , , F ) consisting of a locally constant sheaf HZ of free Z-modules of finite rank on Xlog , a bilinear form  ,  : HQ × HQ → Q, and a decreasing filtration F of log log OX ⊗ HZ by OX -submodules which satisfy the following three conditions (1)–(3). (1) There exist a locally free OX -module M and a decreasing filtration (Mp )p∈Z log log by OX -submodules of M such that OX ⊗Z HZ = OX ⊗τ −1 (OX ) τ −1 (M) and log F p = OX ⊗τ −1 (OX ) τ −1 (Mp ) for all p, and such that Mp = M for p  0, p M = 0 for p  0, and Mp /Mp+1 are locally free for all p. (2) Let x ∈ X, and let (fj )1≤j ≤n be elements of MX,x that are not contained × × × such that (fj mod OX,x )1≤j ≤n generates the monoid (MX /OX )x . Let y ∈ in OX,x −1 log τ (x) ⊂ X . Then if s ∈ sp(y) and if exp(s(log(fj ))) are sufficiently near to 0 for all j , (HZ,y ,  , y , F (s)) is a polarized Hodge structure of weight w. 1,log (3) (d ⊗Z 1HZ )(F p ) ⊂ ωX ⊗Olog F p−1 for all p. X

0.2.20 Elliptic curve (continued). The pair (HZ , F ) in 0.2.18 arising from E →  is an LVPH on . (The Griffiths transversality (3) in 0.2.19 is satisfied automatically.) At a point of ∗ , the fiber of (HZ , F ) is a polarized Hodge structure. However, at 0 ∈ , the fiber of (HZ , F ) should be understood as a family (F (s))s∈sp(y) for some fixed y ∈ τ −1 (0) and for varying s. As is explained in Section 0.4, this family is the so-called nilpotent orbit. This is expressed in the schema (2) in Introduction. 0.2.21 LVPH arising from geometry. By the weakly semistable reduction theorem of Abramovich and Karu [AK], any projective fiber space is modified, by alteration and birational modification, to a projective, toroidal morphism f : Y → X without horizontal divisor, which is equivalent to a projective,vertical, logarithmically smooth gp gp × morphism (2.1.11; see also 0.2.13) with Coker (MX /OX )f (y) → (MY /OY× )y being torsion-free at any y ∈ Y .

24

CHAPTER 0

Then by Kato-Matsubara-Nakayama [KMN], for any m ∈ Z, a variation of polarized logarithmic Hodge structure (HZ ,  , , F ) of weight m on X is obtained in the following way. HZ = R m (f log )∗ Z/(torsion),  ,  : HQ × HQ → Q induced from an ample line bundle, M = R m f∗ (ωY• /X ), ≥p

Mp = R m f∗ (ωY /X ) → M F p = OX ⊗τ −1 (OX ) τ −1 (Mp ) → OX ⊗τ −1 (OX ) τ −1 (M) = OX ⊗Z HZ . log

log

log

There are many other contributors: [F], [Kf2], [Ma1], [Ma2], [U3], etc. This is a generalization of work of Steenbrink [St]. 0.2.22 The nilpotent orbit theorem of Schmid [Sc] is interpreted as follows (see Theorem 2.5.14). Let X be a logarithmically smooth, fs logarithmic analytic space, and let × U = {x ∈ X | MX,x = OX,x } be the open set of X consisting of all points at which the logarithmic structure is trivial. Let H be a VPH on U with unipotent local monodromy along X − U . Then H extends to a LVPH on X. 0.2.23 The theory of logarithmic structure was started in p-adic Hodge theory to construct the logarithmic crystalline cohomology theory for varieties with semistable reduction ([I1], [HK], etc.). Usually, the theory over p-adic fields begins by following its analogue over C. But in the theory of logarithmic structure, applications appeared first in p-adic Hodge theory. We hope that the theory of logarithmic structure will also be useful in Hodge theory.

0.3 GRIFFITHS DOMAINS AND MODULI OF PH In [G1], Griffiths defined and studied classifying spaces D of polarized Hodge structures. We review the definition of D. We regard D as moduli of polarized Hodge structures by discarding the Griffiths transversality from VPH (0.3.5–0.3.7). Fix (w, (hp,q )p,q∈Z , H0 ,  , 0 ) where w is an integer, (hp,q )p,q∈Z is a family of non-negative integers such that hp,q = 0 unless p + q = w, hp,q  = 0 for only finitely many

(p, q), and such that hp,q = hq,p for all p, q, H0 is a free Z-module of rank p,q hp,q , and  , 0 is a nondegenerate bilinear form H0,Q × H0,Q → Q, which is symmetric if w is even and antisymmetric if w is odd.

25

OVERVIEW

Definition 0.3.1 The classifying space D of polarized Hodge structures of type  0 = w, (hp,q )p,q , H0 ,  , 0 is the set of all decreasing filtrations F on H0,C = C ⊗Z H0 such that the triple (H0 ,  , 0 , F ) is a polarized Hodge structure of weight w and of Hodge type (hp,q )p,q . The “compact dual” Dˇ of D is defined to be the set of all decreasing filtrations F on H0,C such that dimC (F p /F p+1 ) = hp,w−p for all p that satisfy the Riemann-Hodge first bilinear relation 0.1.8 (1). The spaces D and Dˇ have natural structures of analytic manifolds, and D is an ˇ open submanifold of D. The space D is called the Griffiths domain and also the period domain. 0.3.2 Examples. (i) Upper half plane. Consider the case w = 1, h1,0 = h0,1 = 1 and hp,q = 0 for other (p, q). Let H0 be a free Z-module of rank 2 with basis e1 , e2 , and define an antisymmetric Z-bilinear form  , 0 : H0 × H0 → Z by e2 , e1 0 = 1. Then D  h, the upper half plane, where we identify a point τ ∈ h with F (τ ) ∈ D defined by F 0 (τ ) = H0,C ,

F 1 (τ ) = C(τ e1 + e2 ),

F 2 (τ ) = {0}.

In this case, Dˇ is identified with P1 (C). (ii) Upper half space (a generalization of Example (i)). Let g ≥ 1 and consider the case w = 1, h1,0 = h0,1 = g and hp,q = 0 for other (p, q). Let H0 be a free Z-module with basis (ej )1≤j ≤2g and define a Z-bilinear form  , 0 : H0 × H0 → Z by   0 −1g . (ej , ek 0 )j,k = 1g 0 Then D  hg , the Siegel upper half space of degree g. Recall that hg is the space of all symmetric matrices over C of degree g whose imaginary parts are positive definite. We identify a matrix τ ∈ hg with F (τ ) ∈ D as follows:    subspace of H0,C spanned τ F 0 (τ ) = H0,C , F 1 (τ ) = , F 2 (τ ) = {0}. by the column vectors of 1g (The symmetry of τ corresponds to the Riemann-Hodge first bilinear relation for F (τ ) and the positivity of Im(τ ) corresponds to the second bilinear relation (cf. 0.1.8).) (iii) Example with w = 2, h2,0 = h0,2 = 2, h1,1 = 1 (a special case of the example investigated in Section 12.2 where the weight w is shifted to 0). Let H0 be a free Z-module of rank 5 with basis (ej )1≤j ≤5 , and let  , 0 : H0,Q × H0,Q → Q be the bilinear form defined by    
cj e j , cj ej = −c1 c1
− c2 c2
− c3 c3
+ c4 c5
+ c5 c4
(cj , cj
∈ Q). 1≤j ≤5

1≤j ≤5

0

26

CHAPTER 0

Let Q := {(z1 : z2 : z3 ) ∈ P2 (C) | z12 + z22 + z32 = 0},     z = (z1 : z2 : z3 ) ∈ Q, a ∈ ( 3 Cej )/C(z1 e1 + z2 e2 + z3 e3 ), j =1  X := (z, a)  . a∈ / Image(Re1 + Re2 + Re3 ) For z ∈ Q, define a decreasing filtration F (z) of H0,C by F p (z) = H0,C for p ≤ 0, F p (z) = 0 for p ≥ 3; F 2 (z) is the two-dimensional C-subspace generated by z1 e1 + z2 e2 + z3 e3 and e5 , and F 1 (z) is the annihilator of F 2 with respect to  , 0 . For a ∈ 3j =1 Cej , let Na : H0,C → H0,C be the nilpotent C-linear map defined by Na (e5 ) = a,

Na (b) = −a, b0 e4 for b ∈

3 

Cej ,

Na (e4 ) = 0.

j =1

Then a  → Na is C-linear, and Na Nb = Nb Na for any a, b. For a, b ∈ 3j =1 Cej and z ∈ Q, exp(Na )F (z) = exp(Nb )F (z) if and only if a ≡ b mod C(z1 e1 + z2 e2 + z3 e3 ). We have ∼

X− → D,

(z, a)  → exp(Na )F (z).

The complex dimension of D is 3. In this example, D is not a symmetric Hermitian domain.

0.3.3 Let GZ = Aut(H0 ,  , 0 ), and for R = Q, R, C let GR = Aut R (H0,R ,  , 0 ) and gR = Lie GR = {A ∈ End R (H0,R ) | A(x), y0 + x, A(y)0 = 0 (∀ x, y ∈ H0,R )}. Then D is a homogeneous space under the natural action of GR . For r ∈ D, let Kr be the maximal compact subgroup of GR consisting of all elements that preserve the Hodge metric ( , )r associated with r (0.1.8). The isotropy subgroup Kr
of GR at r ∈ D is contained in Kr , but they need not coincide for general D (cf. [G1], and also [Sc]). The following conditions (1) and (2) are equivalent for any r ∈ D: (1) D is a symmetric Hermitian domain. (2) dim(Kr ) = dim(Kr
).

0.3.4 Examples. Upper half space (continued). In this case, we have, for R = Q, R, C, GR = Sp(g, R) = {h ∈ GL(2g, R) | t hJg h = Jg },  0 −1  B ∈ Sp(g, R) acts on D by where Jg = 1g 0 g . The matrix CA D F (τ )  → F (τ
),

τ
= (Aτ + B)(Cτ + D)−1 .

27

OVERVIEW


Let r = F (i1g ) ∈ D. Then  AKBr = Kr , and this group is isomorphic to the unitary group U (g) by A + iB  → −B A ∈ Kr ⊂ Sp(g, R).

Example with h2,0 = h0,2 = 2, h1,1 = 1 (continued). As in 0.3.2 (iii), let Q = {(z1 : z2 : z3 ) ∈ P2 (C) | z12 + z22 + z32 = 0}. We have a homeomorphism    3  ∼ θ :Q− → S2 = xj ej ∈ H0,R  xj ∈ R, x12 + x22 + x32 = 1 j =1

characterized as follows. Let z = (a1 + ib1 : a2 + ib2 : a3 + ib3 ) ∈ Q (aj , bj ∈ R). Then (aj )j and (bj )j are orthogonal in R 3 and have the same length. The charac terization of θ is that for θ (z) = 3j =1 cj ej ∈ S2 , (cj )j is orthogonal to (aj )j and (bj )j , and det((aj )j , (bj )j , (cj )j ) > 0. For v ∈ S2 , write r(v) = exp(iNv )F (θ −1 (v)) ∈ D. If we take a basis (fj )1≤j ≤5 of H0,Q given by fj := ej (j = 1, 2, 3), f4 := e5 − f5 := e5 + 12 e4 , then (fj , fk 0 )j,k = −10 4 10 , and hence GR  O(1, 4, R).  p,q Furthermore, for v ∈ S2 , for the Hodge decomposition H0,C = p,q Hr(v) cor2,0 0,2 responding to r(v), (fj )1≤j ≤4 is a C-basis of Hr(v) ⊕ Hr(v) , f5 is a C-basis of 1,1 Hr(v) , and ((fj , fk )r(v) )j,k = 15 . Hence the maximal compact subgroup Kr(v) is O(4, R) × O(1, R) for the basis (fj )1≤j ≤5 and is independent of v. For this basis,
the isotropy subgroup Kr(e is the image of 2)   A B 0 U (2) × O(1, R) → O(4, R) × O(1, R), (A + iB) × (±1)  →−B A 0 . 0 0 ±1 1 e , 2 4


) = 4. (This shows, following 0.3.3, that D We have dimR (Kr(v) ) = 6 > dimR (Kr(v) is not a symmetric Hermitian domain.)

0.3.5 Now we consider the moduli of polarized Hodge structures. We consider what functors D and
\D, for torsion-free subgroups
of GZ , represent as analytic spaces. That is, we ask what are Mor A ( , D) and Mor A ( ,
\D) for the category A of analytic spaces. We consider Mor A ( , D) first. For an analytic space X, a morphism X → D is identified with a decreasing filtration F = (F p )p∈Z on the OX -module OX ⊗Z H0 having the following properties (1) and (2). (1) F p = OX ⊗Z H0 for p  0, F p = 0 for p  0, and the OX -modules F p /F p+1 are locally free for all p ∈ Z. (2) For any x ∈ X, the fiber (H0 ,  , 0 , F (x)) is a PH of weight w and of Hodge type (hp,q )p,q .

28

CHAPTER 0

Here the object (OX ⊗Z H0 , F ) need not satisfy the Griffiths transversality (3) in Section 0.1.10. Corresponding to the identity morphism D → D, there is a universal Hodge filtration F on OD ⊗Z H0 , but this F need not satisfy the Griffiths transverality. Hence, for an analytic space X, we will consider an object (HZ ,  , , F ) which satisfies all conditions of variation of polarized Hodge structure in 0.1.10 except the Griffiths transversality (3). The correct name of such an object might be “the analytic family of polarized Hodge structures parametrized by X without the assumption of Griffiths transversality,” but in this book, for simplicity, we will call this object just a polarized Hodge structure on X, or, simply, a PH on X. For an analytic space X, by a PH on X of type 0 = w, (hp,q )p,q , H0 ,  , 0 , we mean a PH on X of weight w and of Hodge type (hp,q )p,q endowed with an isomorphism (HZ ,  , )  (H0 ,  , 0 ) of local systems on X. Here (H0 ,  , 0 ) is considered as a constant sheaf on X. Let PH0 (X) be the set of all isomorphism classes of PH on X of type 0 . The above interpretation of Mor A ( , D) is rewritten as follows. Lemma 0.3.6 We have an isomorphism PH0  Mor A ( , D) of functors from A to (Sets). If H = (HZ ,  , , F ) is a PH on X of type 0 and ϕ : X → D is the corresponding morphism, ϕ(x) ∈ D for x ∈ X is nothing but the fiber F (x) of F at x regarded as a filtration of H0,C via the endowed isomorphism (HZ,y ,  , )  (H0 ,  , 0 ). Let
be a torsion-free subgroup of GZ . Then
\D is an analytic manifold. Let X be an analytic space and let H = (HZ ,  , , F ) be a PH on X. By a
-level structure of H , we mean a global section of the sheaf
\ Isom((HZ ,  , ), (H0 , , 0 )), on X, where (H0 ,  , 0 ) is considered as a constant sheaf on X. A level structure appears as follows. Let X be a connected analytic space and let H = (HZ ,  , , F ) be a PH on X of weight w and of Hodge type (hp,q ), let x ∈ X, and define (H0 ,  , 0 ) to be the stalk (HZ,x ,  , x ). Assume
contains the image of π1 (X, x) → GZ . Then H has a unique
-level structure µ such that the germ µx is the germ of the identity map of H0 modulo
. Let 1 = w, (hp,q )p,q , H0 ,  , 0 ,
. For an analytic space X, by a PH on X of type 1 , we mean a PH on X of weight w and of Hodge type (hp,q )p,q endowed with a
-level structure. Let PH1 (X) be the set of all isomorphism classes of PH on X of type 1 . Lemma 0.3.7 We have an isomorphism PH1  Mor A ( ,
\D) of functors from A to (Sets). This is deduced from Lemma 0.3.6 as follows. The isomorphism PH0  Mor A ( , D) in 0.3.6 preserves the actions of
. The functor Mor A ( ,
\D) : A → (Sets) is identified with the quotient
\ Mor A ( , D) in the category of sheaf functors, that is, for each object X of A, Mor A (X,
\D) coincides with the set of global sections of the sheaf on X associated with the presheaf U  →
\ Mor A (U, D) on X (here U is an open set of X). Similarly, PH1 is identified with the quotient
\ PH0 in the category of sheaf functors. Hence PH1  Mor A ( ,
\D). 2

29

OVERVIEW

There is a torsion-free subgroup of GZ of finite index. More strongly, there is a neat subgroup of GZ of finite index (see 0.4.1 below). 0.3.8 We emphasize again here that the moduli conditions for PH0 and PH1 do not contain the Griffiths transversality on X (0.1.10 (3)). Griffiths transversality should be understood as an important property of the period map, which is satisfied by period maps coming from geometry. Let X be an analytic manifold, let H be a PH on X of type 0 , and let ϕ : X → D be the morphism corrsponding to H (the period map of H ). Then the following two conditions (1) and (2) are equivalent. (1) H satisfies the Griffths transversality. (2) The image of the morphism of tangent bundles dϕ : TX → TD induced by ϕ is contained in the horizontal tangent bundle TDh ⊂ TD . Here the horizontal tangent bundle is defined by TDh = F −1 (End  ,  (HO ))/F 0 (End  ,  (HO )) ⊂ TD = End  ,  (HO )/F 0 (End  ,  (HO )) where End  ,  (HO ) = {A ∈ EndO (HO ) | Ax, y + x, Ay = 0 (∀ x, y ∈ HO )} and F (End  ,  (HO )) is the filtration on End  ,  (HO ) induced by the universal Hodge filtration on HO = OD ⊗Z HZ ([G2]; also [Sc]). In particular, by applying this equivalence to the identity morphism of D, we see that the universal PH on D satisfies the Griffiths transversality if and only if the tangent bundle of D coincides with the horizontal tangent bundle. These equivalent conditions are satisfied by Example (ii) (and hence by Example (i)) but not by Example (iii) in 0.3.2. In Example (iii) in 0.3.2, TDh is a vector bundle of rank 2 whereas TD is of rank 3. 0.3.9 Let f : Y → X be a projective, smooth morphism of analytic manifolds. Fix a polarization of Y over X. Let (HZ ,  , , F ) be the associated VPH (0.1.11). Assume that X is connected, fix a base point x ∈ X, and let (H0 ,  , 0 ) := (HZ,x ,  , x ), and assume that
:= Image(π1 (X, x) → Aut(H0 ,  , 0 ) is torsion-free. Let ϕ : X →
\D be the associated period map (0.3.7). For the differential dϕ of the period map ϕ, Griffiths obtained the following commutative diagram: TX   K-S R 1 f∗ TY/X

dϕ

h ϕ ∗ T
\D = gr −1 F End  ,  (HO )   ∩ via coupling  p p−1 −−−−−−→ HomOX (R m−p f∗ Y /X , R m−p+1 f∗ Y /X )

−−−−→

p h is the horizontal tangent bundle in the tangent bundle T
\D , K-S on the where T
\D left vertical arrow means the Kodaira-Spencer map, and the right vertical arrow is

30

CHAPTER 0

the canonical map (for details, see [G2]). The bottom horizontal arrow is often more computable than the top horizontal arrow. This gives a geometric presentation of the differential of the period map.

0.4 TOROIDAL PARTIAL COMPACTIFICATIONS OF
\D AND MODULI OF PLH We discuss how to add points at infinity to D to construct a kind of toroidal partial compactification
\D of
\D. Since “nilpotent orbits” appear in degenerations of Hodge structures, it is natural to add “nilpotent orbits” as points at infinity. We do this in 0.4.1–0.4.12. We describe the space
\D in 0.4.13–0.4.19. We then explain in 0.4.20–0.4.34 that this enlarged classifying space is a moduli space of “polarized logarithmic Hodge structures.” We fix (w, (hp,q )p,q∈Z , H0 ,  , 0 ) as in Section 0.3. 0.4.1 We say a subgroup
of GZ is neat if, for each γ ∈
, the subgroup of C× generated by all the eigenvalues of γ is torsion-free. It is known that there exists a neat subgroup of GZ of finite index (cf. [B]). A neat subgroup of GZ is, in particular, torsion-free. Let
be a neat subgroup of GZ . We will construct some toroidal partial compactifications of
\D by adding “nilpotent orbits” as points at infinity. Definition 0.4.2 A subset σ of gR = Lie GR (0.3.3) is called a nilpotent cone if the following conditions (1)–(3) are satisfied. (1) σ = (R≥0 )N1 + · · · + (R≥0 )Nn for some n ≥ 1 and for some N1 , . . . , Nn ∈ σ . (2) Any element of σ is nilpotent as an endomorphism of H0,R . (3) N N
= N
N for any N, N
∈ σ as endomorphisms of H0,R . A nilpotent cone is said to be rational if we can take N1 , . . . , Nn ∈ gQ in (1) above. 0.4.3 For a nilpotent cone σ , a face of σ is a nonempty subset τ of σ satisfying the following two conditions. (1) If x, y ∈ τ and a ∈ R≥0 , then x + y, ax ∈ τ . (2) If x, y ∈ σ and x + y ∈ τ , then x, y ∈ τ . One can show that a face of a nilpotent cone (resp. rational nilpotent cone) is a nilpotent cone (resp. rational nilpotent cone), and that a nilpotent cone has only finitely many faces. commutative nilpotent eleFor example, let Nj ∈ gR (1 ≤ j ≤ n) be mutually n ments that are linearly independent over R. Then σ := j =1 (R≥0 )Nj is a nilpotent

cone, and the faces of σ are j ∈J (R≥0 )Nj for finite subsets J of {1, . . . , n}.

31

OVERVIEW

Definition 0.4.4 A fan in gQ is a nonempty set  of rational nilpotent cones in gR satisfying the following three conditions: (1) If σ ∈ , any face of σ belongs to . (2) If σ, σ
∈ , σ ∩ σ
is a face of σ and of σ
. (3) Any σ ∈  is sharp. That is, σ ∩ (−σ ) = {0}. 0.4.5 Examples. (i) Let  := {(R≥0 )N | N is a nilpotent element of gQ }. Then  is a fan in gQ . (ii) Let σ ∈ gR be a sharp rational nilpotent cone. Then the set of all faces of σ is a fan in gQ . 0.4.6 Nilpotent orbits. Let σ be a nilpotent cone in gR = Lie GR . For R = R, C, we denote by σR the R-linear span of σ in gR .

Definition 0.4.7 Let σ = 1≤j ≤r (R≥0 )Nj be a nilpotent cone. A subset Z of Dˇ is called a σ -nilpotent orbit if there is F ∈ Dˇ which satisfies Z = exp(σC )F and satisfies the following two conditions. (1) N F p ⊂ F p−1 (∀p, ∀N ∈ σ ). (2) exp( 1≤j ≤r zj Nj )F ∈ D if zj ∈ C and Im(zj )  0. In this case, the pair (σ, Z) is called a nilpotent orbit. Definition 0.4.8 Let  be a fan in gQ . We define the space D of nilpotent orbits in the directions in  by D := {(σ, Z) | σ ∈ , Z ⊂ Dˇ is a σ -nilpotent orbit}. Note that we have the inclusion map D → D ,

F  → ({0}, {F }).

For a sharp rational nilpotent cone σ in gR , we denote D{face of σ } by Dσ . Then, for a fan  in gQ , we have D = ∪σ ∈ Dσ . 0.4.9 Upper half plane (continued). Let  be as in 0.4.5. Then D = D ∪ P1 (Q). This is explained as follows. For a ∈ P1 (Q), let Va be the one-dimensional R-vector subspace of H0,R corresponding to a, that is, Va = R(ae1 + e2 ) if a ∈ Q, and V∞ = Re1 .

32

CHAPTER 0

For a ∈ P1 (Q), define a sharp rational nilpotent cone σa ∈  by σa = {N ∈ gR | N (H0,R ) ⊂ Va , N(Va ) = {0}, x, N (x)0 ≥ 0 for any x ∈ H0,R }. We identify a ∈ P1 (Q) with the nilpotent orbit (σa , Za ) ∈ D where Za = {F ∈ Dˇ | F 1  = Va,C }. For example,   0 R≥0 ˇ , Z∞ = C ⊂ P1 (C) = D. σ∞ = 0 0 Definition 0.4.10 Let  be a fan in gQ and let
be a subgroup of GZ . (i) We say
is compatible with  if the following condition (1) is satisfied. (1) If γ ∈
and σ ∈ , then Ad(γ )(σ ) ∈ . Here, Ad(γ )(σ ) = γ σ γ −1 . Note that, if
is compatible with ,
acts on D by γ : (σ, Z)  → (Ad(γ )(σ ), γ Z) (γ ∈
). (ii) We say
is strongly compatible with  if it is compatible with  and the following condition (2) is also satisfied. For σ ∈ , define
(σ ) :=
∩ exp(σ ). (2) The cone σ is generated by log
(σ ), that is, any element of σ can be written as a sum of a log(γ ) (a ∈ R≥0 , γ ∈
(σ )). Note that
(σ ) is a sharp fs monoid and
(σ )gp =
∩ exp(σR ). 0.4.11 Example. If  =  in 0.4.5 and
is of finite index in GZ , then
is strongly compatible with . If  =  and
is just a subgroup of GZ , it is compatible with  but is not necessarily strongly compatible with . 0.4.12 Assume that
and  are strongly compatible. In Chapter 3, we will define a topology of
\D for which
\D is a dense open subset of
\D and which has the following

property. Let (σ, Z) ∈ D , F ∈ D, and Nj ∈ gR (1 ≤ j ≤ n), and assume σ = nj=1 (R≥0 )Nj . Then     n  exp  zj Nj  F mod
 j =1

→ ((σ, Z) mod
)

if zj ∈ C and Im(zj ) → ∞ (∀ j ).

Furthermore, in Chapter 3, we will introduce on
\D a structure of a local ringed space over C and also a logarithmic structure. In 0.4.13 (resp. 0.4.18), we describe what this local ringed structure looks like in the cases of Examples (i) and (ii)

33

OVERVIEW

(resp. Example (iii)) in 0.3.2. If
is neat, the logarithmic structure M of
\D is an fs logarithmic structure and has the form M = {f ∈ O | f is invertible on α
\D} → O for the structure sheaf of rings O of
\D . 0.4.13 Examples. Upper half plane (continued). For    0 1 Z
= ⊂ SL(2, Z), σ = σ∞ = 0 0 1

 R≥0 , 0

we have a commutative diagram of analytic spaces ∗



∩


\D ∩






\Dσ .

Here the upper isomorphism sends e ∈ ∗ (τ ∈ h = D) to (τ mod
); the lower isomorphism extends the upper isomorphism by sending 0 ∈  to the class of the nilpotent orbit ((σ∞ , C) mod
). Next we consider a subgroup
of GZ of finite index. Let σ = σ∞ . Let n ≥ 1 and let
=
(n) be the kernel of SL(2, Z) → SL(2, Z/nZ). Then       1 Z 1 nZ 1 nN gp = .
(σ ) = ,
(σ ) =
∩ 0 1 0 1 0 1 2π iτ


is neat if and only if n ≥ 3. For n ≥ 3, as is well known in the theory of modular curves, we have the following local description of
\D at the boundary point (∞ mod
) in
\D =
\(h ∪ P1 (Q)). We have ∼

local isom

− →
(σ )gp \Dσ −−−−−→
\D , where the first arrow is an isomorphism of analytic spaces which sends q ∈ ∗ to (τ mod
(σ )gp ) with q = e2π iτ/n , and sends 0 ∈  to ((σ∞ , C) mod
(σ )gp ), and the second arrow is the canonical projection and is locally an isomorphism of analytic spaces. Upper half space (continued). We consider the case g = 2, i.e., D = h2 . Let U be the open set of C3 defined by U = {(q1 , q2 , a) ∈ 2 × C | if q1 q2  = 0, then log(|q1 |) log(|q2 |) > (2π Im(a))2 }. Let N1 , N2 ∈ gR = sp(2, R) be the nilpotent elements defined by N1 (e3 ) = e1 , N1 (ej ) = 0 for j  = 3, Then

 exp(z1 N1 + z2 N2 )F

a b

N2 (e4 ) = e2 , N2 (ej ) = 0 for j  = 4.   b a + z1 =F b c

for any z1 , z2 , a, b, c ∈ C. Let σ = (R≥0 )N1 + (R≥0 )N2 ⊂ gR .

b c + z2



34

CHAPTER 0

For
= exp(ZN1 + ZN2 ) = 1 + ZN1 + ZN2 ,
is strongly compatible with the fan {{0}, (R≥0 )N1 , (R≥0 )N2 , σ } of all faces of σ , and we have the following isomorphism of analytic spaces. For qj = e2π iτj , 2 × C ⊃ U 
\Dσ ,   a τ mod
(q1 q2  = 0), (q1 , q2 , a)  → F 1 a τ2    C a (0, q2 , a)  → (R≥0 )N1 , F mod
(q2  = 0), a τ2    τ a (q1 , 0, a)  → R≥0 N2 , F 1 mod
(q1  = 0), a C    C a (0, 0, a)  → σ, F mod
. a C More generally, for any strongly compatible pair (
, ) such that σ ∈  and
is neat and such that
(σ )gp = exp(ZN1 + ZN2 ), the above isomorphism U 
(σ )gp \Dσ induces a morphism of analytic spaces U →
\D which is locally an isomorphism. 0.4.14 We say that we are in the classical situation if D is a symmetric Hermitian domain and the tangent bundle of D coincides with the horizontal tangent bundle. Example (ii) (and hence Example (i)) in 0.3.2 belongs to the classical situation, but Example (iii) does not. In the classical situation,
\D is a toroidal partial compactification as constructed by Mumford et al. [AMRT]. Under the assumption D = ∅, the classical situation is listed as follows. Case (1) w = 2t + 1, ht+1,t = ht,t+1 ≥ 0, hp,q = 0 for other (p, q). Case (2) w = 2t, ht+1,t−1 = ht−1,t+1 ≤ 1, ht,t ≥ 0, hp,q = 0 for other (p, q). For general D, the space
\D is not necessarily a complex analytic space because it may have “slits” caused by “Griffiths transversality at the boundary”. But still it has a kind of complex structure, period maps can be extended to
\D , and infinitesimal calculus can be performed nicely. This was first observed by the simplest example in [U2]. In the terminology of this book,
\D is a “logarithmic manifold,” as explained in 0.4.15–0.4.17 below. 0.4.15 Strong topology. The underlying local ringed space over C of
\D is not necessarily an analytic space in general. Sometimes, it can be something like (1) S := {(x, y) ∈ C2 | x  = 0} ∪ {(0, 0)} = {(x, y) ∈ C2 | if x = 0, then y = 0}

35

OVERVIEW

endowed with a topology that is stronger than the topology as a subspace of C2 , called the “strong topology.” Let Z be an analytic space and S be a subset of Z. A subset U of S is open in the strong topology of S in Z if and only if, for any analytic space Y and any morphism λ : Y → Z such that λ(Y ) ⊂ S, λ−1 (U ) is open on Y . If S is a locally closed analytic subspace of Z, the strong topology coincides with the topology as a subspace of Z. However the strong topology of the set S in (1) is stronger than the topology as a subspace of C2 . For example, (2) Let f : R>0 → R>0 be a map such that for each integer n ≥ 1, there exists εn > 0 for which f (s) ≤ s n if 0 < s < εn . (An example of f (s) is e−1/s .) Then, if s > 0 and s → 0, (f (s), s) converges to (0, 0) for the topology of S as a subspace of C2 , but it does not converge for the strong topology (see 3.1.3). (Roughly speaking, (f (s), s) runs too near to the “bad line” {0} × C.) 0.4.16 Categories A, A(log), B, B(log). We define the categories A ⊂ B,

A(log) ⊂ B(log)

as follows (cf. 3.2.4). We denote by A,

A(log),

the category of analytic spaces and the category of fs logarithmic analytic spaces, respectively. Let B be the category of local ringed spaces X over C (over C means that OX is a C-algebra) having the following property: X has an open covering (Uλ )λ such that, for each λ, there exists an isomorphism Uλ  Sλ of local ringed spaces over C for some subset Sλ of an analytic space Zλ , where Sλ is endowed with the strong topology in Zλ and with the inverse image of OZλ . Let B(log) be the category of objects of B endowed with an fs logarithmic structure. 0.4.17 Logarithmic manifolds. Our space
\D is a very special object in B(log), called a “logarithmic manifold” (cf. Section 3.5). We first describe the idea of the logarithmic manifold by using the example S ⊂ C2 in 0.4.15 (1). Let Z = C2 with coordinate functions x, y, and endow Z with the logarithmic structure MZ associated with the divisor “x = 0.” Then the sheaf ωZ1 of logarithmic differential forms on Z (= the sheaf of differential forms with logarithmic poles along x = 0) is a free OZ -module with basis (d log(x), dy). For each z ∈ Z, let ωz1 be the module of logarithmic differential forms on the point z which is regarded as an fs logarithmic analytic space endowed with the ring C and with the inverse image of MZ . Then, if z does not belong to the part x = 0 of Z, z is

36

CHAPTER 0

just the usual point Spec(C) with the trivial logarithmic structure C× , and ωz1 = 0. If z is in the part x = 0, z is a point Spec(C) with the induced logarithmic structure Mz = n≥0 C× x n  C× × N, and hence ωz1 is a one-dimensional C-vector space generated by d log(x). Thus ωz1 is not equal to the fiber of ωZ1 at z which is a twodimensional C-vector space with basis (d log(x), dy). Now the above set S has a presentation S = {z ∈ Z | the image of yd log(x) in ωz1 is zero}.

(1)

Recall that zeros of a holomorphic function on Z form a closed analytic subset of Z. Here we discovered that S is the set of “zeros” of the differential form yd log(x) on Z, but the meaning of “zero” is not that the image of yd log(x) in the fiber of ωZ1 is zero (the latter “zeros” form the closed analytic subset y = 0 of Z). The set “zeros in the new sense” of a differential form with logarithmic poles is the idea of a “logarithmic manifold.” The precise definition is as follows (cf. 3.5.7). By a logarithmic manifold, we mean a local ringed space over C endowed with an fs logarithmic structure which has an open covering (Uλ )λ with the following property: For each λ, there exist a logarithmically smooth fs logarithmic analytic space Zλ , a finite subset Iλ of
(Zλ , ωZ1 λ ), and an isomorphism of local ringed spaces over C with logarithmic structures between Uλ and an open subset of Sλ = {z ∈ Zλ | the image of Iλ in ωz1 is zero},

(2)

where Sλ is endowed with the strong topology in Zλ and with the inverse images of OZλ and MZλ . 0.4.18 Example with h2,0 = h0,2 = 2, h1,1 = 1 (continued). Let
be a neat subgroup of GZ of finite index. We give a local description of the space
\D and observe that this space has a slit.

Fix v ∈ S2 ∩ ( 3j =1 Qej ) and fix a nonzero element v
of 3j =1 Rej which is linearly independent of v over R. Define  ∈ Q>0 by
∩ exp(QNv ) = exp(ZNv ). Let U = {(q, a, z) ∈ C2 × Q | (q, a, z) satisfies (1) and (2) below}. (1) If q  = 0 and q = e2π iτ/ , then Im(τ )v + Im(a)v
, θ (z)0 < 0 (for θ , see 0.3.4). (2) If q = 0, then θ (z) = v. Endow U with the strong topology in Z := C2 × Q, with the sheaf of rings OZ |U , and with the inverse image of the logarithmic structure of Z associated to the divisor q = 0. By the above condition (2), U has a slit and it is not an analytic space. But U is a logarithmic manifold. Let σ = (R≥0 )Nv . We have morphisms of local ringed spaces over C with logarithmic structures U →
(σ )gp \Dσ →
\D

37

OVERVIEW

which are locally isomorphisms, and the left of which sends (q, a, z) ∈ U to the class of exp(τ Nv + aNv
)F (z) = exp(τ Nv ) exp(aNv
)F (z) if q  = 0 and q = e2π iτ/ , and to the class of (σ, exp(σC ) exp(aNv
)F (z)) if q = 0 and θ(z) = v. The above local description of
\D is obtained from Proposition 12.2.5. Note that e1 , e2 in Section 12.2 are e4 , e5 here, respectively. The slit in
\D corresponding to the slit in U appears by “small Griffiths transversality,” as explained in 0.4.29 below. Theorem 0.4.19 (cf. Theorem A in Section 4.1) Let  be a fan in gQ and let
be a neat subgroup of GZ which is strongly compatible with . (i) Then
\D is a logarithmic manifold. It is a Hausdorff space. (ii) For any σ ∈ , the canonical projection
(σ )gp \Dσ →
\D is locally an isomorphism of logarithmic manifolds. The Hausdorff property of
\D is by virtue of the strong topology. Proposition 12.3.6 gives an example such that
\D is not Hausdorff if we use a naive topology that is weaker than the strong topology. Now we consider a polarized logarithmic Hodge structure and its moduli. 0.4.20 log

For an object X of B(log), a ringed space (Xlog , OX ) is defined just as in the case of fs logarithmic analytic spaces (see Section 2.2). It is described locally as follows. Assume that the logarithmic structure of X is induced from a homomorphism S → OX with S an fs monoid. Let Z = Spec(C[S])an . Then log

log

mult × S1 ), OX = OX ⊗OZ OZ . X log = X ×Z Z log = X ×Hom(S,Cmult ) Hom(S, R≥0

0.4.21 Let X be an object of B(log).Aprepolarized logarithmic Hodge structure ( pre-PLH) on X of weight w is a triple (HZ ,  , , F ) consisting of a locally constant sheaf HZ of free Z-modules of finite rank on Xlog , a bilinear form  ,  : HQ × HQ → Q, log log and a decreasing filtration F on OX ⊗Z HZ by OX -submodules which satisfy the following condition (1). (1) There exist a locally free OX -module M and a decreasing filtration (Mp )p∈Z by OX -submodules of M such that Mp = M for p  0, Mp = 0 for log p  0, and Mp /Mp+1 are locally free for all p, and such that OX ⊗Z HZ = log log OX ⊗τ −1 (OX ) τ −1 (M) and F p = OX ⊗τ −1 (OX ) τ −1 (Mp ) for all p. Furthermore, the annihilator of F p with respect to  ,  coincides with F w+1−p for any p. We give two remarks concerning pre-PLH.

38

CHAPTER 0

(i) If H = (HZ ,  , , F ) is a pre-PLH on X, the OX -modules M and Mp in (1) above are determined by H as log

M = τ∗ (OX ⊗Z HZ ),

Mp = τ∗ (F p ).

This follows from Proposition 2.2.10. (ii) (Proposition 2.3.3 (ii)) If (HZ ,  , , F ) is a pre-PLH on X, then for any x ∈ X and for any y ∈ Xlog lying over x, the action of π1 (x log ) on HZ,y is unipotent. 0.4.22 Let X be an object of B(log) and let H = (HZ ,  , , F ) be a pre-PLH on X. We call H a polarized logarithmic Hodge structure (PLH) on X if for each x ∈ X, it satisfies the following two conditions. Positivity on x. Let y be any element of Xlog lying over x. Take a finite family fj × (1 ≤ j ≤ n) of elements of MX,x which do not belong to OX,x such that the × monoid (MX /OX )x is generated by the images of fj . Then, if s ∈ sp(y) and if exp(s(log(fj ))) are sufficiently near to 0, (HZ,y ,  , y , F (s)) is a polarized Hodge structure. 1,log Griffiths transversality on x. (d ⊗ 1HZ )(F p |x log ) ⊂ ωx ⊗Olog (F p−1 |x log ) for x any p. log

1,log

Here Ox and ωx are those of the point x = Spec(C) endowed with the inverse image of MX , and F |x log denotes the module-theoretic inverse image of F under the log log morphism of ringed spaces (x log , Ox ) → (Xlog , OX ). In other words, PLH is a pre-PLH whose pullback to the fs logarithmic point x for any x ∈ X satisfies the conditions in 0.2.19 for an LVPH (we take x as X in 0.2.19 here). Although x is not logarithmically smooth unless the logarithmic structue of x is trivial, the conditions in 0.2.19 make sense when we replace X there by x. In the case that X is a logarithmically smooth, fs logarithmic analytic space, the validity of the above Griffiths transversality on x for all x ∈ X (we call this the small Griffiths transversality) is much weaker than the Griffiths transversality (3) in 0.2.19 (we call this the big Griffiths transversality). In fact if the logarithmic 1,log × ), ωx = 0 for any point x ∈ X, and structure of X is trivial (that is, MX = OX hence the small Griffiths transversality is an empty condition. Hence an LVPH on X is a PLH on X, but a PLH on X is not necessarily an LVPH on X. 0.4.23 In 0.4.23–0.4.25, we will see that the notion of a “nilpotent orbit” is nothing but a “PLH on an fs logarithmic point”. Let x be an fs logarithmic point. Then x log  Hom(Mx /Ox× , S1 ) and hence gp π1 (x log )  Hom(Mx /Ox× , Z). Let π1+ (x log ) ⊂ π1 (x log ) be the part corresponding gp to the part Hom(Mx /Ox× , N) ⊂ Hom(Mx /Ox× , Z). Then π1+ (x log ) is an fs monoid. gp

39

OVERVIEW

Proposition 0.4.24 (cf. Propositions 2.5.1, and 2.5.5) Let x be an fs logarithmic point and let H = (HZ ,  , , F ) be a pre-PLH on x. Let y ∈ x log . (i) Let hj ∈ Hom(Mx /Ox× , Z) (1 ≤ j ≤ n), let γj ∈ π1 (x log ) be the element corresponding to hj , and let Nj : HQ,y → HQ,y be the logarithm of the (unipotent) action of γj on HQ,y . Let zj ∈ C (1 ≤ j ≤ n), s0 ∈ sp(y), and let s be the element of sp(y) characterized by gp

s((2πi)−1 log(f )) − s0 ((2π i)−1 log(f )) =

n 

zj hj (f ) for any f ∈ Mxgp

j =1

(see 0.2.17). Then

  n  F (s) = exp  zj Nj  F (s0 ). j =1

(ii) Let (γj )1≤j ≤n be a finite family of generators of the monoid π1+ (x log ) and let Nj : HQ,y → HQ,y be the logarithm of the action of γj on HQ,y . Fix s ∈ sp(y). Then H satisfies the positivity on x in 0.4.22 if and only if the following condition is satisfied:     n  HZ,y ,  , y , exp  zj Nj  F (s) is a PH if Im(zj )  0 (∀j ). j =1

(iii) Let Nj (1 ≤ j ≤ n) be as in (ii) and let s ∈ sp(y). Then H satisfies the Griffiths transversality on x in 0.4.22 if and only if Nj F p (s) ⊂ F p−1 (s) for any j and p. 0.4.25

With the notation in (ii) in 0.4.24, let σ = nj=1 (R≥0 )Nj . By (i) of 0.4.24, {F (s)}s∈sp(y) is an exp(σC )-orbit. By (ii) and (iii) of 0.4.24, this exp(σC )-orbit is a σ -nilpotent orbit if and only if H is a PLH on x. In other words, (a PLH on an fs logarithmic point) = (a nilpotent orbit). Hence if H is a PLH on an object X of B(log), for each x ∈ X, the pullback H (x) of H to the fs logarithmic point x is regarded as a nilpotent orbit. This fact is presented in schema (2) in Introduction. 0.4.26 We generalize the functor PH1 : A → (Sets) in 0.3.7 to the logarithmic case. Let
be a neat subgroup of GZ and let  be a fan in gQ . Assume that
and  are strongly compatible (0.4.10). Let  = w, (hp,q )p,q , H0 ,  , 0 ,
,  . For a PLH H = (HZ ,  , , F ) on X, by a
-level structure on H , we mean a global section of the sheaf
\ Isom((HZ ,  , ), (H0 ,  , 0 )) on X log .

40

CHAPTER 0

We define a functor PLH : B(log) → (Sets) as follows. For X ∈ B(log), let PLH (X) be the set of all isomorphism classes of PLH on X of type (w, (hp,q )p,q∈Z ) endowed with a
-level structure µ satisfying the following condition: ∼ → (H0 ,  , 0 ) of For any x ∈ X, any y ∈ x log and a lifting µ˜ y : (HZ,y ,  , y ) − the germ of µ at y, we have the following (1) and (2). (1) There exists σ ∈  such that  by µ˜ y Image π1+ (x log ) → Aut(HZ,y ) −−−→ Aut (H0 ) ⊂ exp(σ ). (2) For the smallest such σ ∈  and for s ∈ sp(y), exp(σC )µ˜ y (F (s)) is a σ nilpotent orbit. Note that the Griffiths transversality that is required for PLH is only the Griffiths transversality. But this definition fits well the moduli problem. Now the precise form of Theorem for Subject I in Introduction is stated as follows. Theorem 0.4.27 (cf. Theorem B in Section 4.2) Let
be a neat subgroup of GZ and let  be a fan in gQ . Assume that
and  are strongly compatible. (i) The logarithmic manifold
\D represents the functor PLH : B(log) → ∼ → Mor( ,
\D ). (Sets), that is, there exists an isomorphism ϕ : PLH − (ii) For any local ringed space Z over C with a logarithmic structure (which need not be fs) and for any morphism of functors h : PLH |A(log) → Mor( , Z)|A(log) (where |A(log) denotes the restrictions to A(log)), there exists a unique morphism f :
\D → Z such that h = (f ◦ ϕ)|A(log) , where f is regarded as a morphism Mor( ,
\D ) → Mor( , Z). 0.4.28 For X ∈ B(log) and for H = (HZ ,  , , F, µ) ∈ PLH (X), the morphism ϕH : X →
\D corresponding to H is called the associated period map, which is settheoretically given by sending x ∈ X to the
-equivalence class of the nilpotent orbit (σ, exp(σC )µ˜ y (F (s))) at x (which is independent of the choices of y, µ˜ y , and s) in 0.4.26 (2). Note that this map is an extension of the classical period map. If U is an open set on which the logarithmic structure of X is trivial (that is, MX |U = OU× ), then the restriction (HZ ,  , , F, µ)|U is a PH on U with a
-level structure, and the period map of H is an extension of the period map U →
\D. Theorem 0.4.27 (ii) characterizes
\D as the universal object among the targets of period maps from objects of A(log) into local ringed spaces over C with logarithmic structures. This indicates that the topology of
\D , its ringed space structure, and the logarithmic structure, that we define in this book are in fact intrinsic structures (not artificial ones) determined by this universality. 0.4.29 The reason that the moduli space
\D of PLH is not necessarily an analytic space but a logarithmic manifold is as follows. We also explain why slits and the strong topology naturally appear.

41

OVERVIEW

Any PLH of type  on an fs logarithmic analytic space X (2.5.8) comes, locally on X, from a universal pre-PLH Hσ on some logarithmically smooth, fs logarithmic analytic space Eˇ σ (3.3.2) for some σ ∈  by pulling back via a morphism X → Eˇ σ (cf. Sections 3.3 and 8.2). Let E˜ σ = {x ∈ Eˇ σ | the inverse image of Hσ on x satisfies Griffiths transversality}, ⊃ Eσ = {x ∈ Eˇ σ | the inverse image of Hσ on x is a PLH}. Locally on Eˇ σ , E˜ σ is the zeros in the new sense (0.4.17) of a finite family of differential forms on Eˇ σ (see Proposition 3.5.10). Hence E˜ σ can have slits. (Note that the Griffiths transversality of the inverse image of Hσ on x ∈ E˜ σ is the small Griffiths transversality (0.4.22).) Furthermore, as in Theorem A (i) stated in Section 4.1, we have Eσ is open in E˜ σ for the strong topology of E˜ σ in Eˇ σ . This openness is not true in general if we use the topology of E˜ σ as a subset of Eˇ σ (12.3.10). Consequently, Eσ is a logarithmic manifold. For a neat subgroup
of GZ and a fan  in gQ that are strongly compatible, the local shape of
\D is similar to that of Eσ . More precisely,
\D is covered by the images of morphisms
(σ )gp \Dσ →
\D (σ ∈ ) which are locally isomorphisms, and Eσ is a σC -torsor over
(σ )gp \Dσ , where σC is the C-vector space spanned by σ (Theorem A (iii) and (iv) in Section 4.1). Thus, slits, the strong topology, and logarithmic manifolds naturally appear in the moduli of PLH. In the nonlogarithmic case where  consists of one element {0} and
= {1}, Eˇ {0} = Dˇ with the universal H , and we have E˜ {0} = Dˇ and E{0} = D. Example with h2,0 = h0,2 = 2, h1,1 = 1 (continued). In this example, the fact that the slit “if q = 0 then v = θ (z)” appears from the small Griffiths transversality is explained as follows. Let U ⊂ C2 × Q be as in 0.4.18. The pullback on U of the universal PLH on
\D extends to a pre-PLH H = (HZ ,  , , F ) on the fs logarithmic analytic space C2 × Q whose logarithmic structure is defined by the divisor {(q, a, z) ∈ C2 × Q | q = 0}. For x = (0, a, z) ∈ C2 × Q, the inverse image of H on x satisfies Griffiths transversality if and only if v = ±θ(z). One of the motivations of the dream of Griffiths to enlarge D was the hope of extending the period map of VPH to the boundary. Concerning this, we have the following result. Theorem 0.4.30 (Theorem 4.3.1) Let X be a connected, logarithmically smooth, × } be the fs logarithmic analytic space and let U = Xtriv = {x ∈ X | MX,x = OX,x open subspace of X consisting of all points of X at which the logarithmic structure of X is trivial. Let H be a variation of polarized Hodge structure on U with unipotent local monodromy along X − U . Fix a base point u ∈ U and let (H0 ,  , 0 ) = (HZ,u ,  , u ). Let
be a subgroup of GZ which contains the global monodromy

42

CHAPTER 0

group Image(π1 (U, u) → GZ ) and assume
is neat. Let ϕ : U →
\D be the associated period map. (i) Assume that X − U is a smooth divisor. Then the period map ϕ extends to a morphism X →
\D of logarithmic manifolds for some fan  in gQ that is strongly compatible with
. (ii) For any x ∈ X, there exist an open neighborhood W of x, a logarithmic modification W
of W , a commutative subgroup

of
, and a fan  in gQ that is strongly compatible with

such that the period map ϕ|U ∩W lifts to a morphism U ∩ W →

\D and extends to a morphism W
→

\D of logarithmic manifolds. Here in (ii) a logarithmic modification is a special kind of proper morphism W
→ W which is an isomorphism over U ∩ W (3.6.12). This theorem can be deduced from the nilpotent orbit theorem of Schmid and some results for fans. Note that (i) can be applied to X = . Elliptic curves (continued). In theorem 0.4.30, consider the case where X = , U = ∗ , and H is the LVPH on  in 0.2.18. Fix a branch of e2 in HZ,u = H0 = Ze1 + Ze2 . The image
of π1 (U, u) → GZ is isomorphic to Z and is generated by the element γ such that γ (e1 ) = e1 , γ (e2 ) = e1 + e2 . The classical period map ∗ →
\D =
\h extends to the period map  →
\Dσ where σ = σ∞ (0.4.13), which coincides with the isomorphism  
\Dσ in 0.4.13. 0.4.31 Infinitesimal period maps. Let f : Y → X be a projective, logarithmically smooth (2.1.11), vertical morphism of logarithmically smooth fs logarithmic  gp × analytic spaces with connected X. Assume, for any y ∈ Y , that Coker (MX /OX )f (y) → gp (MY /OY× )y is torsion-free. Let (HZ ,  , , F ) be the associated LVPH of weight m on X as in 0.2.21. Let
and  be a strongly compatible pair (0.4.10). Assume that
is neat and contains Image(π1 (X log ) → GZ ), and assume that we have the associated period map ϕ : X →
\D (0.4.28). Note that, by Theorem 0.4.30 (ii), these assumptions will be fulfilled locally on X, if we allow a logarithmic modification of it. Then as a generalization of 0.3.9, for the differential dϕ of the period map ϕ, we have the following commutative diagram: θX   K-S R 1 f∗ θY /X

dϕ

h ϕ ∗ θ
\D = gr −1 End  ,  (M)    ∩ via coupling  p p−1 −−−−−−→ HomOX (R m−p f∗ ωY /X , R m−p+1 f∗ ωY /X )

−−−−→

p h is the horizontal logarithmic tangent where θY/X := HomOY (ωY1 /X , OY ), and θ
\D  bundle of the logarithmic tangent bundle θ
\D , K-S on the left vertical arrow means the logarithmic version of the Kodaira-Spencer map, and the right vertical arrow is the canonical map (Section 4.4).

43

OVERVIEW

0.4.32 Note that the classifying space D for polarized Hodge structures on H 2 of surfaces of general type with pg ≥ 2, or on H 3 of Calabi-Yau threefolds, is not classical in the sense of 0.4.14. By the construction in the present book, we can now talk about the extended period maps and their differentials associated with degenerations of all complex projective manifolds. 0.4.33 Moduli of PLH with coefficients. We can generalize the above theorems of the moduli of PLH to the moduli of PLH with coefficients (see Chapter 11). Let A be a finite-dimensional semisimple Q-algebra endowed with a map A → A, a  → a ◦ , satisfying (a + b)◦ = a ◦ + b◦ ,

(ab)◦ = b◦ a ◦

(a, b ∈ A).

By a polarized logarithmic Hodge structure with coefficients in A (A-PLH) we mean a PLH (HZ ,  , , F ) endowed with a ring homomorphism A → End Q (HQ ) satisfying ax, y = x, a ◦ y

(a ∈ A, x, y ∈ HQ ).

The theorems 0.4.19 and 0.4.27 can be generalized to the moduli
\DA of A-PLH (11.1.7, 11.3.1). 0.4.34 In the classical situation 0.4.14, in the work [AMRT], they constructed a fan  which is strongly compatible with GZ such that GZ \D is compact. In our general situation, it can often happen that
\D is not locally compact for any  such that D  = D. However, we can define the notion of a complete fan (a sufficiently big fan, roughly speaking) such that, in the classical situation,  is complete if and only if GZ \D is compact (see Section 12.6). If  is complete and is strongly compatible with
, the classical period map U →
\D in 0.4.30 always extends globally to a morphism X
→
\D of logarithmic manifolds for some logarithmic modification X
→ X (Theorem 12.6.6). One problem which we cannot solve in this book is that of finding a complete fan in general.∗ In Example 0.3.2 (iii) (see also 0.3.4, 0.4.18, and 0.4.29), the fan  in 0.4.5 is complete. But
\D is not compact, not even locally compact, since it has slits.

0.5 FUNDAMENTAL DIAGRAM AND OTHER ENLARGEMENTS OF D We fix (w, (hp,q )p,q∈Z , H0 ,  , 0 ) as in Section 0.3. Let D be the classifying space of polarized Hodge structures, i.e., a Griffiths domain, as in 0.3.1. ∗ See

the end of section 12.7.

44

CHAPTER 0

To prove the main theorems 0.4.19 and 0.4.27, as already mentioned in the Introduction, we need to construct the fundamental diagram (3) in Introduction and to study all the spaces and their relations there. Roughly speaking, in this fundamental diagram, the construction of the four spaces in the right-hand side is based on arithmetic theory of algebraic groups, and that of the four spaces in the left-hand side is based on Hodge theory. They are joined by the central continuous map  D,val → DSL(2) , which is a geometric interpretation of the SL(2)-orbit theorem of Cattan-Kaplan-Schmid [CKS]. We give an overview of our results concerning these spaces. The organization of Section 0.5 is as follows. In 0.5.1, we describe the rough ideas of all the enlargements of D in the fundamental diagram. In 0.5.2, in the case of Example (i) in 0.3.2 (the case of the upper half plane), we give the complete descriptions of all the enlargements of D, other than D which was already described in Section 0.4. After that we explain each of these enlargements (other  than D ) one by one in the general case; D in 0.5.3–0.5.6, DBS in 0.5.7–0.5.10, DSL(2) in 0.5.11–0.5.18, and the “valuative spaces” completing the fundamental diagram in 0.5.19–0.5.29. In particular, explicit descriptions of the central bridge  D,val → DSL(2) in Examples (ii) and (iii) in 0.3.2 are given in 0.5.26 and 0.5.27, respectively. In 0.5.30, we overview -spaces, related to the work of Cattani and Kaplan [CK1]. 0.5.1 First we give some general observations. Recall that D ( is a fan in gQ ) is the set of nilpotent orbits (σ, Z), where σ ∈  and Z is an exp(σC )-orbit in Dˇ satisfying a certain condition (0.4.7). (i) Space D . The space D ( is a fan in gQ ) is a set of pairs (σ, Z) where σ ∈  and Z is an exp(iσR )-orbit in Dˇ satisfying a certain condition (see 0.5.3   below). The space D has a natural topology, D is a dense open subset of D , and,  roughly speaking, the element (σ, Z) ∈ D is the limit point of elements of Z which  “run in the direction of degeneration conducted by σ .” The space D is covered by  

open subsets Dσ = {(σ , Z) ∈ D | σ ⊂ σ }, where σ runs over elements of . (ii) Space DBS . The space DBS is a set of pairs (P , Z) where P is a Q-parabolic subgroup of GR and Z is a subset of D satisfying a certain condition (see 0.5.7 below). The set Z is a torus orbit in the following sense. For (P , Z) ∈ DBS , there is an associated homomorphism of algebraic groups s : (R × )n → P over R such that n Z is an s(R>0 )-orbit in D. The space DBS has a natural topology, D is a dense open subset of DBS , and, roughly speaking, the element (P , Z) ∈ DBS is the limit point of elements of Z which “run in the direction of degeneration conducted by P .” The space DBS is covered by open subsets DBS (P ) = {(P
, Z) ∈ DBS | P
⊃ P }, where P runs over all Q-parabolic subgroups of G. (iii) Space DSL(2) . The space DSL(2) is a set of pairs (W, Z) where W is a compatible family (W (j ) )1≤j ≤r (i.e., distributive families in [K]; see 5.2.12) of rational (j ) weight filtrations W (j ) = (Wk )k∈Z on H0,R and Z is a subset of D satisfying a

45

OVERVIEW

certain condition (see 0.5.11–0.5.13 below). The set Z is a torus orbit in the following sense. For (W, Z) ∈ DSL(2) , there is an associated homomorphism of algebraic groups over R s : (R × )r → GW,R = {g ∈ GR | gWk

(j )

(j )

= Wk

for all j, k}

such that Z is an s(R>0 )n -orbit in D. The space DSL(2) has a natural topology, D is a dense open subset of DSL(2) , and, roughly speaking, the element (W, Z) ∈ DSL(2) is the limit point of elements of Z which “run in the direction of degeneration conducted by W .” The space DSL(2) is covered by open subsets DSL(2) (W ) = {(W
, Z) ∈ DSL(2) | W
is a “subfamily” of W }, where W runs over all compatible families of rational weight filtrations of H0,R . (iv) The other four spaces. The other four spaces are spaces of “valuative” orbits  which are located over D , D , DSL(2) , and DBS , respectively. These upper spaces in the fundamental diagram (3) in Introduction are obtained from the lower spaces as the limits when the directions of degenerations are divided into narrower and narrower directions. We can say also that the vertical arrows in that diagaram are projective limits of kinds of blow-ups. 0.5.2 

Upper half plane (continued). In the easiest case D = h, the sets D , DSL(2) , and DBS are described as follows.  First we describe D . Recall that  = {{0}, σa (a ∈ P1 (Q))} (0.4.9). Recall that   0 R≥0 σ∞ = . 0 0 

The space D is covered by open sets Dσ for σ ∈ . The space Dσ for σ = {0} is identified with D (F ∈ D is identified with the pair (σ, Z) with σ = {0} and Z = {F }). The complement Dσ∞ − D is the set of all pairs (σ∞ , Z) where Z is a subset of C ⊂ Dˇ = P1 (C) of the form x + iR for some x ∈ R. This is a set of all exp(iσ∞,R )-orbits in C. We have a homeomorphism Dσ∞  {x + iy | x ∈ R, 0 < y ≤ ∞},

(σ∞ , x + iR)  → x + i∞,

which extends the identity map of D. Hence (σ∞ , x + iR) is the limit of x + iy ∈ D (y > 0) for y → ∞. Let a ∈ P1 (Q) and let g be any element of SL(2, Q) such that a = g · ∞. Then Dσa − D is the set of all pairs (σa , Z) where Z is a subset of ∼ → Dˇ = P1 (C) such that (σ∞ , g −1 (Z)) ∈ Dσ∞ . We have a homeomorphism Dσ∞ − Dσa , (σ∞ , Z)  → (σa , g(Z)). We describe DBS . A Q-parabolic subgroup of GR = SL(2, R) is either GR itself or Pa (a ∈ P1 (Q)) defined by Pa = {g ∈ SL(2, R) | ga = a} = {g ∈ SL(2, R) | gVa = Va } where Va is as in 0.4.9. For example,    a b  P∞ = a, b, d ∈ R, ad = 1 . 0 d 

46

CHAPTER 0

The space DBS is covered by open sets DBS (P ) for P = GR , Pa (a ∈ P1 (Q)). The space DBS (GR ) is identified with D (F ∈ D is identified with the pair (GR , Z) with Z = {F }). The complement DBS (P∞ ) − D is the set of all pairs (P∞ , Z) where Z is a subset of h = D of the form x + iR>0 for some x ∈ R. We have a homeomorphism DBS (P∞ )  {x + iy | x ∈ R, 0 < y ≤ ∞},

(P∞ , x + iR>0 )  → x + i∞,

which extends the identity map of D. Hence (P∞ , x + iR>0 ) is the limit of x + iy ∈ D (y > 0) for y → ∞. Let a ∈ P1 (Q) and let g be any element of SL(2, Q) such that a = g · ∞. Then DBS (Pa ) − D is the set of all pairs (Pa , Z) where Z is a subset of D such that (P∞ , g −1 (Z)) ∈ DBS (P∞ ). We have a homeomorphism ∼ → DBS (Pa ), (P∞ , Z)  → (Pa , g(Z)). DBS (P∞ ) − We describe DSL(2) . For a ∈ P1 (Q), let W (a) be the increasing filtration on H0,R defined by W1 (a) = H0,R ,

W0 (a) = W−1 (a) = Va ,

W−2 (a) = 0.

For example, W1 (∞) = H0,R ⊃ W0 (∞) = W−1 (∞) = Re1 ⊃ W−2 (∞) = 0. The space DSL(2) is covered by the open subsets DSL(2) (W (a)) where W (a) now denotes the family of weight filtrations consisting of the single member W (a). The space DSL(2) (∅) for the empty family ∅ is identified with D (F ∈ D is identified with the pair (∅, Z) with Z = {F }). The complement DSL(2) (W (∞)) − D is the set of all pairs (W (∞), Z) where Z is a subset of h = D of the form x + iR>0 for some x ∈ R. We have a homeomorphism DSL(2) (W (∞))  {x + iy | x ∈ R, 0 < y ≤ ∞}, (W (∞), x + iR>0 )  → x + i∞, which extends the identity map of D. Hence (W (∞), x + iR>0 ) is the limit of x + iy ∈ D (y > 0) for y → ∞. Let a ∈ P1 (Q) and let g be any element of SL(2, Q) such that a = g · ∞. Then DSL(2) (W (a)) − D is the set of all pairs (W (a), Z) where Z is a subset of D such that (W (∞), g −1 (Z)) ∈ DSL(2) (W (∞)). ∼ → DSL(2) (W (a)), (W (∞), Z)  → We have a homeomorphism DSL(2) (W (∞)) − (W (a), g(Z)). The valuative spaces in this case are naturally identified with the spaces under  them respectively in the fundamental diagram. That is, the canonical maps D,val →  D , DBS,val → DBS , DSL(2),val → DSL(2) are homeomorphisms and the canonical map D,val → D is bijective.  The identity map of D extends to GQ -equivariant homeomorphisms D   DSL(2)  DBS , which induce homeomorphisms Dσa  DSL(2) (W (a))  DBS (Pa ) for each a ∈ P1 (Q) described as (σa , Z
) ↔ (W (a), Z) ↔ (P (a), Z),

Z
= exp(iσa,R )Z,

Z = Z
∩ D.

Thus the fundamental diagram in this case becomes like (4) in Introduction.

47

OVERVIEW

0.5.3 



Space D . In 0.5.3–0.5.6, we consider the space D which is on the left-hand side (the Hodge side) of the fundamental diagram, next to the space D of nilpotent orbits considered in Section 0.4.

A nilpotent i-orbit is a pair (σ, Z) consisting of a nilpotent cone σ = ˇ 1≤j ≤r (R≥0 )Nj and a subset Z ⊂ D which satisfy, for some F ∈ Z,    Z = exp(iσR )F, NF p⊂ F p−1 (∀p,∀N ∈ σ ), 

 exp 1≤j ≤r iyj Nj F ∈ D (∀ yj  0). Let  be a fan in gQ . As a set, we define  ˇ D := {(σ, Z) nilpotent i-orbit | σ ∈ , Z ⊂ D}. 

Note that we have the inclusion map D → D , F  → ({0}, {F }). There is a canonical surjection D → D , (σ, Z)  → (σ, exp(σC )Z). For a rational nilpotent cone    σ in gR , we denote D{face of σ } by Dσ . Then, for a fan  in gQ , we have D =   σ ∈ Dσ . 0.5.4 

In Chapter 3, we will define a topology of D that has the following property. Let

(σ, Z) ∈ D , let Nj ∈ gQ (1 ≤ j ≤ n), F ∈ Z, and assume σ = nj=1 (R≥0 )Nj . Then   n  iyj Nj  F → (σ, Z) if yj ∈ R and yj → ∞. exp  j =1

Theorem 0.5.5 (Theorem A in Section 4.1) 

(i) The space D is Hausdorff. (ii) Assume that
is strongly compatible with . Then
\D is Hausdorff. (iii) Assume that
is strongly compatible with  and is neat. Then the canonical   projection D →
\D is a local homeomorphism. (iv) Assume that
is strongly compatible with  and is neat. Then we have a canonical homeomorphism 


\D  (
\D )log which is compatible with the projections to
\D . 

By (iv), for
as in (iv), the canonical map
\D →
\D is proper, and the fibers are products of finite copies of S1 .

48

CHAPTER 0

0.5.6 



Here we give local descriptions of D →
\D →
\D for Example (i), Example (ii) (g = 2), and Example (iii) in 0.3.2 (for some choices of  and
).  Upper half plane (continued). Let
= 10 Z1 , σ = σ∞ . Then we have a commutative diagram of topological spaces: {x + iy | x ∈ R, 0 < y ≤ ∞}  ↓

Dσ ↓


\Dσ

log



↓

↓ 
\Dσ .



Here the lower horizontal isomorphism is that in 0.4.13, the upper horizontal isomorphism is the one described in 0.5.2, and the upper left vertical arrow sends τ = x + iy (0 < y < ∞) to e2π iτ ∈ ∗ ⊂ log , and x + i∞ to (0, e2π ix ) ∈ log = || × S1 . Upper half space (continued). Let g = 2 and D = h2 . Let U be the open set of 2 × C defined in 0.4.13, and let
= exp(ZN1 + ZN2 ) = 1 + ZN1 + ZN2 , σ = (R≥0 )N1 + (R≥0 )N2 . We describe Dσ and
\Dσ . We have a commutative diagram of topological spaces: (|| × R)2 × C

⊃

↓ (|| × S ) × C 1 2

↓ ⊃

↓  ×C 2

U˜ log  U

log

↓ ⊃

U

Dσ ↓


\Dσ ↓ 
\Dσ .

Here the upper left vertical arrow is induced by R → S1 , x  → e2π ix , the lower left vertical arrow is induced by || × S1 → , (r, u)  → ru, and U˜ log is the inverse image of U in (|| × R)2 × C. The space U log is identified with the inverse image of U in (|| × S1 )2 × C. The inclusions ⊃ in this diagram are open immersions. Let rj = e−2πyj . The upper horizontal isomorphism of this diagram sends ((r1 , x1 ), (r2 , x2 ), a) ∈ U˜ log to   x + iy1 a F 1 if r1 r2  = 0, a x2 + iy2    a x + iR (R≥0 )N1 , F 1 if r1 = 0 and r2  = 0, a x2 + iy2    a x + iy1 (R≥0 )N2 , F 1 if r1  = 0 and r2 = 0, a x2 + iR    x + iR a if r1 = r2 = 0. σ, F 1 a x2 + iR Example with h2,0 = h0,2 = 2, h1,1 = 1 (continued). Let the notation be as in 0.4.18. Let
be a neat subgroup of GZ of finite index. We have a commutative

49

OVERVIEW

diagram of topological spaces ((R≥0 ) × R) × C × Q

⊃

↓ ((R≥0 ) × S ) × C × Q 1

→

↓ ⊃

↓ C×C×Q

U˜ log U

log

↓ 

→
\D

↓ ⊃

U



D

↓ →


\D

in which the three horizontal arrows are local homeomorphisms. Here the upper left vertical arrow is induced by R → S1 , x  → e2π ix , the lower left vertical arrow is induced from (R≥0 ) × S1 → C, (r, u)  → ru, and U˜ log is the inverse image of U in ((R≥0 ) × R) × C × Q. The space U log is identified with the inverse image of U in ((R≥0 ) × S1 ) × C × Q. Recall that U is endowed with the strong topology. The spaces U log and U˜ log are endowed here with the topologies as fiber products by left squares. The upper horizontal arrow sends (r, x, a, z) ∈ U˜ log (r  = 0) to exp((x + iy)Nv + aNv
)F (z), where r = e−2πy/ with , v, and v
as in 0.4.18, and (0, x, a, θ −1 (v)) to ((R≥0 )Nv , exp(iRNv ) exp(xNv + aNv
)F (θ −1 (v))). 0.5.7 Space DBS . In 0.5.7–0.5.10, we consider the space DBS which is on the right-hand side (algebraic group side) of the fundamental diagram. DBS is a real manifold with corners. The definition of DBS will be reviewed in Section 5.1. Here we give an explicit presentation of the open set DBS (P ) of DBS , for simplicity, under the assumptions that the Q-parabolic subgroup P of GR is an R-minimal parabolic subgroup of GR and that the largest R-split torus in the center of P /Pu (Pu is the unipotent radical of P ) is Q-split. In this case, DBS (P ) is described by using the Iwasawa decomposition of GR . Upper half plane (continued). We first observe the easiest case D = h. In this case, we have a homeomorphism   1 R × (R>0 ) × SO(2, R)  GR = SL(2, R), 0 1   1/t 0 (g, t, k)  → gs(t)k, where s(t) = . 0 t ∼

This is an Iwasawa decomposition of SL(2, R). By SL(2, R)/ SO(2, R) − → h, g  → g · i, this Iwasawa decomposition induces a homeomorphism   ∼ 1 x R × R>0 − → h = D, (x, t)  → s(t) · i = x + t −2 i. 0 1 This homeomorphism extends to a homeomorphism R × R≥0  DBS (P∞ ),

(x, 0)  → (P∞ , x + iR>0 ).

50

CHAPTER 0

In general, let K be a maximal compact subgroup of GR and let P be an Rminimal parabolic subgroup of GR . Denote by Pu the unipotent radical of P . Then we have a homeomorphism (Iwasawa decomposition) n Pu × (R>0 ) × K  GR ,

(g, t, k)  → gs(t)k,

for a unique pair (n, s) where n ≥ 0 is an integer and s is a homomorphism (R × )n → P of R-algebraic groups satisfying the following conditions (1)–(3). s

(1) The composition (R × )n − → P → P /Pu induces an isomorphism from (R × )n onto the largest R-split torus in the center of P /Pu . ∼ → GR associated with K (see below) sends s(t) (2) The Cartan involution GR − to s(t)−1 . (3) For t = (tj )1≤j ≤n ∈ (R × )n , |tj | < 1 for any j if and only if all eigenvalues of Ad(s(t)) on Lie(Pu ) have absolute values > 1. Here the Cartan involution associated with a maximal compact subgroup K of GR is the unique homomorphism ι : GR → GR of R-algebraic groups such that ι2 = id and such that K = {g ∈ GR | ι(g) = g}. For r ∈ D, the Cartan involution of GR associated with Kr coincides with the map g  → Cr gCr−1 where Cr is the operator in 0.1.8 (2). Now let P be a Q-parabolic subgroup of GR which is an R-minimal parabolic subgroup of GR such that the largest R-split torus in the center of P /Pu is Q-split. Let r ∈ D, let Kr be the maximal compact subgroup of GR corresponding n to r (0.3.3), and consider the Iwasawa decomposition Pu × (R>0 ) × Kr  GR with respect to (P , Kr ) satisfying (1)–(3) as above. It induces a homeomorphism n Pu × (R>0 ) × (Kr /Kr
)  D,

(g, t, k)  → gs(t)k · r.

This extends to a homeomorphism n Pu × (R≥0 ) × (Kr /Kr
)  DBS (P ). n The element of DBS (P ) corresponding to (g, 0, k) ∈ Pu × (R≥0 ) × (Kr /Kr
) (g ∈ n Pu , k ∈ Kr ) coincides with the pair (P , Z) where Z = {gs(t)k · r | t ∈ R>0 }. More n , generally, the element of DBS (P ) corresponding to (g, t, k) (g ∈ Pu , t ∈ R≥0 k ∈ KR ) coincides with the element (Q, Z) of DBS (P ) where Q is the Q-parabolic subgroup of GR containing P which corresponds to the subset J = {j | 1 ≤ j ≤ n, tj  = 0} of the set {1. . . . , n} (there is a bijection between the set of all subsets of {1, . . . , n} and the set of all Q-parabolic subgroups of GR containing P ), and n , tj
= tj if j ∈ J } ⊂ D. Z = {gs(t
)k · r | t
∈ R>0 Thus DBS is understood by the theory of algebraic groups, rather than by Hodge theory.

Theorem 0.5.8 (i) DBS is a real manifold with corners. For any p ∈ DBS , there are an open neighborhood U of p in DBS , integers m, n ≥ 0, and a homeomorphism n U  R m × R≥0

which sends p to (0, 0). The point p belongs to D if and only if n = 0.

51

OVERVIEW

(ii) For any subgroup
of GZ ,
\DBS is Hausdorff. (iii) If
is of finite index in GZ ,
\DBS is compact. (iv) If
is a neat subgroup of GZ , the projection DBS →
\DBS is a local homeomorphism. The definition of DBS and the proof of this theorem were given in [KU2] and [BJ] independently. The definition of DBS is a modification of the definition in [BS] of the original Borel-Serre space XBS , which is an enlargement of the symmetric Hermitian space X of all maximal compact subgroups of GR . There is a canonical surjection DBS → XBS which sends F ∈ D to KF ∈ X . The proof of the above theorem is a reduction to the similar properties of the original Borel-Serre space proved in [BS]. In 0.5.9 and 0.5.10 below, we give local descriptions of DBS for Example (ii) (g = 2) and Example (iii) in 0.3.2, respectively. 0.5.9 Upper half space (continued). Let g = 2 and D = h2 . Let P be the Q-parabolic subgroup of GR consisting of elements that preserve the R-subspaces Re1 , Re1 + Re2 , and Re1 + Re2 + Re4 of H0,R . This is an R-minimal parabolic subgroup of GR . There is a homeomorphism R 4  Pu (not an isomorphism of groups, for Pu is noncommutative). Let s : (R × )2 → P be the homomorphism of algebraic groups given by s(t)e1 = t1−1 t2−1 e1 , s(t)e2 = t2−1 e2 , s(t)e4 = t2 e4 , s(t)e3 = t1 t2 e3 .  Put r = F 0i 0i . Then we have Kr = Kr
= Sp(2, R) ∩ O(4, R)  U (2). We have a homeomorphism Pu × (R × )2  P , (g, t)  → gs(t). We have a homeomorphism (Iwasawa decomposition) 2 Pu × (R>0 ) × Kr  GR ,

(g, t, k)  → gs(t)k,

which satisfies the conditions (1)–(3) in 0.5.7. This induces a homeomorphism 2 Pu × R>0  D,

(g, t)  → gs(t) · r,

which extends to a homeomorphism 2 Pu × R≥0  DBS (P ).

0.5.10 Example with h2,0 = h0,2 = 2, h1,1 = 1 (continued). We use the notation in 0.3.4. Let G◦R be the kernel of the determinant map GR → {±1} (for the action on H0,R ), and let P be the Q-parabolic subgroup of GR consisting of all elements

of G◦R which preserve the subspaces Re4 and 4j =1 Rej of H0,R . This is an Rminimal parabolic subgroup of GR . We have an isomorphism of R-algebraic groups ∼ → Pu , a  → exp(Na ). Let s : R × → P be the homomorphism of R-algebraic R3 −

52

CHAPTER 0

groups defined by s(t)e4 = t −1 e4 ,

s(t)ej = ej (1 ≤ j ≤ 3),

s(t)e5 = te5 .

∼

→ P , (g, t)  → gs(t). Then we have a homeomorphism Pu × R × − Let v ∈ S2 . We have a homeomorphism (Iwasawa decomposition) ∼

Pu × (R>0 ) × Kr(v) − → GR ,

(g, t, k)  → gs(t)k

which satisfies the conditions (1)–(3) in 0.5.7. We have a homeomorphism ∼

→ Kr(v) · r(v), {±1} × S2 −

(±1, v
)  → s(±1) · r(v
).

Hence this Iwasawa decomposition induces a homeomorphism ∼

→ D, R 3 × (R>0 ) × {±1} × S2 −

(a, t, ±1, v)  → exp(Na )s(±t) · r(v),

which extends to a homeomorphism ∼

→ DBS (P ). R 3 × (R≥0 ) × {±1} × S2 − In this example, all Q-parabolic subgroups of GR other than G◦R are conjugate to P under GQ , and hence DBS is covered by open sets DBS (gP g −1 ) (g ∈ GQ ) each of which is homeomorphic to DBS (P ) via the homeomorphism that extends g −1 : D → D. 0.5.11 The space DSL(2) . In general, D and DBS are still far from each other in nature. We find an intermediate existence DSL(2) to connect them. We consider this space DSL(2) in 0.5.11–0.5.18. Hodge theory and algebraic group theory are unified on this space. This unification is based on a fundamental property of the SL(2)-action on the upper half plane h:     √ y 0 0 1 √ (i). exp iy (0) = 0 1/ y 0 0 When y > 0 varies, the left-hand side produces a nilpotent i-orbit, while the righthand side produces a torus orbit in the Borel-Serre space. We define DSL(2) as follows. A pair (ρ, ϕ), consisting of a homomorphism ρ : SL(2, C)r → GC of algebraic ˇ is called groups which is defined over R and a holomorphic map ϕ : P1 (C)r → D, an SL(2)-orbit of rank r if it satisfies the following conditions (1)–(4) ([CKS, Chapter 4], [KU2, Chapter 3]): (1) (2) (3) (4)

ϕ(gz) = ρ(g)ϕ(z) for all g ∈ SL(2, C)r and all z ∈ P1 (C)r . The Lie algebra homomorphism ρ∗ : sl(2, C)⊕r → gC is injective. ϕ(hr ) ⊂ D. Let z ∈ hr , let Fz• (sl(2, C)⊕r ) be the Hodge filtration of sl(2, C)⊕r induced • (gC ) be the by the Hodge filtration of (C2 )⊕r corresponding to z, and let Fϕ(z) Hodge filtration of gC induced by the Hodge filtration ϕ(z) of H0,C . Then p p ρ∗ : sl(2, C)⊕r → gC sends Fz (sl(2, C)⊕r ) into Fϕ(z) (gC ) for any p.

53

OVERVIEW

Here in (4), for F ∈ D, the Hodge filtration on gC induced by F is defined as p

FF (gC ) = {h ∈ gC | h(F s ) ⊂ F s+p (∀ s)}.  The Hodge filtration of (C2 )⊕r = rj =1 (Ce1j ⊕ Ce2j ) corresponding to z is defined  as F 0 (z) = (C2 )⊕r , F 1 (z) = rj =1 C(zj e1j + e2j ), and F 2 (z) = 0. Let i = (i, . . . , i) ∈ hr . Then, if the condition (1) is satisfied, (3) is satisfied if p and only if ϕ(i) ∈ D, and (4) is satisfied if and only if ρ∗ sends Fi (sl(2, C)⊕r ) into p Fϕ(i) (gC ) for any p. 0.5.12 For an SL(2)-orbit (ρ, ϕ) of rank r, let Nj ∈ gR be the image under ρ∗ of sl(2, R) in the j th factor. Let W (j ) = W (N1 + · · · + Nj )

0 1 00

∈

(1 ≤ j ≤ r),

where W (N) for a nilpotent linear operator N denotes the monodromy weight filtration associated with N (Deligne [D5]; see 5.2.4). The family W = (W (j ) )1≤j ≤r is called the family of weight filtrations associated with (ρ, ϕ). Definition 0.5.13 ([KU2, 3.6], 5.2.6) Two SL(2)-orbits (ρ1 , ϕ1 ) and (ρ2 , ϕ2 ) of r rank r are equivalent if there exists (t1 , . . . , tr ) ∈ R>0 such that   −1    −1 0 t t1 0 ,..., r ◦ ρ1 , ρ2 = Int ρ1 0 tr 0 t1  −1   −1  0 t t1 0 ,..., r · ϕ1 . ϕ2 = ρ1 0 tr 0 t1 Here Int(g) means the inner automorphism by g. Define DSL(2),r to be the set of all equivalence classes of SL(2)-orbits (ρ, ϕ) of rank r whose associated family of weight filtrations is defined over Q. Define DSL(2) = r≥0 DSL(2),r where DSL(2),0 = D. For an SL(2)-orbit (ρ, ϕ) of rank r, the family W of weight filtrations associated with (ρ, ϕ) and the set Z = {ϕ(iy1 , . . . , iyr ) | yj > 0 (1 ≤ j ≤ r)} are determined by the class [ρ, ϕ] of (ρ, ϕ) in DSL(2) . Conversely, [ρ, ϕ] is determined by the pair (W, Z) (see [KU2, 3.10]). We will denote [ρ, ϕ] ∈ DSL(2) also by (W, Z). 0.5.14 If the condition (2) in 0.5.11 is omitted, a pair (ρ, ϕ) is called an SL(2)-orbit in r variables. For an SL(2)-orbit (ρ, ϕ) in n variables, there exists a unique SL(2)-orbit (ρ
, ϕ
) such that for some J ⊂ {1, . . . , n}, (ρ, ϕ) = (ρ
, ϕ
) ◦ πJ , where πJ : (SL(2, C) × P1 (C))n → (SL(2, C) × P1 (C))r is the projection to the J -factor and such that (ρ
, ϕ
) is an SL(2)-orbit of rank r := (J ). We denote the point [ρ
, ϕ
] of DSL(2) also by [ρ, ϕ].

54

CHAPTER 0

The notion of the SL(2)-orbit generalizes the (H1 )-homomorphism in the context of equivariant holomorphic maps of symmetric domains (cf. [Sa2, II § 8]). In the classical situation (0.4.14), the Satake-Baily-Borel compactification of
\D for a subgroup
of GZ of finite index is a quotient of
\DSL(2) , and is philosophically close to
\DSL(2) . 0.5.15 In 5.2.13, we will review the definition of the topology of DSL(2) given in [KU2]. In this topology, for [ρ, ϕ] ∈ DSL(2) , we have ϕ(iy1 , . . . , iyn ) → [ρ, ϕ]

if yj ∈ R>0 and yj /yj +1 → ∞ (yn+1 denotes 1).

Theorem 0.5.16 (5.2.16, 5.2.15) (i) Let p ∈ DSL(2) be an element of rank r. Then there are an open neighborhood U of p in DSL(2) , a finite dimensional vector space V over R, R-vector subspaces VJ of V given for each subset J of the set {1, . . . , r}, satisfying VJ ⊃ VJ
if J ⊂ J
⊂ {1, . . . , r} and V∅ = V , and a homeomorphism r U  {(a, t) ∈ V × R≥0 | a ∈ VJ where J = {j | tj = 0}}

which sends p to (0, 0). (ii) For any subgroup
of GZ ,
\DSL(2) is Hausdorff. (iii) If
is a neat subgroup of GZ , the projection DSL(2) →
\DSL(2) is a local homeomorphism. 0.5.17 We consider the relation between DBS and DSL(2) . In DBS , the direction of degeneration is determined by a parabolic subgroup of GR . On the other hand, in DSL(2) , it is determined by a family of weight filtrations. Let [ρ, ϕ] ∈ DSL(2) be an element of rank r and let W = (W (j ) )1≤j ≤r be the family of weight filtrations associated with (ρ, ϕ). Let G◦R be the kernel of the determinant map GR → {±1}, and let G◦W,R be the (j ) subgroup of G◦R consisting of all elements which preserve Wk for any j, k. If ◦ r = 1, i.e., if W consists of one weight filtration, GW,R is a Q-parabolic subgroup of GR . Let DSL(2),≤1 be the part of DSL(2) consisting of all elements of rank ≤ 1. Then DSL(2),≤1 is an open set of DSL(2) . The identity map of D is extended to a continuous map DSL(2),≤1 → DBS which has the form (W, Z)  → (P , Z
), where P = G◦W,R and Z
is a certain subset of D containing Z (5.1.5). Even for r ≥ 2, in the case hp,q = 0 for any (p, q)  = (1, 0), (0, 1) (the case D = hg ), the associated family W = (W (j ) )1≤j ≤r of weight filtrations is so simple that all filters in this family are linearly ordered, (j )

(j )

(1) (r) 0 = W−2 ⊂ W−1 · · · ⊂ W−1 ⊂ W0(r) ⊂ · · · ⊂ W0(1) ⊂ W1

= HQ ,

55

OVERVIEW

for any j , i.e., they form a single long filtration. With one exception below, the same is true for other classical situations in 0.4.14. Hence W and P are related directly by P = G◦W,R , and we have DSL(2) (W ) = DBS (G◦W,R ), DSL(2) = DBS (which also coincides with the classical Borel-Serre space XBS ) as in (5) in Introduction (cf. 12.1.2, [KU2, 6.7]). Exceptional Case ([KU2, 6.7]). The weight w is even, rank H0 = 4, and there exists a Q-basis (ej )1≤j ≤4 of H0,Q such that ej , ek 0 = 1 if j + k = 5, and = 0 otherwise. In general, we have the following criterion. Criterion ([KU2, 6.3]). The following are equivalent. (i) The identity map of D extends to a continuous map DSL(2) → DBS . (j ) (ii) At any point of DSL(2) , the filters Wk which appear in the associated family (j ) W = (W )j of weight filtrations are linearly ordered by inclusion. For examples with no continuous extension DSL(2) → DBS of the identity map of D, see [KU2, 6.10] and 12.4.7. 0.5.18 We have the following criterion for the local compactness of DSL(2) . Criterion (Theorem 10.1.6). Let p = [ρ, ϕ] ∈ DSL(2) . The following (i) and (ii) are equivalent. (i) There exists a compact neighborhood of p in DSL(2) . (ii) The following conditions (1) and (2) hold. (j )

(1) All filters Wk appeared in the associated compatible family W = (W (j ) )j of weight filtrations at p are linearly ordered by inclusion. (2) Lie(Kr ) ⊂ Lie(GW,R ) + Lie(Kr
), where r = ϕ(i) and GW,R = {g ∈ GR | (j ) (j ) gWk = Wk (∀j, ∀k)}. By this criterion, DSL(2) for the example with h2,0 = h0,2 = 2, h1,1 = 1 in 0.3.2 ∼ → DBS (iii) is locally compact, and hence has no slit. We have DSL(2) = DSL(2),≤1 − in this case. But even for the examples of similar kind in 12.2.10, DSL(2) can have slits in general. 0.5.19 Valuative spaces. In the general case, the family W of weight filtrations associated with p ∈ DSL(2) becomes more complicated and we do not have a direct relation between the W and the parabolic subgroups P (Criterion in 0.5.17). We have to introduce the valuative spaces DSL(2),val and DBS,val to relate the spaces DSL(2) and DBS . These are the projective limits of certain kinds of blow-ups of the respective spaces.

56

CHAPTER 0

To relate the spaces D and DSL(2) , we also have to introduce the valuative space    D,val of D . This is the projective limit of a kind of blow-up of D corresponding  to rational subdivisions of the fan . We have a continuous map D,val → DSL(2) which is a geometric interpretation of the SL(2)-orbit theorem [CKS] as in 0.5.24 below. In all cases, the valuative spaces are projective limits over the corresponding original spaces so as to divide the directions of degenerations into narrower and narrower. 

Theorem 0.5.20 Let X be one of D,val , DSL(2),val , DBS,val . Then (i) X is Hausdorff.  (ii) Let
be a subgroup of GZ . In the case X is D,val , assume
is strongly compatible with . Then
\X is Hausdorff. If furthermore
is neat, the canonical projection X →
\X is a local homeomorphism. The definitions of the four valuative spaces in the fundamental diagram are given in Chapter 5. Here we just give in 0.5.22–0.5.23 explicit local descriptions of them in the case of Example (ii) with g = 2 in 0.3.2. (The cases of Examples (i) and (iii) are not interesting concerning valuative spaces, for, in these cases, the valuative spaces are identified with the spaces under them in the fundamental diagram.) 0.5.21 2 Example (C2 )val and (R≥0 )val . In 0.5.22 and 0.5.23, we give explicit descriptions of some valuative spaces in the case D = h2 . For this, we introduce here the spaces 2 2 (C2 )val and (R≥0 )val obtained as projective limits of blow-ups from C2 and R≥0 , respectively. In general, for any object X of B(log), we will define in Section 3.6 (see 3.6.18 and 3.6.23) a space Xval obtained from X by taking blow-ups along the logarithmic structure. The space (C2 )val is Xval for X = C2 which is endowed with the logarithmic structure associated with the normal crossing divisor C2 − (C× )2 . Let X = X0 = C2 , and let X1 be the blow-up of X at the origin (0, 0). Then (C× )2 ⊂ X1 , and the complement X1 − (C× )2 is the union of three irreducible divisors C0 , C1 , C∞ where C0 is the closure of {0} × C× , C∞ is the closure of C× × {0}, and C1 is the inverse image of (0, 0). Next let X2 be the blow-up of X1 at two points, the intersection of C0 and C1 and the intersection of C1 and C∞ . Then (C× )2 ⊂ X2 , and the complement X2 − (C× )2 is the union of five irreducible divisors C0 , C1/2 , C1 , C2 , and C∞ . Here C1/2 is the inverse image of the intersection of C0 and C1 in X1 , C2 is the inverse image of the intersection of C1 and C∞ , and we denote the proper transformations of C0 , C1 , and C∞ in X1 simply by C0 , C1 , and C∞ , respectively. In this way, we have a sequence of blow-ups

· · · → X3 → X2 → X1 → X0 = X, where Xn+1 is obtained from Xn by blow-up the intersections of different irreducible components of Xn − (C× )2 . We define (C2 )val = lim Xn . ← − n

57

OVERVIEW

This (C2 )val is obtained also as the inverse limit of the toric varieties [KKMS, Od] 2 corresponding to finite rational subdivisions of the cone R≥0 in R 2 . For example, 2 the above X2 is the toric variety corresponding to the finite subdivision of R≥0 consisting of the subcones {0}, σ0 , σ0,1/2 , σ1/2 , σ1/2,1 , σ1 , σ1,2 , σ2 , σ2,∞ , and σ∞ 2 of R≥0 . Here σs (s = 0, 1/2, 1, 2) is the half line {(x, sx) | x ∈ R≥0 } of slope s, σ∞ is the half line {0} × R≥0 , and σs,t is the cone generated by σs and σt . The open subvariety of X2 corresponding to the cone σs is X2 − ∪s
=s Cs
, and the open subvariety of X2 corresponding to the cone σs,t is X2 − ∪s
=s,t Cs
. Let q1 and q2 2 be the coordinate functions of C2 . Then for a finite rational subdivision of R≥0 , the corresponding toric variety is the union of the open subvarieties Spec(C[P (σ )])an for cones σ in this subdivision, where P (σ ) = {q1m q2n | (m, n) ∈ Z, am + bn ≥ 0 ∀(a, b) ∈ σ }. The projection f : (C2 )val → C2 is proper and surjective, and f induces a homeo∼ → C2 − {(0, 0)}. We regard C2 − {(0, 0)} as an open morphism f −1 (C2 − {(0, 0)}) − subspace of (C2 )val via f −1 . The complement f −1 ((0, 0)) is described as / Q>0 } f −1 ((0, 0)) = {(0, 0)s | s ∈ [0, ∞], s ∈ ∪ {(0, 0)s,z | s ∈ Q>0 , z ∈ P1 (C)}. Here if s ∈ Q>0 and z ∈ C× and if s is expressed as m/n with m, n ∈ Z, m > 0, n > 0 and GCD(m, n) = 1, then (0, 0)s,z is the unique point of f −1 ((0, 0)) at which both q1m q2−n and q1−m q2n are holomorphic and the value of q1m q2−n is z. For any 2 finite rational subdivision of R≥0 containing the half line σ = {(x, sx) | x ∈ R≥0 } 2 of slope s, the map from (C )val to the toric variety corresponding to this subdivision induces a bijection from {(s, z) | z ∈ C× } onto the fiber over (0, 0) ∈ C2 of the open subvariety corresponding to σ . For s ∈ Q>0 , (0, 0)s,0 (resp. (0, 0)s,∞ ) is the limit of (0, 0)s,z (z ∈ C× ) for z → 0 (resp. z → ∞). The point (0, 0)0 (resp. (0, 0)∞ ) is the limit of (0, z) ∈ C2 (resp. (z, 0) ∈ C2 ) (z ∈ C× ) for z → 0. Finally, (0, 0)s for s ∈ R>0 − Q>0 is the unique point of f −1 ((0, 0)) at which q1m q2−n (m, n ∈ Z, m > 0, n > 0) is holomorphic if and only if m/n > s and q1−m q2n is holomorphic if and only if s > m/n. A point of f −1 ((0, 0)) has the form (0, 0)s (s ∈ [0, ∞] − Q>0 ) if and only if for any n ≥ 0, its image in Xn is the intersection of two different irreducible components of the divisor Xn − (C× )2 . These points of f −1 ((0, 0)) are characterized as the limits of points of (C× )2 ⊂ (C2 )val as follows. If s ∈ [0, ∞] − Q>0 , (q1 , q2 ) ∈ (C× )2 converges to (0, 0)s if and only if (q1 , q2 ) → (0, 0) and log(|q2 |)/ log(|q1 |) → s. If s ∈ Q>0 and z ∈ P1 (C), and if s is expressed as m/n with m, n ∈ Z, m > 0, n > 0 and GCD(m, n) = 1, (q1 , q2 ) ∈ (C× )2 converges to (0, 0)s,z if and only if (q1 , q2 ) → (0, 0), log(|q2 |)/ log(|q1 |) → s, and q1m /q2n → z. 2 2 − {(0, 0)} (regarded as a )val ⊂ (C2 )val be the closure of the subset R≥0 Let (R≥0 2 2 2 2 subset of C − {(0, 0)} ⊂ (C )val ). That is, (R≥0 )val is the union of R≥0 − {(0, 0)} −1 and the part of f ((0, 0)) consisting of elements (0, 0)s for s ∈ [0, ∞] such that s∈ / Q>0 , and elements (0, 0)s,z with s ∈ Q>0 and with z ∈ R≥0 ∪ {∞} ⊂ P1 (C). 2 2 The canonical projection (R≥0 )val → R≥0 is proper and surjective. The inverse 2 image of (0, 0) in (R≥0 )val is regarded as a very long totally ordered set by the

58

CHAPTER 0

following rule: (0, 0)s < (0, 0)s
,z < (0, 0)s
,z
< (0, 0)s

if 0 ≤ s < s
< s

≤ ∞, s∈ / Q>0 , s
∈ Q>0 , s

∈ / Q>0 , 0 ≤ z < z
≤ ∞. Closed intervals form a base of closed sets in this totally ordered set. 0.5.22 Upper half space (continued). Let g = 2 and D = h2 . We have the following commutative diagrams of topological spaces in which all inclusions are open immersions. (2 )val × C

⊃

↓

Uval




\Dσ,val

↓

 ×C

↓

U

(||2 )val × R 2 × C

⊃

log U˜ val

⊃

↓ U˜ log



DBS,val (P )

↓ || × R × C 2

2

2 )val Pu × (R≥0

↓ 2 Pu × R≥0



(1)

⊃

2


\Dσ . 

↓ 

↓ 



Dσ,val (2)

Dσ .

(3)

DBS (P ).

Here in (1), (2 )val ⊂ (C2 )val is the inverse image of 2 under (C2 )val → C2 , U and
are as in 0.4.13, and Uval denotes the inverse image of U ⊂ 2 × C in (2 )val × C. 2 2 In (2), (||2 )val ⊂ (R≥0 )val is the inverse image of ||2 under (R≥0 )2val → R≥0 , U˜ log log log 2 2 is as in 0.5.6, and U˜ val denotes the inverse image of U˜ in (|| )val × R × C. In (3), P is as in 0.5.9, and DBS,val (P ) denotes the inverse image of DBS (P ) in DBS,val . The lower rows of these diagrams are those obtained in 0.4.13, 0.5.6, and 0.5.9, respectively. 2 The first diagram is obtained as follows. We identify σ with the cone R≥0 via 2 2 R≥0  σ , (a1 , a2 )  → a1 N1 + a2 N2 . For a rational subdivision S of R≥0 , we have the corresponding subdivision of σ . If B(S) denotes the toric variety corresponding to S with a proper birational morphism B(S) → C2 , we have an isomorphism U (S) 
\ Dσ (S), where U (S) is the inverse image of U in B(S) × C and
\ Dσ (S) is a blow-up of
\ Dσ corresponding to this subdivision of σ . The upper row of the diagram (1) is obtained as the projective limit of B(S) × C ⊃ U (S) 
\Dσ (S). Roughly speaking, in the first diagram, we are dividing the direction of degeneration exp(z1 N1 + z2 N2 ) with Im(z1 ), Im(z2 ) → ∞ into narrower and narrower directions. A narrow direction that appears here is the direction exp(z1 (N1 + sN2 ) + z2 (N1 + s
N2 )) with Im(z1 ), Im(z2 ) → ∞ for some s, s
∈ Q≥0 or the direction exp(z1 (N1 + sN2 ) + z2 N2 ) with Im(z1 ), Im(z2 ) → ∞ for some s ∈ Q>0 . When the directions become infinitely narrow, we obtain points at infinity in
\Dσ,val . The second diagram is obtained in a similar manner. For the third diagram, see [KU2, 2.14]. Note that DSL(2),val = DBS,val in this case [KU2, 6.7].

OVERVIEW

59

0.5.23 Upper half space (continued). Let the notation be as in 0.5.22 and consider exp(iy1 N1 + iy2 N2 )F (0) ∈ D for y1 , y2 ∈ R>0 . We observe how this point converges or diverges when y1 and y2 move in special ways. This point correlog sponds to (e−2πy1 , e−2πy2 , 0, 0, 0) ∈ U˜ val in the diagram (2), and to (1, (y2 /y1 )1/2 , (1/y2 )1/2 ) ∈ Pu × (R>0 )2 in the diagram (3). From this, we have (i) When t → ∞, exp(it (2 + sin(t))N1 + itN2 )F (0) converges in Dσ to the  image of (0, 0, 0, 0, 0) ∈ U˜ log , but diverges in Dσ,val , DBS , DBS,val . We show here the divergence in DBS . The corresponding point is p(t) := (1, (2 + 2 , and sin(t))−1/2 , t −1/2 ) ∈ Pu × R≥0  2 (1, 2−1/2 , t −1/2 ) → (1, 2−1/2 , 0) ∈ Pu × R≥0     when t = π n, n = 1, 2, 3, . . . , p(t) = 2  (1, 1, t −1/2 ) → (1, 1, 0) ∈ Pu × R≥0    when t = 2π n − π/2, n = 1, 2, 3, . . . . Hence p(t) diverges in DBS . This (i) shows that the image of (0, 0, 0, 0, 0) ∈ U˜ log in Dσ has no neighborhood V such that the inclusion map V ∩ D → D extends to a continuous map V → DSL(2) = DBS . We also see at the end of 0.5.26, concerning a map in the converse direction, 2 that the image of (1, 0, 0) ∈ Pu × R≥0 in DBS has no neighborhood V such that the inclusion map V ∩ D → D extends to a continuous map V → Dσ , and even that it has no neighborhood V such that the canonical map V ∩ D →
\D extends to a continuous map V →
\Dσ . Thus, to connect the world of Dσ and Dσ to the world of DSL(2) = DBS , we have  to climb to the valuative space Dσ,val . 2 )val explained in Similarly, by using the topological natures of ||2val and (R≥0 0.5.21, we can show (ii) When t → ∞, exp(it c(2+sin(t)) N1 + itN2 )F (0), for a fixed c > 1 converges in log  Dσ,val to the image of ((0, 0)0 , 0, 0, 0) ∈ U˜ val , but diverges in DBS,val . Hence  the image of ((0, 0)0 , 0, 0, 0) in Dσ,val has no neighborhood V such that the inclusion map V ∩ D → D extends to a continuous map V → DBS,val . When t → ∞, exp(i(t + sin(t))N1 + itN2 )F (0) converges in DBS,val (P ) to  2 )val , but diverges in Dσ,val . Hence the the image of (1, (0, 0)1,1 ) ∈ Pu × (R≥0 image of (1, (0, 0)1,1 ) in DBS,val (P ) has no neighborhood V such that the  inclusion map V ∩ D → D extends to a continuous map V → Dσ,val . Thus, the views of the infinity of various enlargements of D in the fundamental diagram are rather different from each other. (iii) When y2 → ∞ and y1 /y2 → ∞, exp(iy1 N1 + iy2 N2 )F (0) converges in log  Dσ,val to the image of ((0, 0)0 , 0, 0, 0) ∈ U˜ val , and also converges in DBS (P )

60

CHAPTER 0 2 to the image of (1, 0, 0) ∈ Pu × R≥0 . As is explained in 0.5.26 below, these two limit points at infinity are related by the following continuous map  ψ : Dσ,val → DSL(2) .

0.5.24  SL(2)-orbit Theorem and continuous map D,val → DSL(2) . The SL(2)-orbit the orem in several variables in [CKS] is interpreted as the relation between D,val and DSL(2) . Let N1 , . . . , Nn ∈ gR be mutually commutative nilpotent elements, and F

∈ D. Assume that (N1 , . . . , Nn , F ) generates a nilpotent orbit, i.e., for σ = 1≤j ≤n (R≥0 )Nj , exp(σC )F is a σ -nilpotent orbit (0.4.7, 1.3.7). Cattani, Kaplan, and Schmid [CKS] defined an SL(2)-orbit (ρ, ϕ) in n variables associated to the family (N1 , . . . , Nn , F ) (cf. Section

6.1). Here the order of N1 , . . . , Nn is important. They showed that two maps exp( nj=1 iyj Nj )F and ϕ(iy1 , . . . , iyn ) into D behave asymptotically when yj /yj +1 → ∞ (yn+1 means 1). Our geometric interpretation of the SL(2)-orbit Theorem is as follows.  There is a unique continuous map ψ : D,val → DSL(2) which extends the identity

 map of D (we will prove this in Chapter 6), exp( nj=1 iyj Nj )F converges in D,val when yj /yj +1 → ∞ (1 ≤ j ≤ n) (we saw this in 0.5.23 (iii) in a special case), and ψ sends this limit point to [ρ, ϕ].  This continuous map ψ : D,val → DSL(2) is the most important bridge in the fundamental diagram (3) in Introduction, which joins the four spaces D , D,val ,   D , and D,val of orbits under nilpotent groups in the left-hand side and the four spaces DSL(2) , DSL(2),val , DBS , and DBS,val of orbits under tori in the right-hand side.

0.5.25 Fundamental Diagram. We thus have the diagram that relates D and DBS ((3) in Introduction; see also 5.0.1): DSL(2),val


\D,val

←

↓
\D

 D,val

→

→

DBS,val

↓

↓

DSL(2)

DBS

↓ ←



D

where all maps are continuous, and all vertical maps are proper surjective. The theorems on these spaces introduced in this Section 0.5 are proved by starting at DBS and moving in this diagram from the right to the left. The spaces DBS , DBS,val , DSL(2),val , and DSL(2) were already studied in [KU2]. By using the results on these spaces (which are reviewed in Chapter 5), in this book, we prove the results on the other spaces.

61

OVERVIEW 

We now give explicit descriptions of ψ : D,val → DSL(2) in some examples. In the case of Example (i) (the case D = h) in 0.3.2, for any fan  in gQ , ψ is just the    ∼ → DSL(2) in 0.5.2. In the following 0.5.26 and canonical map D,val = D ⊂ D − 0.5.27, we consider the cases of Example (ii) with g = 2 and Example (iii) in 0.3.2, respectively.

0.5.26 Upper half space (continued). Let g = 2 and D = h2 . In this case, DSL(2) = DBS . Let N1 , N2 , and σ be as before (0.4.13). The triple (N1 , N2 , F (0)) generates a σ nilpotent orbit, and the associated SL(2)-orbit in two variables (ρ, ϕ) is given (see 6.1.4) by   a 0 b 0       0 e 0 f  a b e f z 0 , ρ , = ϕ(z, w) = F . c 0 d 0  c d g h 0 w 0 g 0 h 

That is, the continuous map ψ : Dσ,val → DSL(2) sends the limit of exp(iy1 N1 +  iy2 N2 )F (0) for y1 /y2 , y2 → ∞ in Dσ,val (see 0.5.23) to [ρ, ϕ] ∈ DSL(2) . Furthermore, (ρ, ϕ) is of rank 2, and “the N1 and N2 of ρ” defined in 0.5.12 coincide with N1 and N2 here, respectively. Let W = (W (1) , W (2) ) be the family of weight filtrations associated with [ρ, ϕ] ∈ DSL(2) , that is, W (1) = W (N1 ) and W (2) = W (N1 + N2 ). Then (1) (1) ⊂ W−1 = Re1 ⊂ W0(1) = Re1 + Re2 + Re4 ⊂ W1(1) = H0,R , 0 = W−2 (2) (2) ⊂ W−1 = Re1 + Re2 = W0(2) ⊂ W1(2) = H0,R . 0 = W−2

We have GW,R = P , DSL(2) (W ) = DBS (P ). The element [ρ, ϕ] ∈ DSL(2) is also  written as (W, Z) ∈ DSL(2) with Z = F iR0>0 iR0>0 , and is also written as (P , Z) ∈ DBS with the same Z. log  In the homeomorphism U˜ val  Dσ,val in 0.5.22, the above limit point of log  is the image of ((0, 0)0 , 0, 0, 0) ∈ U˜ ⊂ (||2 )val × R 2 × C. In the homeD val

σ,val

2  DBS (P ), the above [ρ, ϕ] ∈ DSL(2) = DBS is the image omorphism Pu × R≥0 of (1, 0, 0).  We give an explicit description of ψ : Dσ,val → DSL(2) for some open neighborlog log  . Let (U˜ )
be the open set of U˜ consisting hood of the above limit point of D σ,val

val

val

of all points (p, x1 , x2 , a) (p ∈ (||2 )val , x1 , x2 ∈ R, a ∈ C) such that p  = (0, 0)∞ log and such that p  = (r, 0) for any r ∈ || − {0}. Then (U˜ val )
contains the point log ((0, 0)0 , 0, 0, 0) of U˜ val . Define N3 , N4 ∈ Lie(Pu ) by     0 0 0 1 0 −1 0 0 0 0 1 0 0 0 0 0    N3 := e23 + e14 =  0 0 0 0 , N4 := e43 − e12 = 0 0 0 0 . 0 0 0 0 0 0 1 0

62

CHAPTER 0

We have a homeomorphism ∼

R4 − → Pu ,

(xj )1≤j ≤4  → exp



3 j =1

 xj Nj exp(x4 N4 ).

The restriction of ψ : Dσ,val → DSL(2) (W ) = DBS (P ) to (U˜ val )
is explicitly described by the commutative diagram log



(||2 )val × R 2 × C ⊃ (U˜ val )
→ Dσ,val  ψ     DSL(2) (W ) log

2 ∼ − → Pu × R≥0



 DBS (P ).

log Here the left vertical arrow sends p = (r, x1 , x2 , x3 + iy3 ) ∈ (U˜ val )
with r ∈ 2 2 (|| )val and x1 , x2 , x3 , y3 ∈ R to the following element of Pu × R≥0 .

(1) When the image of r in ||2 is not (0, 0), if we write r = (e−2πy1 , e−2πy2 ) with 0 < y1 ≤ ∞ and 0 < y2 < ∞, p is sent to # "      −1 3 y32 y3 y2 1 2 ∈ Pu × R≥0 exp 1− xj Nj exp − N4 , , . j =1 y2 y1 y1 y 2 y2 (2) When the image of r in ||2 is (0, 0) and r has the form (0, 0)s or (0, 0)s,z , p is sent to     3  2 exp  xj Nj  , s, 0 ∈ Pu × R≥0 . j =1

Note that, when p ∈ (U˜ val )
as in (1) converges to a point of (U˜ val )
as in (2) whose r has the form (0, 0)s or (0, 0)s,z , then x1 , x2 , x3 , and y3 converge, y1 , y2 → ∞, and y2 /y1 → s, and hence the terms y3 /y2 and 1/y2 in (1) converge to 0 and the term   y32 −1 y2 in (1) converges to s. 1 − y1 y 2 y1 Let
= exp(ZN1 + ZN2 ) = 1 + ZN1 + ZN2 . We show that, for any neighborhood V of [ρ, ϕ] ∈ DSL(2) (W ) = DBS (P ), there is no continuous map V →
\Dσ that extends the projection V ∩ D →
\D. In fact, for any c ∈ R, the image log  pc ∈ Dσ,val of ((0, 0)0 , 0, 0, ic) ∈ (U˜ val )
is sent by ψ to [ρ, ϕ] ∈ DSL(2) which is independent of c. On the other hand, the image pc
of pc in
\Dσ is the class of the nilpotent orbit    0 ic σ, exp(σC )F ic 0 log

log

modulo
, and we have pc
 = pd
if c, d ∈ R and c  = d. Hence there is no V with V →
\Dσ as above.

63

OVERVIEW

0.5.27 Example with h2,0 = h0,2 = 2, h1,1 = 1 (continued). In this example, the fundamental diagram becomes (see Criteria in 0.5.17 and in 0.5.18, and Theorem 10.1.6). DSL(2),val

=

 ←

D,val

 D,val



→

DSL(2)

DBS,val 

=

DBS .

 ←

D



D .

Let W be the Q-rational increasing filtration of H0,R defined by Wk = 0 (k ≤ −1),

W0 = W1 = Re4 , W2 = W3 = 4j =1 Rej , Wk = H0,R (k ≥ 4). On the other hand, let the Q-parabolic subgroup P of GR be as in 0.5.10. Then, we have DSL(2) (W ) =   DBS (P ) (cf. 0.5.10), and the map ψ : D = D,val → DSL(2) is injective.

 Let v ∈ S2 ∩ ( 3j =1 Qej ) and let σ = (R≥0 )Nv . Then ψ : D → DSL(2)  sends Dσ into DSL(2) (W ). We consider the map ψ : Dσ = Dσ,val → DSL(2) (W ) = DBS (P ).

Consider the exterior product × in 3j =1 Rej , i.e., the bilinear map ( 3j =1 Rej ) ×

( 3j =1 Rej ) → 3j =1 Rej characterized by e1 × e2 = e3 ,

e2 × e3 = e1 , e3 × e1 = e2 , ej × ek = −ek × ej .

3 For z ∈ Q and u ∈ j =1 Cej such that u is not contained in Cz + 3j =1 Rej , we have exp(Nu )F (z) = exp(Nb )s(t) · r(θ (z)) with b = Re(u) − Im(u) × θ(z), t = (−Im(u), θ (z)0 )−1/2 (cf. Section 12.2). Fix v
∈ S2 which is orthogonal to v. From 0.5.6, 0.5.10, and 0.5.18, we have a commutative diagram U˜ log

((R≥0 ) × R) × C × Q ⊃

→

  

3

(

Dσ  ψ  DSL(2) (W )

∼



2 → DBS (P ). j =1 Rej ) × (R≥0 ) × {±1} × S −

The left vertical arrow sends (r, x, a, z) ∈ U˜ log with r  = 0 to (b, t, 1, θ (z)) where b = xv + Re(a)v
− (yv + Im(a)v
) × θ(z), t = (−yv + Im(a)v
, θ (z)0 )−1/2 with y ∈ R defined by r = e−2πy/ ( is as in 0.4.18), and sends (0, x, a, θ −1 (v)) ∈ U˜ log to (xv + Re(a)v
+ Im(a)v × v
, 0, 1, v). We show the following two results. 



(i) If we embed D in DSL(2) by the injection ψ, the topology of D does not coincide with the topology as a subspace of DSL(2) .

64

CHAPTER 0

(ii) Let us call the topology of U˜ log as a subspace of ((R≥0 ) × R) × C × Q the ψ → DSL(2) (W ) = DBS (P ) naive topology. Then the composition U˜ log → Dσ − is not continuous for the naive topology. Let p ∈ Dσ be the image of (0, 0, 0, θ −1 (v)) ∈ U˜ log . Take c > 0 and consider the elements f (s) = (exp(−2π/(s c )), 0, 0, θ −1 ((1 − s 2 )1/2 v + sv
)) ∈ U˜ log (s > 0, s → 0). In U˜ log , when s → 0, f (s) converges to (0, 0, 0, θ −1 (v)) for the naive topology. However, in U with the strong topology, the image (exp(−2π/(s c )), 0, θ −1 ((1 − s 2 )1/2 v + sv
)) of f (s) does not converge to the image (0, 0, θ −1 (v)) of (0, 0, 0, θ −1 (v)) (0.4.15 (2)). Hence the image of f (s) in Dσ does not converge to p (0.5.6).

The image of f (s) in ( 3j =1 Rej ) × (R≥0 ) × {±1} × S2 is (s 1−c v
× v, s c/2 (1 − s 2 )−1/4 , 1, (1 − s 2 )1/2 v + sv
). This converges to (0, 0, 1, v) if c < 1, but does not converge if c > 1 (by the existence of the term s 1−c ). For c < 1, this proves (i) because the image of f (s) in Dσ does not converge to p but the image of f (s) in DSL(2) converges to ψ(p). For c > 1, this proves (ii) because f (s) converges to (0, 0, 0, θ −1 (v)) for the naive topology but the image of f (s) in DSL(2) does not converge to the image ψ(p) of (0, 0, 0, θ −1 (v)). 0.5.28 For an object X of B(log), we will define a logarithmic local ringed space Xval over X in 3.6.18 and 3.6.23, by using the projective limit of blow-ups along the logarithmic structure. Though Xval need not belong to B(log), we can define the log topological space (Xval )log (we denote it as Xval ) in the same way as before. If 2 X = C with the logarithmic structure associated with the normal crossing divisor C2 − (C× )2 , then Xval = (C)2val ,

log

2 Xval = (R≥0 )val × (S1 )2 .

For a fan  in gQ and for a neat subgroup
of GZ which is strongly compatible with , we have (
\D )val =
\D,val ,

log



(
\D )val =
\D,val .

(See 8.4.3.) In the classical situation 0.4.14, except for the unique exceptional case 0.5.17, the fundamental diagram and the fact that DSL(2) = DBS in this situation show that there is a unique continuous map log

(
\D )val →
\DBS which extends the identity map of D. Chikara Nakayama (unpublished) proved that in the classical situation 0.4.14, if we take  and
such that
\D is a toroidal compactification of
\D, then
\D,val coincides with the projective limit of all toroidal compactifications of
\D.

65

OVERVIEW

In the general situation, we have Theorem 0.5.29 Let X be a connected, logarithmically smooth, fs logarithmic × analytic space, and let U = Xtriv = {x ∈ X | MX,x = OX,x } be the open subspace of X consisting of all points of X at which the logarithmic structure of X is trivial. Let H be a variation of polarized Hodge structure on U with unipotent local monodromy along X − U . Fix a base point u ∈ U and let (H0 ,  , 0 ) = (HZ,u ,  , u ). Let
be a subgroup of GZ which contains the global monodromy group Image(π1 (U, u) → GZ ) and assume
is neat. Then the associated period map ϕ : U →
\D extends to a continuous map log

Xval →
\DSL(2) . Here Xval is the projective limit of certain blow-ups of X at the boundary (see Section 3.6). As is explained in Section 8.4, this theorem 0.5.29 is obtained from the period  maps in 0.4.30 (ii) and the map ψ : D,val → DSL(2) . (The period map in 0.4.30 (ii) is obtained only locally on X, but the composition globalizes.)

0.5.30 -spaces. Cattani and Kaplan [CK1] generalized Satake-Baily-Borel compactifications of
\D for a symmetric Hermitian domain D, to the case where D is a Griffiths domain of weight 2 under certain assumptions, and showed that period maps from a punctured disc ∗ extend over the unit disc . This was the first successful attempt to enlarge D beyond the classical situation 0.4.14. In Chapter 9 and Section 10.4, we consider the relationship between our theory and their theory and discuss related subjects.   We define the quotient topological spaces DBS := DBS / ∼, DBS,val := DBS,val / ∼ divided by the action of the unipotent radical of the parabolic subgroup of GR associated to each point (9.1.1).  In the case of D being symmetric Hermitian domain, DBS was studied by Zucker in [Z1, Z4], and is called the “reductive Borel-Serre space” by him.  Similarly we have the quotient space DSL(2),≤1 of the part DSL(2),≤1 of DSL(2) of points of rank ≤ 1 by the unipotent part GW,R,u of GW,R associated to each point of DSL(2),≤1 (cf. 5.2.6). The space D ∗ of Cattani and Kaplan in [CK1] (defined for  special D) is essentially this DSL(2),≤1 (9.1.5). Let X be an analytic manifold and U be the complement of a smooth divisor on X. Assume that we are given a variation of polarized Hodge structure H on U with unipotent local monodromy which has a
-level structure for a neat subgroup
of GZ of finite index, then the associated period map U →
\D extends to a  continuous map X →
\DSL(2),≤1 (see Section 9.4). This is obtained as the composition of X →
\D (0.4.30 (i)) and the con   tinuous map
\D →
\DSL(2),≤1 which is induced from ψ : D = D,val → DSL(2),≤1 ⊂ DSL(2) .

66

CHAPTER 0

0.6 PLAN OF THIS BOOK The plan of this book is as follows. Chapters 1–3 are preliminaries to state the main results of the present book, Theo rems A and B in Chapter 4. In Chapter 1, we define the sets D and D . In Chapter 2, we describe the theory of polarized logarithmic Hodge structures. In Chapter 3, we discuss the strong topology, logarithmic manifolds, the spaces Eσ , E˜ σ , Eˇ σ , the categories B, B(log), and other enlargements of the category of analytic spaces. In Chapter 4, we state Theorems A and B without proofs. Theorems 0.4.19 and 0.5.5 are contained in Theorem A, and Theorem 0.4.27 is contained in Theorem B. Theorem 0.5.8 is contained in 5.1.10 and Theorem 5.1.14, Theorem 0.5.16 is contained in Theorem 5.2.15 and Proposition 5.2.16, and Theorem 0.5.20 is contained in Theorems 7.3,2, 7.4.2, 5.1.14, and 5.2.15. We also discuss, in Chapter 4, extensions of period maps over boundaries, and infinitesimal properties of extended period maps. In Chapters 5–8, we prove Theorems A and B by moving from the right to the left in fundamental diagram (3) in Introduction (also in 0.5.25). In Chapter 5, we review the spaces DSL(2) , DBS , DSL(2),val , and DBS,val defined in [KU2], and then we  define D,val and D,val . By using the work [CKS] of Cattani, Kaplan, and Schmid on SL(2)-orbits in several variables, in Chapters 5 and 6 we connect the spaces  D,val and DSL(2) as in Fundamental diagram (3) in Introduction (also in 0.5.25). In Chapter 7, we prove Theorem A, and in Chapter 8, Theorem B. In Chapters 9–12, we give complements, examples, generalizations, and open problems. In Chapter 9, we consider the relationship of the present work with the enlargements of D studied by Cattani and Kaplan [CK1]. In Chapter 10, we describe local structures of DSL(2) . In Chapter 11, we consider the moduli of PLHs with coefficients. Although the case with coefficients is more general than the case without coefficients, we have chosen not to show the coefficients everywhere in this book (which would make the notation too complicated), but to describe the theory without coefficients except in Chapter 11, where we show that the results with coefficients can be simply deduced from those without. In Chapter 12, we give examples and discuss open problems. Corrections to Previous Work We indicate three mistakes in our previous work [KU1, KU2]. (i) In [KU1, (5.2)], there is a mistake in the definition of the notion of polarized logarithmic Hodge structures of type . This mistake and its correction are explained in 2.5.16. ˜ · (ii) In [KU2, Lemma 4.7], the definition of B(U, U
, U

) is written as {g ρ(t)k r | · · · }, which is wrong. The correct definition is {ρ(t)gk ˜ · r | · · · }. This point will be explained in 5.2.17. (iii) In [KU2, Remarks 3.15, and 3.16], we indicated that we would consider a   space DSL(2) in this book. However, we actually consider only a part DSL(2),≤1 of   DSL(2) (Chapter 9). We realized that DSL(2) is not necessarily Hausdorff and seems

67

OVERVIEW 

not to be a good object to consider, but that the part DSL(2),≤1 is Haussdorff and is certainly a nice object. The present work was announced in [KU1] under the title “Logarithmic Hodge Structures and Classifying Spaces” and in [KU2] under the title “Logarithmic Hodge Structures and Their Moduli,” but we have changed the title.

0.7 NOTATION AND CONVENTION Throughout this book, we use the following notation and terminology. 0.7.1 As usual, N, Z, Q, R, and C mean the set of natural numbers {0, 1, 2, 3, . . . }, the set of integers, the set of rational numbers, the set of real numbers, and the set of complex numbers, respectively. 0.7.2 In this book, a ring is assumed to have the unit element 1, a subring shares 1, and a ring homomorphism respects 1. 0.7.3 Let L be a Z-module. For R = Q, R, C, we denote LR := R ⊗Z L. We fix a 4-tuple 0 = (w, (hp,q )p,q∈Z , H0 ,  , 0 ) where w is an integer, (hp,q )p,q∈Z is a set of non-negative integers satisfying  p,q  h = 0 for almost all p, q, hp,q = 0 if p + q  = w,   p,q h = hq,p for any p, q,

H0 is a free Z-module of rank p,q hp,q , and  , 0 is a Q-rational nondegenerate C-bilinear form on H0,C which is symmetric if w is even and antisymmetric if w is odd. It is easily checked that the associated classifying space D of Griffiths (1.2.1) is nonempty if and only if either w is odd or w = 2t and the signature (a, b) of (H0,R ,  , 0 ) satisfies  a (resp. b) = ht+j,t−j , j

where j ranges over all even (resp. odd) integers. Let GZ := Aut(H0 ,  , 0 ),

68

CHAPTER 0

and for R = Q, R, C, let GR := Aut(H0,R ,  , 0 ), gR := Lie GR = {N ∈ End R (H0,R ) | N x, y0 + x, Ny0 = 0 (∀ x, ∀ y ∈ H0,R )}. Following [BS], a parabolic subgroup of GR means a parabolic subgroup of (G◦ )R , where G◦ denotes the connected component of G in the Zariski topology containing the unity. (Note that G◦ = G if w is odd, and G◦ = {g ∈ G | det(g) = 1} if w is even.) 0.7.4 We refer to a complex analytic space as an analytic space for brevity. We use the definition of analytic space (due to Grothendieck) in which the structure sheaf OX of an analytic space X can have nonzero nilpotent sections. Precisely speaking, in this definition, an analytic space means a local ringed space over C which is locally isomorphic to the ringed space (V , OU /(f1 , . . . , fm )), where U is an open set of Cn for some n ≥ 0, f1 , . . . , fm are elements of
(U, OU ) for some m ≥ 0, and V = {p ∈ U | f1 (p) = · · · = fm (p) = 0}. We denote by A,

A(log)

the category of analytic spaces and the category of fs logarithmic analytic spaces, i.e., analytic spaces endowed with an fs logarithmic structure, respectively. 0.7.5 Throughout this book, compact spaces and locally compact spaces are already Hausdorff as in Bourbaki [Bn]. Throughout this book, proper means “proper in the sense of Bourbaki [Bn] and separated.” Here, for a continuous map f : X → Y , f is proper in the sense of Bourbaki [Bn] if and only if for any topological space Z the map X ×Y Z → Z induced by f is closed. (For example, if X and Y are locally compact, f is proper if and only if for any compact subset K of Y , the inverse image f −1 (K) is compact.) On the other hand, f is separated if and only if the diagonal map X → X ×Y X is closed. That is, f is separated if and only if, for any a, b ∈ X such that f (a) = f (b), there are open sets U and V of X such that a ∈ U , b ∈ V , and U ∩ V = ∅. In particular, in this book, a topological space X is compact if and only if the map from X to the one point set is proper. 0.7.6 For a continuous map f : X → Y and a sheaf F on Y , we denote the inverse image of F on X by f −1 (F), not by f ∗ (F). This is to avoid confusion with the moduletheoretic inverse image. For f a morphism of ringed spaces (X, OX ) → (Y, OY )

OVERVIEW

69

and a sheaf F of OY -modules on Y , we denote by f ∗ (F) the module-theoretic inverse image OX ⊗f −1 (OY ) f −1 (F) on X. 0.7.7 Concerning monoids and cones, we use the following concise terminology in this book, for simplicity. We call a commutative monoid just a monoid. A monoid is assumed (as usual) to have the neutral element 1. A submonoid is assumed to share 1, and a homomorphism of monoids is assumed to respect 1. Concerning cones, as explained in Section 1.3, a convex cone in the sense of [Od] is called just a cone in this book. A convex polyhedral cone in [Od] is called a finitely generated cone. A strongly convex cone in [Od] is called a sharp cone. In [Od], for a finitely generated cone σ (in our sense), the topological interior of σ in the vector space σR is called the relative interior of σ . (Here σR denotes the R-vector space generated by σ .) We call the relative interior of σ just the interior of σ . The interior of σ coincides with the complement of the union of all proper faces of σ . (Here a proper face of σ is a face of σ that is different from σ .) For an fs monoid S (0.2.11 and 2.1.4), similarly, we call the complement of the union of all proper faces of S the interior of S.

Chapter One Spaces of Nilpotent Orbits and Spaces of Nilpotent i-Orbits

We recall polarized Hodge structures in Section 1.1, and the classifying space D of polarized Hodge structures in Section 1.2 (cf. [G1]). In Section 1.3, we introduce the  space of nilpotent orbits D and the space of nilpotent i-orbits D in the directions in  as enlargements of D. The space D is the main object in this book.

1.1 HODGE STRUCTURES AND POLARIZED HODGE STRUCTURES Let w and (hp,q ) = (hp,q )p,q∈Z be as in Section 0.7. 1.1.1 A Hodge structure of weight w and type (hp,q ) is a pair (HZ , F ) consisting

of Hodge p,q of a free Z-module HZ of rank p,q h and of a decreasing filtration F on HC = C ⊗ HZ , called a Hodge filtration, which satisfies the following two conditions. dimC F p /F p+1 = hp,w−p for all p.  HC = F p ∩ F¯ w−p .

(1) (2)

p

1.1.2 A polarized Hodge structure of weight w and of Hodge type (hp,q ) is a triple (HZ ,  , , F ) consisting of a Hodge structure (HZ , F ) and of a nondegenerate Q-bilinear form  ,  on HQ = Q ⊗ HZ , symmetric for even w and antisymmetric for odd w, which satisfies the following two conditions. (1) F p , F q  = 0 for p + q > w. (2) The Hermitian form HC × HC → C,

(x, y)  → CF (x), y, ¯

is positive definite. Here  ,  is regarded as the natural extension to the C-bilinear form, the overbar indicates complex conjugation with respect to HZ , and CF is the Weil operator, which is defined by CF (x) := i p−q x for x ∈ F p ∩ F¯ q with p + q = w. The condition (1) (resp. (2)) is called the Riemann-Hodge first (resp. second) bilinear relation.

SPACES OF NILPOTENT ORBITS AND SPACES OF NILPOTENT i-ORBITS

71

1.2 CLASSIFYING SPACES OF HODGE STRUCTURES  Let 0 = w, (hp,q ), H0 ,  , 0 be as in Section 0.7. Definition 1.2.1 The classifying space D of polarized Hodge structures of type  0 = w, (hp,q ), H0 ,  , 0 is the set of all decreasing filtrations F on H0,C = C ⊗ H0 such that the triple (H0 ,  , 0 , F ) is a polarized Hodge structure of weight w and of Hodge type (hp,q ). 1.2.2 The compact dual Dˇ of D is the set of all decreasing filtrations F on H0,C such that the triple (H0 ,  , 0 , F ) satisfies the conditions 1.1.1 (1) and 1.1.2 (1). 1.2.3 Example. Let g ≥ 1 and consider the case w = 1, h1,0 = h0,1 = g and hp,q = 0 otherwise. Let H0 be a free Z-module with basis (ej )1≤j ≤2g and define a Z-bilinear form  , 0 : H0 × H0 → Z ⊂ Q by   0 −1g (ej , ek 0 )j,k = . 1g 0 Then D  hg , the Siegel upper half space of degree g. Recall that hg is the space of all symmetric matrices over C of degree g whose imaginary parts are positive definite (corresponding to the Riemann-Hodge first and second bilinear relations, respectively). We identify a matrix τ ∈ hg with F (τ ) ∈ D as follows:    subspace of H0,C spanned τ 0 1 , F (τ )2 := {0}. F (τ ) := H0,C , F (τ ) := by the column vectors of 1g Furthermore, if g = 1 then D  h1 =: h, the Poincaré upper half plane, where we identify a point τ ∈ h with F (τ ) ∈ D under F (τ )0 = H0,C , F (τ )1 = C(τ e1 + e2 ), F (τ )2 = {0}. In this case, we have Dˇ = P1 (C), the space of all one-dimensional C-subspaces of H0,C . 1.2.4 As in Section 0.7, let GZ = Aut(H0 ,  , 0 ), and, for R = Q, R, C, let GR = Aut(H0,R ,  , 0 ) and gR = Lie GR . Then, Dˇ (resp. D) is homogeneous under GC (resp. GR ). 1.2.5 Example. In the example 1.2.3, we have, for R = Q, R, C, GR = Sp(g, R) = {M ∈ GL(2g, R) | t MJg M = Jg },

72 where Jg =

CHAPTER 1



0 1g

−1g  A B 0 . C D ∈ Sp(g, R) acts on D by

F (τ )  → F (τ
),

τ
= (Aτ + B)(Cτ + D)−1 .

1.3 EXTENDED CLASSIFYING SPACES Definition 1.3.1 Let A be a finite-dimensional R-vector space. A subset σ of A is called a cone of A if it is nonempty and the following condition (1) is satisfied: (1) If x, y ∈ σ and a ∈ R≥0 , then x + y, ax ∈ σ . A cone σ in A is said to be sharp if σ ∩ (−σ ) = {0}. For a cone σ in A, we denote the R-linear span of σ in A by σR , and the C-linear span of σ in AC = C ⊗R A by σC . A cone σ in A is said to be finitely generated if the following condition (2) is satisfied: (2) There is a finite family (Nj )1≤j ≤n of elements of σ such that σ = (R≥0 )N1 + · · · + (R≥0 )Nn . Definition 1.3.2 Let A be as in 1.3.1, and let σ be a cone in A. A face of σ is a subcone τ of σ in A satisfying the following condition (1): (1) If x, y ∈ σ and x + y ∈ τ , then x, y ∈ τ . The following is known: If σ is a finitely generated cone in A, then σ has only finitely many faces, and any face of σ is finitely generated (cf. [Od]). Definition 1.3.3 Let A be as in 1.3.1. A fan in A is a nonempty set  of finitely generated sharp cones in A satisfying the following conditions (1) and (2): (1) If σ ∈ , any face of σ belongs to . (2) If σ, τ ∈ , then σ ∩ τ is a face of σ and of τ . Definition 1.3.4 Now let A be a finite-dimensional vector space over Q. A finitely generated cone σ in AR = R ⊗Q A is said to be rational if we can take N1 , . . . , Nn ∈ A in 1.3.1 (2). It is known that any face of a rational finitely generated cone is also rational (cf. [Od]). A fan in AR is said to be rational if all members of  are rational. Let 0 = (w, (hp,q ), H0 ,  , 0 ) be as in Section 0.7. Definition 1.3.5 A nilpotent cone in gR is a finitely generated cone σ in gR satisfying the following conditions (1) and (2): (1) Any element of σ is nilpotent as an endomorphism of H0,R . (2) NN
= N
N for any N, N
∈ σ as endomorphisms of H0,R .

SPACES OF NILPOTENT ORBITS AND SPACES OF NILPOTENT i-ORBITS

73

 Definition 1.3.6 Let Dˇ orb (resp. Dˇ orb ) be the set of all pairs (σ, Z) where σ is a ˇ that is, nilpotent cone in gR and Z is an exp(σC )-orbit (resp. exp(iσR )-orbit) in D, ˇ Z is a subset of Dˇ having the form exp(σC )F (resp. exp(iσR )F ) for some F ∈ D.

Definition 1.3.7 By a nilpotent orbit (resp. nilpotent i-orbit), we mean an element  (σ, Z) of Dˇ orb (resp. Dˇ orb ) satisfying the following conditions (1) and (2) for some F ∈ Z: (1) NF p ⊂ F p−1 (∀ p ∈ Z, ∀ N ∈ σ ).  (2) Write σ = (R≥0 )N1 + · · · + (R≥0 )Nn . Then exp 1≤j ≤n zj Nj F ∈ D if zj ∈ C and Im(zj )  0. (In fact, if (1) (resp. (2)) is satisfied for some F ∈ Z, then it is satisfied for any F ∈ Z.) If (σ, Z) is a nilpotent orbit (resp. nilpotent i-orbit), we say Z is a σ -nilpotent orbit (resp. σ -nilpotent i-orbit). Definition 1.3.8 In this book, by a fan in gQ , we mean a rational fan in gR consisting of nilpotent cones.   For a fan  in gQ , we define the space D ⊂ Dˇ orb (resp. D ⊂ Dˇ orb ) of nilpotent orbits (resp. nilpotent i-orbits) in the directions in  by D = {(σ, Z) ∈ Dˇ orb | (σ, Z) is a nilpotent orbit and σ ∈ },  D = {(σ, Z) ∈ Dˇ orb | (σ, Z) is a nilpotent i-orbit and σ ∈ }.

1.3.9 We have the inclusion maps 

D → D ;

D → D ,

F  → ({0}, F ),

and the canonical surjection 

D → D ,

(σ, Z)  → (σ, exp(σC )Z),

which is compatible with the above inclusion maps. For a rational sharp nilpotent cone σ in gR , we denote Dσ := D{face of σ } , Then, for a fan  in gQ , we have $ D = Dσ , σ ∈



Dσ := D{face of σ } . 

D =

$

Dσ .

σ ∈

Definition 1.3.10 Let  be a fan in gQ and let
be a subgroup of GZ . (i) We say that
is compatible with  if the following condition (1) is satisfied. (1) If γ ∈
and σ ∈  then Ad(γ )(σ ) ∈ .  Note that, if
is compatible with ,
acts on D and on D by (σ, Z)  → (Ad(γ )(σ ), γ Z).

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CHAPTER 1

(ii) We say that
is strongly compatible with  if it is compatible with  and the following condition (2) is satisfied. For σ ∈ , define a sharp fs monoid
(σ ) by
(σ ) =
∩ exp(σ ). (2) If σ ∈ , any element of σ can be written as a sum of a log(γ ) (a ∈ R≥0 , γ ∈
(σ )). 1.3.11 Example. Let  := {(R≥0 )N | N is a nilpotent element of gQ }. Then  is a fan in gQ . If
is of finite index in GZ ,
is strongly compatible with .

Chapter Two Logarithmic Hodge Structures

In Section 2.1, we recall basic facts about logarithmic structures (cf. [Kk1]). In log Section 2.2, we recall the ringed spaces (X log , OX ) introduced in [KkNc], with some generalizations. We study properties of local systems on Xlog in Section 2.3. In Section 2.4, we introduce the notion of the polarized logarithmic Hodge struclog ture, which is defined on the ringed space (X log , OX ). In Section 2.5, we observe the relationship of polarized logarithmic Hodge structures and nilpotent orbits. In particular, we interpret the nilpotent orbit theorem of Schmid in the language of polarized logarithmic Hodge structures. Further, we introduce the notion of the polarized logarithmic Hodge structure of type  and study the associated period map. Finally, in Section 2.6, we introduce the notion of logarithmic mixed Hodge structures.

2.1 LOGARITHMIC STRUCTURES We review the theory of logarithmic structures of Fontaine and Illusie briefly for our later use (cf. [Kk1,I2]). 2.1.1 Prelogarithmic structures and logarithmic structures. Let X be a ringed space with structure sheaf OX .Aprelogarithmic structure on X is a sheaf of monoids M together with a homomorphism α : M → OX , where OX is regarded as a sheaf of monoids by multiplication. (See our terminology 0.7.7. In particular, a monoid in this book means a commutative monoid.) A logarithmic structure on X is a prelogarithmic structure (M, α) on X that satisfies ∼

× × α −1 (OX )− → OX

via α.

× If (M, α) is a logarithmic structure on X, we regard OX as a subsheaf of M via this isomorphism. × The simplest example of a logarithmic structure on X is M = OX with α the inclusion map. This logarithmic structure is called the trivial logarithmic structure.

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Let (M, α) be a prelogarithmic structure on X. The associated logarithmic ˜ α) structure (M, ˜ is defined as the push-out M˜ of ⊂

× α −1 (OX ) −−−−→ M   α × OX

in the category of sheaves of monoids on X, together with the homomorphism α˜ : × → OX . More explicitly, M˜ → OX induced by α : M → OX and the inclusion OX × M˜ is the sheafification of the presheaf (M × OX )/ ∼, where (m, f ) ∼ (m
, f
) × ) such that mg1 = m
g2 and f α(g2 ) = if and only if there exist g1 , g2 ∈ α −1 (OX
f α(g1 ). A ringed space endowed with a logarithmic structure is called a logarithmic ringed space. 2.1.2 Standard example. Let X be a complex manifold, let Y be a divisor on X with normal crossings, and let M := {f ∈ OX | f is invertible outside Y } ⊂ OX . Then, M with the inclusion map α : M → OX is a logarithmic structure, and is called the logarithmic structure on X associated with Y . 2.1.3 Inverse image. Let f : (X, OX ) → (Y, OY ) be a morphism of ringed spaces and let (M, α) be a logarithmic structure on Y . Then the sheaf-theoretic inverse image f −1 M together with the composite morphism f −1 M → f −1 OY → OX form a prelogarithmic structure on X. The inverse image f ∗ (M, α) of (M, α) is defined as the logarithmic structure on X associated with the above prelogarithmic structure. We have ∼

× f −1 (MY /OY× ) − → f ∗ (MY )/OX .

2.1.4 Fs monoids. Here we briefly recall some facts about monoids. For a monoid S, we denote by S × the group of all invertible elements of S. By an fs monoid, we mean a monoid (commutative always, see 0.7.7) having the following three properties: (1) S is finitely generated. (2) If a, b, c ∈ S and ab = ac, then b = c. (Hence S is embedded in the group S gp = { ab | a, b ∈ S}.) (3) If a ∈ S gp and a n ∈ S for some integer n ≥ 1, then a ∈ S.

77

LOGARITHMIC HODGE STRUCTURES

(The terminology “fs monoid” comes from “finitely generated and saturated monoid.” If the reader feels this terminology strange, we propose the terminology “integral cone.”) 2.1.5 Fs logarithmic structures and charts. A logarithmic structure (M, α) on a ringed space X is fs if there exist an open covering (Uλ )λ of X and a family of pairs (Sλ , θλ )λ consisting of an fs monoid Sλ , regarded as a constant sheaf on Uλ , and of a homomorphism θλ : Sλ → M|Uλ of sheaves of monoids which induces an isomor∼ phism S˜λ − → M|Uλ . Here S˜λ denotes the logarithmic structure associated with the prelogarithmic structure Sλ → M|Uλ → OUλ . In this case, (Sλ , θλ ) is called a chart of M|Uλ . We give basic facts about fs logarithmic structures. × (i) If M is an fs logarithmic structure, then, for any x ∈ X, (MX /OX )x is a sharp gp × × gp fs monoid and (MX /OX )x = ((MX /OX )x ) is a finitely generated free abelian group. (ii) If M is an fs logarithmic structure and S → M is a chart, then for each ∼ × → )x is surjective and S(x)gp S/S(x)gp − x ∈ X the induced map S → (MX /OX × × (MX /OX )x where S(x) is the face of S defined as the inverse image of OX,x under S → MX,x . (iii) If M is an fs logarithmic structure and x ∈ X, then for some open neighbor∼ × )x . In particular, → (MX /OX hood U of x there is a chart S → M|U such that S − for an fs logarithmic structure M, locally there is a chart S → M with S a sharp fs monoid. (iv) If f : X → Y is a morphism of ringed spaces and (M, α) is an fs logarithmic structure on Y , the inverse image f ∗ (M, α) is also an fs logarithmic structure. If S → M is a chart, the induced homomorphism S → f ∗ M is also a chart.

A ringed space endowed with an fs logarithmic structure is called an fs logarithmic ringed space. In particular, an analytic space endowed with an fs logarithmic structure is called an fs logarithmic analytic space. 2.1.6 Examples. (i) The logarithmic structure M in Example 2.1.2 is an fs logarithmic structure.

In fact, let 1≤j ≤r zj = 0 be a local equation of Y in X such that each {zj = 0} is smooth. Then we have a chart % a(j ) θ zj . S := Nr −→ M, (a(j ))1≤j ≤r  → 1≤j ≤r

(ii) Let S be an fs monoid and let C[S] be the monoid ring (i.e., semigroup ring) of S over C. Then the toric variety X := Spec(C[S])an has the canonical fs logarithmic structure associated with the prelogarithmic structure S → C[S] ⊂ OX .

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2.1.7 Logarithmic differential forms. For an analytic space X, let 1X := ∗ (I/I 2 ) be the sheaf of Kähler differentials on X, where I is the sheaf of ideals of OX×X defining the image of the diagonal morphism  : X → X × X. For f ∈ OX , the class of pr ∗1 (f ) − pr ∗2 (f ) in 1X is denoted by df . Let X be an fs logarithmic analytic space. The sheaf of logarithmic differential 1-forms on X is defined by    gp 1 1 /N, ωX := X ⊕ OX ⊗Z MX where N is the OX -submodule generated by {(−dα(f ), α(f ) ⊗ f ) | f ∈ MX }. gp MX ,

1 the image of (0, 1 ⊗ f ) in ωX is denoted by d log(f ). For f ∈ For a morphism f : X → Y of fs logarithmic analytic spaces, define 1 1 := Coker(f ∗ ωY1 → ωX ). ωX/Y

Let q

ωX/Y :=

q &

1 ωX/Y

OX

for q ∈ N. q These sheaves ωX/Y are coherent as OX -modules. • We have the logarithmic de Rham complex (ωX/Y , d) of X over Y , where the differential d is characterized by the following properties. This d is compatible with the differential d of the usual de Rham complex ( •X/Y , d), d(d log(f )) = 0 gp q r for f ∈ MX , and d(f ∧ g) = df ∧ g + (−1)q f ∧ dg for f ∈ ωX/Y and g ∈ ωX/Y (q, r ∈ Z). 1 1 In the case Y = Spec(C) with the trivial logarithmic structure, ωX/Y = ωX , and • • (ωX/Y , d) is denoted as (ωX , d).

2.1.8 Examples. 1 is nothing but the sheaf 1X (log(Y )) of (i) In the standard example 2.1.2, ωX differential forms with logarithmic poles along Y .

(ii) If X is the toric variety Spec(C[S])an (S is an fs monoid) with the canonical logarithmic structure in 2.1.6 (ii), we have an isomorphism ∼

1 , → ωX OX ⊗Z S gp −

f ⊗ g  → f d log(g).

1 is free in this case (cf. [Od, 3.1]). In particular, ωX

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LOGARITHMIC HODGE STRUCTURES

2.1.9 By an fs logarithmic point, we mean an fs logarithmic analytic space whose underlying ringed space over C is Spec(C). If x is an fs logarithmic point, there is an fs monoid S such that S × = {1} and such that the logarithmic structure Mx of x is S × C× with α : Mx → C which sends (f, u) (f ∈ S, u ∈ C× ) to 0 if f  = 1 and to u if f = 1. We have ∼

→ ωx1 , c ⊗ f  → cd log(f ). C ⊗Z S gp  C ⊗Z (Mxgp /Ox× ) −

2.1.10 Fiber products. The category A(log) of fs logarithmic analytic spaces has fiber products. For fs logarithmic analytic spaces X, Y, Z and for morphisms X → Z and Y → Z, the fiber product X ×Z Y in this category is obtained as follows. Locally on X, Y, Z there are fs monoids Sj (j = 1, 2, 3), charts S1 → MX , S2 → MY , S3 → MZ , and homomorphisms S3 → S1 and S3 → S2 for which the following diagram is commutative. X   

−−−−→

Z   

←−−−−

Y   

Spec(C[S1 ])an −−−−→ Spec(C[S3 ])an ←−−−− Spec(C[S2 ])an In this local situation, let S4 be the push-out of S1 ← S3 → S2 in the categp gory of monoids, let S5 be the image of S4 in the group S4 = {ab−1 | a, b ∈ gp S4 } associated with S4 (note that S4 → S4 need not be injective), and let gp gp S6 = {a ∈ S4 = S5 | a n ∈ S5 for some n ≥ 1}. (This S6 is the push-out of S1 ← S3 → S2 in the category of fs monoids.) Let (X ×Z Y )
be the fiber product of the underlying analytic spaces of X, Y, Z taken in the category A of analytic spaces. Then we have the induced morphism of analytic spaces (X ×Z Y )
→ Spec(C[S4 ])an . The fiber product X ×Z Y in the category A(log) is the fiber product (X ×Z Y )
×Spec(C[S4 ])an Spec(C[S6 ])an in the category A endowed with the inverse image of the canonical logarithmic structure of Spec(C[S6 ])an . In general, the fiber product X ×Z Y in A(log) is obtained by gluing this local construction. The underlying analytic space of the fiber product X ×Z Y in A(log) need not coincide with the fiber product in the category A of analytic spaces. For example, if X and Y denote copies of Spec(C[T1 , T2 ])an which is endowed with the logarithmic structure associated with N2 → C[T1 , T2 ], (m, n)  → T1m T2n , and if X → Y denotes the morphism induced by C[T1 , T2 ] → C[T1 , T2 ], T1  → T1 T2 , T2  → T2 , then the fiber product X ×Y X in the category A(log) coincides with X (the diagonal morphism X → X ×Y X is an isomorphism) although the fiber product (X ×Y X)
of underlying spaces taken in A does not coincide with the diagonal. Here X ×Y X = X because the morphism of Mor( , X) → Mor( , Y ) of functors from A(log) to (Sets) is injective. In fact, for an fs logarithmic analytic space S,

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the map Mor(S, X) → Mor(S, Y ) is identified with the injective map from the set {(a, b) | a, b ∈
(S, MS )} to itself given by (a, b)  → (ab, b). To avoid confusion, when we consider in this book a fiber product in A(log) whose underlying analytic space differs from the fiber product in A, we will always state explicitly that we take the fiber product in A(log). The underlying analytic space of the fiber product X ×Z Y in A(log) coincides with the fiber product in the category A of analytic spaces if one of the following conditions (1) and (2) is satisfied. (1) At least one of the morphisms X → Z, Y → Z is strict. Here we say a morphism X → Z is strict if the logarithmic structure of X is the inverse image of that of Z via this morphism. If X → Z is strict, the fiber product X ×Z Y in A(log) coincides with the usual fiber product in A endowed with the inverse image of the logarithmic structure of Y . (2) The logarithmic structure of Z is trivial. In general, the canonical map from the fiber product X ×Z Y in A(log) to the fiber product (X ×Z Y )
is proper as a map of topological spaces, and the fibers of this map are finite. This fact is reduced to the case X = Spec(C[S1 ])an , Y = Spec(C[S2 ])an , Z = Spec(C[S3 ])an , X ×Z Y = Spec(C[S6 ])an , (X ×Z Y )
= Spec(C[S4 ])an in the above argument.

2.1.11 For a morphism f : X → Y of fs logarithmic analytic spaces, we say f is logarithmically smooth if locally on X and Y there are charts P → MX and Q → MY with P , Q fs monoids, and a homomorphism Q → P of monoids satisfying the following conditions (1)–(3). (1) The following diagram is commutative. Q   

−−−−→

P   

f −1 (MY ) −−−−→ MX . (2) The map Q → P is injective. (3) The morphism X → Y ×Spec(C[Q])an Spec(C[P ])an of analytic spaces is smooth. An fs logarithmic analytic space X is said to be logarithmically smooth if the evident morphism X → Spec(C) is logarithmically smooth for the trivial logarithmic structure of Spec(C). An fs logarithmic analytic space X is logarithmically smooth if and only if there are an open covering (Uλ )λ of X and an fs monoid Sλ for each λ

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LOGARITHMIC HODGE STRUCTURES

such that each Uλ is isomorphic to an open subset of Zλ := Spec(C[Sλ ])an endowed with the restrictions of OZλ and MZλ . log

2.2 RINGED SPACES (Xlog , OX ) log

In [KkNc], a ringed space (X log , OX ) is constructed associated with an fs logarithmic analytic space X. (This space X log was also considered independently in [KyNy].) Here we generalize the definition to logarithmic local ringed spaces X over C which belong to the category A1 (log) introduced below. The construction log of (X log , OX ) in this generalized situation is just similar to that in [KkNc]. log The reason why we generalize the definition of (X log , OX ) is the following. As is described in Chapter 0, the moduli spaces of polarized logarithmic Hodge structures, which are the main subjects of this book, need not belong to the category A(log) of fs logarithmic analytic spaces. They belong to B(log) (3.2.4 below), which seems to us the best category to work with in the theory of moduli of polarized logarithmic Hodge structures. We have A(log) ⊂ B(log) ⊂ A1 (log) For the definition of

log (X log , OX ),

(see Section 3.2).

A1 (log) seems to be the most natural category.

2.2.1 First, let A1 be the full subcategory of the category of local ringed spaces over C consisting of objects X satisfying the following condition (A1 ). (A1 ) For any open set U of X and for any n ≥ 0, the canonical map Mor(U, Cn ) → O(U )n

(1)

is bijective. Here Cn is regarded as an analytic space (and hence a local ringed space over C) in the standard way, and Mor means the set of morphisms in the category of local ringed spaces over C. The map (1) sends a morphism f : U → Cn to (f ∗ (zj ))1≤j ≤n ∈ O(U )n where the zj are the coordinate functions of Cn . For example, analytic spaces belong to A1 . Also, any topological space endowed with the sheaf of complex valued continuous functions belongs to A1 . Let A1 (log) be the full subcategory of the category of fs logarithmic local ringed spaces over C consisting of objects whose underlying local ringed spaces satisfy (A1 ). That is, an object of A1 (log) is an object of A1 endowed with an fs logarithmic structure. The condition (A1 ) is equivalent to the following condition: (A
1 ) For any open set U of X and any scheme Z locally of finite type over C, the canonical map Mor(U, Zan ) → Mor(U, Z) is bijective.

(2)

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CHAPTER 2

Here Zan denotes the analytic space associated with Z, and the map (2) is induced from the canonical morphism Zan → Z. The equivalence of (A1 ) and (A
1 ) is shown as follows. (A
1 ) implies (A1 ) because O(U )n = Mor(U, Spec(C[T1 , . . . , Tn ])), Spec(C[T1 , . . . , Tn ])an = Cn . Assume (A1 ) is satisfied. To prove that (A
1 ) is satisfied, we may assume Z is affine, and hence Z = Spec(C[T1 , . . . , Tn ]/(f1 , . . . , fm )) for some n, m ≥ 0 and for some f1 , . . . , fm ∈ C[T1 , . . . , Tn ]. Then Mor(U, Zan ) = {ϕ ∈ Mor(U, Cn ) | f1 (ϕ) = · · · = fm (ϕ) = 0}, Mor(U, Z) = {ϕ ∈ O(U )n | f1 (ϕ) = · · · = fm (ϕ) = 0}. This shows that (A
1 ) is satisfied.

2.2.2 We give remarks on objects X of A1 . First, for an open set U of X, the addition (resp. multiplication) of the ring O(U ) coincides with O(U ) × O(U ) = Mor(U, C2 ) → Mor(U, C) = O(U ), where the arrow is induced from the addition (resp. multiplication) C × C → C. From this we see that, for each x ∈ U , the map Map(U, C) → C, f  → f (x), induces a ring homomorphism O(U ) = Mor(U, C) → Map(U, C) → C, which we denote also by f  → f (x), and, in the limit, a ring homomorphism over C OX,x → C, which we denote again by f  → f (x). The kernel of this homomorphism is the unique maximal ideal of OX,x . Hence (1) For any x ∈ X, the residue field of the local ring OX,x is C. Next, exp : C → C× induces exp : OX = Mor(

exp

, C) −→ Mor(

× , C× ) = OX .

We have an exact sequence of sheaves (2)

2π i

exp

× → 1. 0 → Z −→ OX −→ OX

83

LOGARITHMIC HODGE STRUCTURES

Lastly, let S be an fs monoid. Then for an object X of A1 we have a canonical bijection Mor A1 (X, Spec(C[S])an )  Hom(S,
(X, OX )), where
(X, OX ) is regarded as a monoid by multiplication. In fact, for any local ringed space X over C, a morphism of local ringed spaces X → Spec(C[S]) (without ( )an ) over C corresponds bijectively to a homomorphism C[S] →
(X, OX ) of rings over C, and hence bijectively to a homomorphism S →
(X, OX ). If X belongs to A1 , we can add ( )an to Spec(C[S]). From this, we can deduce easily that for an object X of A1 (log) we have a canonical bijection Mor A1 (log) (X, Spec(C[S])an )  Hom(S,
(X, MX )). Here Spec(C[S]) is endowed with the canonical logarithmic structure (2.1.6 (ii)). 2.2.3 Let X be an object of A1 (log). As a set, X log is defined to be the set of all pairs (x, h) consisting of a point gp x ∈ X and an argument function h which is a homomorphism MX,x → S1 whose × is u  → u(x)/|u(x)|. Here S1 := {z ∈ C | |z| = 1}. Note that the restriction to OX,x × value u(x) ∈ C is defined as explained in (2.2.2). We endow Xlog with the weakest topology satisfying the following conditions (1) and (2). (1) The canonical map τ : Xlog → X, (x, h)  → x, is continuous. gp (2) For any open set U of X and for any f ∈
(U, MX ), the map τ −1 (U ) → 1 S , (x, h)  → h(f ) is continuous. The canonical map τ : Xlog → X is proper and surjective. The surjectivity is clear and the properness is seen as follows. Locally on X, take a chart S → MX . Then we have an injective map Xlog → X × Hom(S gp , S1 ), (x, h)  → (x, hS ), gp

where hS denotes the composite map S gp → MX,x → S1 . The image of this injection is closed, and the topology of X log coincides with the topology as a closed subset of X × Hom(S gp , S1 ). Since Hom(S gp , S1 ) is compact, the map τ is proper. For x ∈ X, the inverse image τ −1 (x) is homeomorphic to (S1 )r where r is the gp × rank of the abelian group (MX /OX )x . 2.2.4 log

We define the sheaf of rings OX on X log as follows. We will define first the sheaf gp log of logarithms L = LX of MX on Xlog , and will then define OX as a sheaf of τ −1 (OX )-algebras generated by L.

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CHAPTER 2

Let L be the fiber product of τ −1 (MX )    gp

Cont(

exp

, iR) −−−−→ Cont(

, S1 ),

where Cont( , T ), for a topological space T , denotes the sheaf on Xlog of contingp uous maps to T , and τ −1 (MX ) → Cont( , S1 ) comes from the definition of X log . We define OX := (τ −1 (OX ) ⊗Z SymZ (L))/a, log

(1)

where SymZ (L) denotes the symmetric algebra of L over Z, and a is the ideal of τ −1 (OX ) ⊗Z SymZ (L) generated by the image of τ −1 (OX ) → τ −1 (OX ) ⊗Z SymZ (L),

f  → f ⊗ 1 − 1 ⊗ ι(f ).

Here the map ι : τ −1 (OX ) → L is the one induced by τ −1 (OX ) → Cont(

, iR),

f  → 12 (f − f¯),

and exp

× ) ⊂ τ −1 (MX ). τ −1 (OX ) −→ τ −1 (OX gp

× In the above, the overbar indicates complex conjugation. Note that exp : OX → OX is defined as explained in 2.2.2. gp gp We denote the projection L → τ −1 (MX ) by exp, and the inverse τ −1 (MX ) → L/(2πiZ) by log. Then we have a commutative diagram with exact rows: exp

2π i

× 0 −−−−→ Z −−−−→ τ −1 (OX ) −−−−→ τ −1 (OX ) −−−−→ 1 '   '   ι ∩ ' ∩ 2π i

0 −−−−→ Z −−−−→

exp

(2)

−−−−→ τ −1 (MX ) −−−−→ 1.

L

gp

We have the evident morphism of ringed spaces over C log

τ : (X log , OX ) → (X, OX ). A morphism f : X → Y in A1 (log) induces a morphism log

log

f log : (Xlog , OX ) → (Y log , OY ) of ringed spaces over C in the obvious way. 2.2.5 log

For y ∈ Xlog , the stalk OX,y is described as follows. Let x = τ (y) ∈ X and gp × r = rank Z (MX /OX )x . Let (j )1≤j ≤r be a family of elements of Ly whose images gp × in (MX /OX )x under exp form a system of free generators. Then we have an

85

LOGARITHMIC HODGE STRUCTURES

isomorphism of OX,x -algebras ∼

log

→ OX,y , Tj  → j . OX,x [T1 , . . . , Tr ] − log

Note that OX,y is not a local ring if r ≥ 1. 2.2.6 For a morphism f : X → Y of fs logarithmic analytic spaces, we denote ωX/Y := τ ∗ (ωX/Y ) = OX ⊗τ −1 (OX ) τ −1 (ωX/Y ) q,log

log

q

q

•,log

for q ∈ Z (cf. 2.1.7). We have the logarithmic de Rham complex (ωX/Y , d) of Xlog over Y log , where the differential d is characterized by the following properties (1)–(3). • (1) This d is compatible with the differential d of (ωX/Y , d). gp

log

(2) For f ∈ MX , d sends log(f ) ∈ L/(2π iZ) ⊂ OX /(2π iZ) to d log(f ) ∈ 1,log 1 ⊂ ωX/Y (2.1.7). ωX/Y q,log

r,log

(3) d(f ∧ g) = df ∧ g + (−1)q f ∧ dg for f ∈ ωX/Y and g ∈ ωX/Y (q, r ∈ Z). gp

q

More explicitly, for f ∈ OX ,  ∈ L, g = exp() ∈ MX , and η ∈ ωX/Y , we have d(f n ⊗ η) = n ⊗ df ∧ η + nf n−1 ⊗ d log(g) ∧ η + f n ⊗ dη d(f ⊗ η) = df ∧ η + f dη.

for n ≥ 1,

2.2.7 Examples. (i) Let X := C and let z be the coordinate of X. Let M be the fs logarithmic structure on  X associated with the divisor {0} (see 2.1.2). We can take a chart × n N → MX = n≥0 OX z , n  → zn . We have an isomorphism ∼

Xlog − → (R≥0 ) × S1 , (x, h)  → (|z(x)|, h(z)). We have OX = τ −1 (OX )[log(z)] and ωX log

1,log

= τ −1 (OX )[log(z)]d log(z).

(ii) More generally, for a toric variety X = Spec(C[S])an with the canonical logarithmic structure in 2.1.6 (ii), we have X  Hom(S, Cmult ), mult X log  Hom(S, R≥0 ) × Hom(S gp , S1 ), τ : Xlog → X is identified with the map induced by (R≥0 ) × S1 → C, (r, u)  → ru, log log OX = τ −1 (OX )[log(f ); f ∈ S] ⊂ j∗ OU (j log : U = Spec(C[S gp ])an → X log ), ∼ log 1,log → ωX , f ⊗ g  → f d log(g). OX ⊗Z S gp −

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CHAPTER 2 log

(iii) We have local descriptions of Xlog and OX for a general object X of A1 (log) by using the example in (ii). Locally on X, take a chart S → MX with S an fs monoid. Then as is explained in 2.2.2, we have the corresponding morphism f : X → Z := Spec(C[S])an of objects of A1 (log). This induces a continuous map X log → Z log , and we have a homeomorphism ∼

Xlog − → X ×Z Z log . Furthermore, we have an isomorphism log ∼

log

OX ⊗OZ OZ − → OX , log

where we denote the inverse images of OX , OZ , and OZ on X log simply by OX , log OZ , and OZ . We have ∼

q

q

OX ⊗OZ ωZ − → ωX log OX

q ∼ → ⊗OZ ωZ −

on X,

log OX

q,log ∼

⊗Olog ωZ Z

q,log

− → ωX

on Xlog .

2.2.8 gp

Let x be an fs logarithmic point (2.1.9). Then x log  (S1 )r where r = rank(Mx / log Ox× ). The sheaf Ox is a locally constant sheaf on x log which is locally isomorphic to C[T1 , . . . , Tr ]. For an object X of A1 (log) and for x ∈ X, we usually regard x as an fs logarithmic point whose logarithmic structure is the inverse image of the logarithmic structure of X. We identify x log = τ −1 (x). 2.2.9 Let x be an fs logarithmic point and let r = rank(Mx /Ox× ). Since x log  (S1 )r , the fundamental group π1 (x log ) of x log is isomorphic to Zr . We define a canonical isomorphism gp

π1 (x log )  Hom(Mxgp /Ox× , Z) as follows. Taking a base point h0 ∈ x

log

(1)

, h0 : Mx → S , we have a homeomorphism 1

∼

x log − → Hom(Mxgp /Ox× , S1 ), h  → hh−1 0 . The exact sequence 2π i

exp

0 → Z −→ iR −→ S1 → 1 induces an exact sequence 2π i

exp

0 → Hom(Mxgp /Ox× , Z) −→ Hom(Mxgp /Ox× , iR) −→ Hom(Mxgp /Ox× , S1 ) → 1. This shows that the r-dimensional R-vector space Hom(Mx /Ox× , iR) is the unigp versal covering of the real torus x log  Hom(Mx /Ox× , S1 ) whose fundamental gp group is π1 (x log )  Hom(Mx /Ox× , Z), which acts on the universal covering as gp

87

LOGARITHMIC HODGE STRUCTURES

translations by (2πi)-multiples of integral points. Note that the last isomorphism is independent of the choice of h0 . We define π1+ (x log ) := Hom(Mx /Ox× , N) ⊂ π1 (x log ).

(2)

Then, π1 (x log ) = π1+ (x log )gp . For f ∈ Mx =
(x, Mx ), since logarithms of f in the sheaf Lx on x log form a locally constant sheaf on x log , the fundamental group π1 (x log ) acts on local sections log(f ) of Lx . For γ ∈ π1 (x log ), we have γ (log(f )) = log(f ) − 2πi[γ , f ] gp where [ , ] : π1 (x log ) × Mx → gp log × π1 (x )  Hom(Mx /Ox , Z).

(not log(f ) + 2π i[γ , f ], cf. Appendix A1),

Z is the pairing given by the above isomorphism

For example, if x = 0 ∈ C where C is endowed with the logarithmic structure associated to the divisor {0} ⊂ C, then Mx /Ox×  N is generated by the image of the coordinate function q of C, and the generator of π1+ (x log ) = Hom(Mx /Ox× , N)  N, which sends q to 1, is the class of the counterclockwise directed circle γ : [0, 1] → x log = {0} × S1 , γ (t) = (0, exp(2π it)). We have γ (log(q)) = log(q) − 2π i (not log(q) + 2πi). Some reader may feel that the sign here is strange. See Appendix A1 for a discussion of the sign problem. Proposition 2.2.10 For any object X of A1 (log), we have log

τ∗ (OX ) = OX ,

log

R m τ∗ (OX ) = 0 for m ≥ 1.

Proof. (Proofs for fs logarithmic analytic spaces X are given in [Ma1, 4.6] and [IKN, (3.7) (3)].) Let x ∈ X. Replacing X by an open neighborhood of x in X, gp gp × take a family (qj )1≤j ≤r of elements of
(X, MX ) whose image in (MX /OX )x is a gp × log Z-basis of (MX /OX )x . Consider the sheaf of rings on X log

R = C[log(q1 ), . . . , log(qr )] ⊂ OX . log

Here log(qj ) are determined only modulo 2π i · Z, but the subsheaf R of OX is independent of the local choices of the branches of log(qj ) and hence defined globally. ( log Then, by 2.2.5, the germ of the homomorphism OX C R → OX at any y ∈ Xlog −1 log lying over x is bijective. Let ι : τ (x) → X be the inclusion map. Then, by the proper base change theorem (Appendix A2),    log m m (R τ∗ (OX ))x  R τ∗ OX ⊗C R x

 OX,x ⊗C (R τ∗ R)x  OX,x ⊗C H m (τ −1 (x), ι−1 R) m

It is sufficient to prove that H 0 (τ −1 (x), ι−1 R) = C and H m (τ −1 (x), ι−1 R) = 0 for m ≥ 1. Since ι−1 R is a locally constant sheaf on τ −1 (x) = x log , H m (τ −1 (x), ι−1 R)  H m (π1 (x log ), Ry ) where y is a point of X log lying over x, Ry is the stalk of R at y, and H m (π1 (x log ), Ry ) is the mth cohomology of the module Ry over the group π1 (x log ). Let (γj )1≤j ≤r be

88

CHAPTER 2

× )x , Z) which is dual to a Z-basis of the abelian group π1 (x log )  Hom((MX /OX gp × × the Z-basis (qj mod OX,x )j of (MX /OX )x . Then γj (log(qk )) = log(qk ) − 2π iδj k where δj k is the Kronecker symbol (see 2.2.9 and Appendix A1). For 0 ≤ j ≤ r, let
j be the subgroup of π1 (x log ) generated by γk (j < k ≤ r) and let Rj = C[log(q1 ), . . . , log(qj )] ⊂ Ry . Hence
0 = π1 (x log ) and R0 = C. We prove R
(
j , Ry ) = Rj by downward induction on j . The case j = r is clear (note that
r = {1}). We go from j (1 ≤ j ≤ r) to j − 1. Assume R
(
j , Ry ) = Rj . Then we have gp

R
(
j −1 , Ry ) = R
(
j −1 /
j , R
(
j , Ry )) = (the complex [γj − 1 : Rj → Rj ]). As is easily seen, γj − 1 : Rj → Rj is surjective and the kernel is Rj −1 .

2

2.3 LOCAL SYSTEMS ON X log In this section, we consider local systems on Xlog for objects X of A1 (log) (2.2.1). We prove that, if L is a locally constant sheaf on Xlog of free Z-modules of finite rank with “unipotent local monodromy” (2.3.1 below), then locally on X, L is embedded in log

OX ⊗ L0 in a special way, where L0 is a stalk of L regarded as a constant sheaf. From the next section, we will consider “logarithmic Hodge structures” and “polarized logarithmic Hodge structures” by putting Hodge filtrations on local systems on Xlog . 2.3.1 Let L be a locally constant sheaf on X log . For x ∈ X and y ∈ Xlog lying over x, we call the action of π1 (x log ) = π1 (τ −1 (x)) on Ly the local monodromy of L at y. Assume that L is a locally constant sheaf of abelian groups on Xlog . We say the local monodromy of L is unipotent if the local monodromy of L at y is unipotent for any y ∈ X log . Proposition 2.3.2 Let X be an object of A1 (log), let A be a subring of C, and let L be a locally constant sheaf on X log of free A-modules of finite rank. Let x ∈ X, let y be a point of X log lying over x, and assume that the local monodromy of L at gp y is unipotent. Let (qj )1≤j ≤n be a finite family of elements of MX,x whose image in gp × (MX /OX )x is a Z-basis, and let (γj )1≤j ≤n be the dual Z-basis of π1 (x log ) in the duality in 2.2.9. Then if we replace X by some open neighborhood of x, we have an log isomorphism of OX -modules log

∼

log

→ OX ⊗A L0 , ν : OX ⊗A L −

L0 = the stalk Ly ,

where L0 is regarded as a constant sheaf, satisfying the following condition (1). Let Nj = log(γj ) : L0,Q → L0,Q ,

89

LOGARITHMIC HODGE STRUCTURES gp

lift qj in
(X, MX ) (by replacing X by an open neighborhood of x), and let   n  ∼ log log → OX ⊗A L0 . ξ = exp  (2πi)−1 log(qj ) ⊗ Nj  : OX ⊗A L0 − j =1

n

Note that the operator ξ = exp(

j =1 (2π i)

−1

log(qj ) ⊗ Nj ) depends on the local

choices of the branches of log(qj ), but that the subsheaf ξ −1 (1 ⊗ L0 ) of OX ⊗A L0 , which we consider in (1) below, is independent of the choices and hence is defined globally on Xlog . log

(1) The restriction of ν to L = 1 ⊗ L induces an isomorphism of locally constant sheaves ∼

→ ξ −1 (1 ⊗ L0 ). ν:L− If we fix branches of the germs log(qj )y at y (1 ≤ j ≤ n), we can take an isomorphism ν satisfying (1) as above which satisfies furthermore the following condition (2). (2) The branch of ξy defined by the fixed branches of log(qj )y satisfies νy (1 ⊗ v) = ξy−1 (1 ⊗ v) for any v ∈ L0 . Proof. Let L
be the locally constant subsheaf ξ −1 (1 ⊗ L0 ) of OX ⊗ L0 . Fix a branch of log(qj )y at y for 1 ≤ j ≤ n, and let ν : Ly → (L
)y be the isomorphism of A-modules v  → ξy−1 (1 ⊗ v) where ξy is defined by the fixed branches of log(qj )y . Then ν preserves the local monodromy actions of π1 (x log ) on these stalks of the locally constant sheaves L and L
. In fact, for v ∈ L0 and for 1 ≤ k ≤ n, log

γk (ξy−1 (1 ⊗ v) in L
y ) = γk (ξy )−1 · (1 ⊗ v)    n  = exp −  ((2π i)−1 log(qj )y − δj k ) ⊗ Nj  · (1 ⊗ v) j =1

=

ξy−1

exp(1 ⊗ Nk )(1 ⊗ v) = ξy−1 (1 ⊗ γk (v in Ly ))

(δjk is the Kronecker symbol). Hence there is a unique isomorphism ν : L|x log → L
|x log between the pullbacks of L and L
to x log which induces the above isomorphism ν on the stalks at y. By the proper base change theorem (Appendix A2) applied to the proper map τ : Xlog → X and to the sheaf F of isomorphisms from L to L
on Xlog , the iso∼ → L
if we replace X by some morphism ν extends to an isomorphism ν : L − open neighborhood of x in X. This isomorphism ν induces an isomorphism of log OX -modules ∼

ν : OX ⊗A L − → OX ⊗A L
= OX ⊗A L0 . log

log

log

2

Example. Let f : E →  be the degenerating family of elliptic curves in 0.2.10 log and consider the locally constant sheaf L = R 1 f∗ (Z) on log . In 2.3.2, take

90

CHAPTER 2

X = , x = 0 ∈ , A = Z, and take the coordinate function q of  as q1 (n = 1 in this situation). Then the element γ1 , which we denote here by γ , is the positive generator of π1 (log ) (represented by a circle in ∗ in the counterclockwise direction; cf. Appendix A1). As is explained in 0.2, L has a Z-basis (e1 , e2 ) locally (e1 is defined globally but e2 is determined by a local choice of the branch of log(q)). Fix a branch of log(q) at y and take the corresponding e2,y . We have γ (e1,y ) = e1,y , γ (e2,y ) = e1,y + e2,y , N (e1,y ) = 0, N (e2,y ) = e1,y , where N = log(γ ). log

log

The OX -module OX ⊗Z L has a global base (1 ⊗ e1 , ω) as in 0.2.15. We have an log isomorphism of OX -modules log

∼

log

ν : OX ⊗ L − → OX ⊗ L0 ,

1 ⊗ e1 → 1 ⊗ e1,y , ω  → e2,y .

This ν has the property stated in Proposition 2.3.2 globally on . In fact, since ω = (2πi)−1 log(q)e1 + e2 , ν sends e1 to 1 ⊗ e1,y = ξ −1 (1 ⊗ e1 ) and e2 to −(2πi)−1 log(q)e1,y + e2,y = ξ −1 (1 ⊗ e2,y ). Proposition 2.3.3 Let X be an object of A1 (log) and let L be a locally constant sheaf of finite-dimensional C-vector spaces on X log . (i) If the local monodromy of L is unipotent, the OX -module   log M := τ∗ OX ⊗C L is locally free of finite rank, and we have an isomorphism ∼

OX ⊗τ −1 (OX ) τ −1 (M) − → OX ⊗C L. log

log

(ii) Conversely, assume that there are a locally free OX -module M of finite rank log log log on X and an isomorphism of OX -modules OX ⊗τ −1 (OX ) τ −1 (M)  OX ⊗C L. ∼

log

Then the local monodromy of L is unipotent and M − → τ∗ (OX ⊗C L). Proof. We prove (i). By Proposition 2.3.2, locally on X, we have an isomorphism of log log log OX -modules OX ⊗C L  OX ⊗C L0 for some finite-dimensional C-vector space L0 regarded as a constant sheaf on X log . Hence     log log τ∗ OX ⊗C L  τ∗ OX ⊗C L0 = OX ⊗C L0 by proposition 2.2.10. ∼ log → τ∗ (OX ⊗C L) is proved as We prove (ii). The fact that M −     log log log τ∗ OX ⊗C L  τ∗ OX ⊗τ −1 (OX ) τ −1 (M) = τ∗ (OX ) ⊗OX M = M by proposition 2.2.10. It remains to prove that the local monodromy of L is unipotent. For this we may assume that X is an fs logarithmic point. In this log case τ −1 (M) is a constant sheaf, L is embedded in OX ⊗C τ −1 (M), and

91

LOGARITHMIC HODGE STRUCTURES

(γ − 1)n (log(q1 ) · · · log(qm )) = 0 if n > m for any γ ∈ π1 (x log ) and any gp q1 , . . . , qm ∈ MX,x . Hence the action of π1 (x log ) in L is unipotent. 2 Proposition 2.3.4 Let X be an object of A1 (log) (2.2.1), let A be a subring of C containing Q, and let L be a locally constant sheaf on Xlog of free A-modules of finite rank. Assume that the local monodromy of L is unipotent. (i) There exists a unique A-homomorphism × )⊗L N : L → (MX /OX gp

× satisfying the following condition (1). Here we denote the inverse image of MX /OX gp × on X log simply by MX /OX . gp

(1) For any x ∈ X, any y ∈ Xlog lying over x, and for any γ ∈ π1 (x log ), if hγ : gp × (MX /OX )x → Z denotes the homomorphism corresponding to γ (2.2.9), the N

hγ

× composition Ly − → (MX /OX )x ⊗ Ly −→ Ly coincides with the logarithm of the action of γ on Ly . gp

(ii) Assume that X is an fs logarithmic point {x}. Let N
: L → ωx1 ⊗A L be the composition of N and the C-linear map (Mx /Ox× ) ⊗ L → ωx1 ⊗ L, f ⊗ log 1,log v  → (2πi)−1 d log(f ) ⊗ v, and let 1 ⊗ N
: Ox ⊗A L → ωx ⊗A L be the log log Ox -linear homomorphism induced by N
. Let M := H 0 (x log , Ox ⊗A L) = log log τ∗ (OX ⊗A L). Then the restriction M → ωx1 ⊗C M of d ⊗ 1L : Ox ⊗A L → 1,log
ω ⊗A L coincides with the restriction of 1 ⊗ N to M. gp

Proof. (i) We may assume that L = ξ −1 (1 ⊗ L0 ) with the notation in 2.3.2. Define gp × N : L → (MX /OX ) ⊗ L by N (ξ −1 (1 ⊗ v)) =

n 

qj ⊗ ξ −1 (1 ⊗ Nj (v))

j =1

(then N is independent of the choice of the branch of ξ ). We show that N has the property stated in (i). Let x
∈ X, y
being a point of X log lying over x
, γ ∈ gp × )x
→ Z be the homomorphism corresponding to π1 ((x
)log ), and let hγ : (MX /OX
γ . Then, for v ∈ L0 and for any branch of ξy−1
at y , the local monodromy action of −1 −1 γ on Ly
satisfies log(γ )(ξy
(1 ⊗ v)) = (log(γ )(ξy−1
))(1 ⊗ v) and log(γ )(ξy
) =

n −1 j =1 hγ (qj )ξy
(1 ⊗ Nj ). log

(ii) Since Ox is the union of C-subsheaves that are locally constant sheaves of finite dimensional C-vector spaces, the collection of N of these subsheaves gives log gp log log 1,log N : Ox → (Mx /Ox× ) ⊗ Ox and N
: Ox → ωx . It is easy to see that the last homomorphism coincides with −d. Since the local monodromy acts trivially log log on the images of M in the stalks of Ox ⊗A L, (N
of Ox ) ⊗ 1 + 1 ⊗ (N
of L)

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CHAPTER 2

on Ox ⊗A L induces the zero map on M ⊂ Ox ⊗A L. Hence −d ⊗ 1L + 1 ⊗ N
is zero on M. 2 Proposition 2.3.5 (Ogus [Og, Theorem 3.1.2]) Let X be a logarithmically smooth, fs logarithmic analytic space, let U := Xtriv be the open set of X consisting of all points at which the logarithmic structure is trivial (that is, U = {x ∈ × X | MX,x = OX,x }), and let j log : U → Xlog be the inclusion map. Then the restriction to U gives an equivalence from the category of locally constant sheaves on Xlog to the category of locally constant sheaves on U . The inverse functor is given by (j log )∗ . ∼ → π1 (X log , u) for any u ∈ U . If X is connected, π1 (U, u) − log

log

Proof. It is sufficient to prove that for any y ∈ Xlog and any neighborhood V of y, there is a contractible open neighborhood W ⊂ V of y such that W ∩ U is contractible. We may assume X = Spec(C[S])an for some fs monoid S such that S × = {1}, endowed with the canonical logarithmic structure, and y = (0, 1) ∈ mult ) × Hom(S gp , S1 ) where 0 denotes the homomorphism which Xlog = Hom(S, R≥0 sends all elements of S − {1} to 0 ∈ R≥0 and 1 denotes the homomorphism which sends all elements of S gp to 1. Take a finite family (aj )1≤j ≤n of elements of S − {1} which generates S, and a sufficiently small contractible open neighborhood C ⊂ Hom(S gp , S1 ) of 1 ∈ Hom(S gp , S1 ). If we take c > 0 sufficiently small and define mult ) × C | r(aj ) < c (∀j )} ⊂ X log , W := {(r, u) ∈ Hom(S, R≥0

then W is a contractible open neighborhood of y such that W ⊂ V , and W ∩ U = 2 {(r, u) ∈ W | r(aj )  = 0 (∀j )} is contractible. In [KjNc], it is shown that, under the assumption of Proposition 2.3.5, Xlog is a topological manifold with the boundary Xlog − U . 2.3.6 The canonical extension of Deligne in [D1] can be understood from our logarithmic point of view, as follows. Let X be a logarithmically smooth fs logarithmic analytic space. Let L be a locally constant sheaf of finite-dimensional C-vector spaces on U = Xtriv with unipotent local monodromy along X-U . Let L
be the unique locally constant sheaf on Xlog whose restriction on U is L (2.3.5). Then the local monodromy of L
is unipotent in the sense of 2.3.1. The canonical extension of Deligne, M, of log OU ⊗C L to X is τ∗ (OX ⊗C L
). The canonical extension M has a connection with log 1 ⊗OX M. This ∇ is induced by d ⊗ 1 : OX ⊗C logarithmic poles ∇ : M → ωX 1,log

L → ωX ⊗C L . 2.3.7 The isomorphism ν in 2.3.2 appears locally on X depending on the local choice of (qj )j . Here we see that, in the case X = Spec(C[S])an , a canonical ν exists globally on X.

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LOGARITHMIC HODGE STRUCTURES

Let S be an fs monoid, X = Spec(C[S])an with the canonical logarithmic structure, and let U = Xtriv = Spec(C[S gp ])an . Then U = Hom(S gp , C× ), and via the exact sequence 0 → Hom(S gp , Z) → Hom(S gp , C) → Hom(S gp , C× ) → 0 (the third arrow is induced from C → C× , z  → exp(2π iz)), Hom(S gp , C) is regarded as a universal covering of U and the fundamental group of U is identified with Hom(S gp , Z). Let A be a subring of C, let L be a locally constant sheaf on Xlog of free A-modules of finite rank with unipotent local monodromy, and let L0 be the stalk of L at the unit point 1 ∈ U = Hom(S gp , C× ) regarded as a constant sheaf on X log . Then there log is a unique isomorphism of OX -modules log

∼

log

→ OX ⊗A L0 ν : OX ⊗A L − satisfying the following conditions (1) and (2) for any finite family (qj )1≤j ≤n of elements of S gp which is a Z-basis of S gp /(torsion). Let (γj )1≤j ≤n be the Z-basis of π1 (U, 1) = Hom(S gp , Z) which is dual to (qj )1≤j ≤n , and let Nj : L0,Q → L0,Q be the logarithm of γj .

(1) ν(1 ⊗ L) = ξ −1 (1 ⊗ L0 ) with ν = exp( nj=1 (2π i)−1 log(qj ) ⊗ Nj ). (2) Let log(qj )1,0 be the branch of the germ of log(qj ) at 1 ∈ U which has the value 0 at 1, and let ξ1,0 = exp( nj=1 (2π i)−1 log(qj )1,0 ⊗ Nj ). Then the map ξ1,0 ◦ νy : 1 ⊗ Ly → 1 ⊗ L0 is the identity map. The proof is similar to that of 2.3.2. First fix (qj )1≤j ≤n . For the locally constant log −1 subsheaf L
= ξ −1 (1 ⊗ L0 ) of OX ⊗ L0 , the isomorphism ξ0,1 : L1 → L
1 of stalks preserves the actions of π1 (X log , 1)  π1 (U, 1), and it is extended uniquely to an ∼ log → L
on Xlog . This induces an isomorphism of OX -modules isomorphism ν : L − ∼ log log → OX ⊗A L
= OX ⊗A L0 . It is easy to check that ν is independent ν : Olog ⊗A L − of the choice of (qj )1≤j ≤n . 2.3.8 The following variant of 2.3.7 appears in 2.5.15. Let X = n , U = (∗ )n , and endow X with the logarithmic structure associated with the divisor X − U . Let L be a locally constant sheaf on Xlog of free A-modules of finite rank and assume that the local monodromy of L along X − U is unipotent. Let p ∈ U and fix a lifting p˜ ∈ hn of p (h is the upper half plane) for the surjection hn → (∗ )n , (zj )1≤j ≤n  → (exp(2πizj ))1≤j ≤n . Let L0 be the stalk Lp regarded as a constant sheaf on X log . log Then there is a unique isomorphism of OX -modules log

∼

log

→ OX ⊗ L0 ν : OX ⊗A L − satisfying the following conditions (1) and (2).

(1) ν(1 ⊗ L) = ξ −1 (1 ⊗ L0 ) where ξ = exp( nj=1 (2π i)−1 log(qj ) ⊗ Nj ) with (qj )1≤j ≤n the standard coordinate of n , Nj = log(γj ) with (γj )1≤j ≤n the standard basis of π1 (X log )  π1 ((∗ )n ) (in the counterclockwise direction).

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CHAPTER 2 ∼

→ 1 ⊗ L0 is the identity map where ξp,0 is the (2) The map ξp,0 ◦ νp : 1 ⊗ Lp − branch of ξp defined by the branch of ((2π i)−1 log(qj )p )1≤j ≤n whose values at p is p. ˜ The proof is essentially the same as that of 2.3.7.

2.4 POLARIZED LOGARITHMIC HODGE STRUCTURES In this section, X denotes an object of A1 (log) (2.2.1). Here we define the notion “polarized logarithmic Hodge structure (PLH) on X.” We will define the notion of a “logarithmic Hodge structure on X” later in Section 2.6. Let w and (hp,q ) = (hp,q )p,q∈Z be as in Section 0.7. Definition 2.4.1 A prelogarithmic Hodge structure (pre-LH) on X of weight w pair (HZ , F ) consisting of a locally constant sheaf of and of Hodge type (hp,q ) is a free Z-modules HZ of rank p,q hp,q on X log and of a decreasing filtration F of log

log

the OX -module OX ⊗ HZ , which satisfy the following condition. (1) There exist an OX -module M on X and a decreasing filtration (Mp )p∈Z on M by OX -submodules such that Mp /Mp+1 is locally free of rank hp,w−p log for each p, τ ∗ M = OX ⊗Z HZ , τ ∗ (Mp ) = F p . Here τ ∗ is the module-theoretic inverse image OX ⊗τ −1 (OX ) τ −1 ( ). log

Proposition 2.4.2 Let (HZ , F ) be a pre-LH on X. Then the local monodromy of HZ is unipotent. 2

Proof. This follows from 2.3.3 (ii).

Definition 2.4.3 A prepolarized logarithmic Hodge structure (pre-PLH) on X of weight w and of Hodge type (hp,q ) is a triple (HZ ,  , , F ) consisting of a pre-LH (HZ , F ) and of a nondegenerate Q-bilinear form  ,  on HQ = Q ⊗ HZ , symmetric for even w and antisymmetric for odd w, which satisfy the following condition. (1) F p , F q  = 0 if p + q > w. log

Here  ,  is regarded as the natural extension to OX -bilinear form. For a morphism Y → X in A1 (log) and for a pre-LH (resp. pre-PLH) on X, its inverse image on Y , which is a pre-LH (resp. pre-PLH) on Y , is defined evidently. Proposition 2.4.4 (i) Let H be the category of pre-LH on X of weight w and of Hodge type (hp,q ), and let H
be the category of triples

(HZ , M, ι) where HZ is a locally constant sheaf of free Z-modules HZ of rank p,q hp,q on X log , M is an OX -module on X endowed with a decreasing filtration (Mp )p by OX -submodules such that Mp /Mp+1 is

95

LOGARITHMIC HODGE STRUCTURES log

locally free of rank hp,w−p for each p, and ι is an isomorphism of OX -modules ∼ log → OX ⊗Z HZ . Then we have an equivalence of categories τ ∗ (M) − ∼

→ H, H
−

(HZ , M, ι)  → (HZ , F ),

F p = ι(τ ∗ (Mp )). log

The inverse functor is given as (HZ , F )  → (HZ , M, ι) where M = τ∗ (OX ⊗Z HZ ), ∼ log → Mp = τ∗ (F p ), and ι is the canonical isomorphism OX ⊗τ −1 (OX ) τ −1 (M) − log OX ⊗Z HZ . (ii) Let P be the category of pre-PLH on X of weight w and of Hodge type (hp,q ), and let P
be the category of 4-tuples (HZ ,  , , M, ι) where (HZ , M, ι) is an object of H
and  ,  is a nondegenerate Q-bilinear form on HQ = Q ⊗ HZ , symmetric for even w and antisymmetric for odd w, which satisfy the condition τ ∗ Mp , τ ∗ Mq  = 0 if p + q > w. Then we have an equivalence of categories ∼

→ P, P
−

F p = ι(τ ∗ (Mp )).

(HZ ,  , , M, ι)  → (HZ ,  , , F ),

The inverse functor is given in a similar manner as in (i). 2

Proof. This follows from 2.4.2 and 2.3.3 (i). 2.4.5

Let X be an fs logarithmic analytic space. Let H = (HZ , F ) be a pre-LH on X. Let log

1,log

d ⊗ 1 = d ⊗ 1HC : OX ⊗C HC → ωX

⊗C H C .

We say that H satisfies the Griffiths transversality over X if 1,log

(d ⊗ 1)(F p ) ⊂ ωX

⊗Olog F p−1

for all p.

X

2.4.6 Let x ∈ X (X is an object of A1 (log)) and let Mx → Ox = C be the fs logarithmic log structure on x induced from that on X. Let (x log , Ox ) be the associated ringed log −1 space. For y ∈ x = τ (x), we define log sp(y) := HomC-alg (Ox,y , C).

(1)

For s ∈ sp(y), we have a well-defined homomorphism s˜ : Mxgp → C× ,

a  → exp(s(log(a))),

(2)

log

where log(a) is defined in Ly /(2π iZ) and its image in Ox,y /(2π iZ) is also denoted by log(a) (see 2.2.4). For a pre-LH (HZ , F ) on X, for x ∈ X, for y ∈ x log and for s ∈ sp(y), we have a decreasing filtration  log F (s) := C ⊗Olog (F p |x log )y p∈Z , with s : Ox,y → C, (3) x,y

on the C-vector space HC,y = C ⊗Z HZ,y , where F p |x log = Ox ⊗k−1 (Olog ) k −1 (F p ) log

with k : x log → X log .

X

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Definition 2.4.7 Let x be an fs logarithmic point (2.1.9). A pre-PLH (HZ ,  , , F ) on x of weight w is called a polarized logarithmic Hodge structure (PLH) on x of weight w, if it satisfies the following two conditions. (1) (HZ , F ) satisfies the Griffiths transversality over x (cf. 2.4.5). (2) (Positivity) Let y ∈ x log and s ∈ sp(y). If s˜ : Mx → C (see 2.4.6 (2)) is sufficiently near to the structure morphism of the logarithmic structure α : Mx → C in the topology of simple convergence of C-valued functions, then (HZ,y ,  , y , F (s)) is a polarized Hodge structure of weight w in the usual sense 1.1.2. Note that the validity of the condition 2.4.7 (2) is independent of the choice of log y. In fact, let y, y
∈ x log . Since Ox is a locally constant sheaf, a path from y to y
log ∼ log → Ox,y by rewinding the path (cf. Appendix in x log induces an isomorphism Ox,y
− A1). This induces a bijection sp(y) → sp(y
) which preserves the corresponding gp morphisms Mx → C× (see 2.4.6 (2)). Furthermore, if (HZ , F ) is a pre-LH on x, ∼ → HZ,y . The induced isomorphism the path also induces an isomorphism HZ,y
− ∼ HC,y
− → HC,y sends F (s
) to F (s), if sp(y) → sp(y
) sends s to s
. Definition 2.4.8 A pre-PLH (HZ ,  , , F ) on X is a polarized logarithmic Hodge structure (PLH) on X if, for any x ∈ X, the inverse image of (HZ ,  , , F ) on x is a PLH. Here we regard x as an fs logarithmic point endowed with the inverse image of the logarithmic structure of X. 2.4.9 Assume that X is an fs logarithmic analytic space. Let (HZ , F ) be a pre-LH on X. We have two types of Griffiths transversality for (HZ , F ). One is the Griffiths transversality over X as in 2.4.5, which we call the big Griffiths transversality. The other is the condition that the inverse image of (HZ , F ) on each x ∈ X satisfies the Griffiths transversality over x, which we call the small Griffiths transversality. If the logarithmic structure of X is trivial, then, for any x ∈ X, the sheaf ωx1 of logarithmic 1-forms on x is 0 and hence any pre-LH on x satisfies the small Griffiths transversality automatically. For a pre-LH (HZ , F ) on X, the big Griffiths transversality is much stronger than the small Griffiths transversality. When X is a smooth analytic space with trivial logarithmic structure, a PLH on X satisfying the big Griffiths transversality is nothing but a variation of polarized Hodge structure. So, when X is a logarithmically smooth fs logarithmic analytic space (2.1.11), we call a PLH on X satisfying the big Griffiths transversality a logarithmic variation of polarized Hodge structure (LVPH). 2.4.10 We already explained in 0.2.21 how LVPH appears in geometry. The LVPH in 0.2.21 is a higher direct image of a constant LVPH. Higher direct images of more general LVPHs are discussed in [KMN].

97

LOGARITHMIC HODGE STRUCTURES

2.5 NILPOTENT ORBITS AND PERIOD MAPS Here we show that the notion “PLH on an fs logarithmic point” is essentially equivalent to the notion “nilpotent orbit.” We also discuss period maps of PLH. At the end of this section, we give a PLH-theoretic interpretation of the nilpotent orbit theorem of Schmid [Sc]. In this section, we mainly work on an object of A1 (log) (2.2.1). Proposition 2.5.1 Let x be an fs logarithmic point (2.1.9) and let H = (HZ ,  , , F ) be a pre-PLH on x. Let y ∈ x log , let s, s
∈ sp(y), and write the homomorphism Mxgp /Ox× = Mxgp /C× → C, f  → (2π i)−1 (s
(log(f )) − s(log(f ))),

gp in the form nj=1 zj hγj with zj ∈ C and γj ∈ π1 (x log ), where hγj : Mx /Ox× → Z is the homomorphism corresponding to γj (2.2.9). Then   n  zj Nj  F (s), F (s
) = exp  j =1

where Nj : HQ,y → HQ,y is the logarithm of the action of γj . Proof. Take a family (qj )1≤j ≤n of elements of Mx whose image in Mx /Ox× is a Zbasis, and let (γj )1≤j ≤n be the basis of π1 (x log ). Then the above homomorphism

dual n × Mx /Ox → C has the form j =1 (2π i)−1 (s
(log(qj )) − s(log(qj )))hγj . Hence it is sufficient to prove gp

gp

F (s
) = ξy (s
)ξy (s)−1 F (s) log

(1) ∼

log

→ Ox ⊗ HZ,y be as in where ξ is as in 2.3.2. Let the isomorphism ν : Ox ⊗ HZ − −1 2.3.2 and let ξy,0 be the branch of ξy for which 1 ⊗ v = ν −1 ◦ ξy,0 (1 ⊗ v) for any v ∈ HZ,y . By 2.4.4, the filtration F has the form ν −1 (Ox ⊗C F0 ) for some filtration F0 on the C-vector space HC,y . Let ν(s) : HC,y → HC,y be the isomorphism obtained −1 (s) : HC,y → HC,y is the identity map, we from ν by applying s. Since ν −1 (s) ◦ ξy,0 have log

F (s) = ν(s)−1 F0 = ξ0,y (s)F0 and (1) follows from this.

(2) 2

2.5.2
-level structure. Let 0 = (w, (hp,q ), H0 ,  , 0 ) be as in Section 0.7 and let
be a subgroup of GZ . Let X be an object of A1 (log) (2.2.1), and let H = (HZ ,  , , F ) be a pre-PLH on X of weight w and of Hodge type (hp,q ). A
-level structure on H is a global section of the sheaf
\ Isom((HZ ,  , ), (H0 ,  , 0 )) on Xlog . Here H0 is regarded as a constant sheaf on X log , Isom is the sheaf of isomorphisms, and γ ∈
acts on Isom(· · · ) by h  → γ ◦ h.

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Example 1. Assume that w is odd and  , 0 is a perfect pairing H0 × H0 → Z of ∼ → HomZ (H0 , Z) by this pairing), let n ≥ 1, and let Z-modules (i.e., H0 −
= {γ ∈ GZ | γ acts on H0 /nH0 trivially}. Then, if (HZ ,  , ) has a
-level structure,  ,  is a perfect pairing HZ × HZ → Z of local systems of Z-modules. If  ,  is a perfect pairing HZ × HZ → Z of local systems of Z-modules, a
-level structure on (HZ ,  , ) is equivalent to an isomorphism of local systems HZ /nHZ  H0 /nH0 (usually called an n-level structure) which sends the pairing (HZ /nHZ ) × (HZ /nHZ ) → Z/nZ induced by  ,  to the pairing (H0 /nH0 ) × (H0 /nH0 ) → Z/nZ induced by  , 0 . (This follows from the surjectivity of Sp(g, Z) → Sp(g, Z/nZ).) Example 2. Let X be an object of A1 (log), let H be a pre-PLH on X, let x ∈ X, let y be a point of Xlog lying over x, and let (H0 ,  , 0 ) = (HZ,y ,  , y ). Let
be the image of the local monodromy action Image(π1 (x log ) → GZ ). Then, by 2.3.2, if we replace X by an open neighborhood of x, we have the isomorphism ξ ◦ ν from (HZ ,  , ) to (H0 ,  , 0 ) determined modulo the choices of the branches of ξ . This ξ ◦ ν is regarded as a
-level structure of H . 2.5.3 We define the period map associated with a pre-PLH endowed with a
-level structure. (See 2.5.10 for another more sophisticated period map.) Let the notation be as in 2.5.2, and let H = (HZ ,  , , F ) be a pre-PLH on X of weight w and of Hodge type (hp,q ) endowed with a
-level structure. Let Dˇ orb be as in 1.3.6. We define the period map ϕˇ : X →
\Dˇ orb associated with H . Here the quotient
\Dˇ orb is defined with respect to the action of
on Dˇ orb , γ : (σ, Z)  → (Ad(γ )σ, γ Z)

(γ ∈
, (σ, Z) ∈ Dˇ orb ).

Note that this is only a set-theoretic map. ∼ → Let x ∈ X, and take y ∈ Xlog lying over x, and let µ˜ y : (HZ,y ,  , y ) − (H0 ,  , 0 ) be a lifting of the germ µy of the
-level structure µ at y. Then, via µ˜ y , the action of π1 (x log ) on HZ,y defines a unipotent action of π1 (x log ) on H0 preserving  , 0 . Let σ be the cone in gR generated by the logarithms of the actions of elements of π1+ (x log ) on H0 . By Proposition 2.5.1, the set {µ˜ y (F (s)) | s ∈ sp(y)} ⊂ Dˇ of filtrations on H0,C is an exp(σC )-orbit. We define ϕ(x) ˇ := ((σ, Z) mod
) ∈
\Dˇ orb . In fact, ϕ(x) ˇ is independent of the choices of y and µ˜ y : If we fix y and change µ˜ y , the element (σ, Z) of Dˇ orb is changed by the action of
, and hence ((σ, Z) mod
) does not change. If we replace y by another element y
of τ −1 (x), a path from y to

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LOGARITHMIC HODGE STRUCTURES

y
sends µ˜ y to a lifting µ˜ y
of µ at y
and sends s to an element s
of sp(y
), and we have µ˜ y (F (s)) = µ˜ y
(F (s
)). The proof of the following lemma is straightforwards and we omit it. Lemma 2.5.4 Let X and H be as in 2.5.3, let X
be an object of A1 (log) over X, and let H
be the inverse image of H on X
. Consider the period maps ϕˇ : X →
\Dˇ orb ,

ϕˇ
: X
→
\Dˇ orb

associated with H and H
, respectively. Let x
∈ X
and let x ∈ X be the image of x
. (i) Let ϕ(x) ˇ = ((σ, Z) mod
),

ϕˇ
(x) = ((σ
, Z
) mod
)

where (σ
, Z
) ∈ Dˇ orb is defined by using a point y
∈ (X
)log lying over x
and using a lifting µ˜
y
of the germ of the inverse image µ
of µ on (X
)log at y
, and (σ, Z) ∈ Dˇ orb is defined by using the image y ∈ Xlog of y
and using the unique lifting µ˜ y of µy which induces µ˜
y
. Then σ
⊂ σ,

and Z
⊂ Z = exp(σC )Z
.

(ii) If X
→ X is strict (i.e., the logarithmic structure of X
coincides with the inverse image of that of X), then ϕ(x) ˇ = ϕˇ
(x
). Proposition 2.5.5 Let X be an object of A1 (log), let
be a subgroup of GZ , let H be a pre-PLH on X of weight w and of Hodge type (hp,q ) endowed with a
-level structure, and let ϕˇ : X →
\Dˇ orb be the associated period map. Let x ∈ X and write ϕ(x) ˇ = ((σ, Z) mod
) with (σ, Z) ∈ Dˇ orb . Take F ∈ Z. (i) The inverse image of H on the fs logarithmic point x satisfies the Griffiths transversality 2.4.7 (1) if and only if NF p ⊂ F p−1 for all N ∈ σ and for all p.

(ii) Write σ = nj=1 (R≥0 )Nj . Then the inverse image of H on x satisfies the positivity condition 2.4.7 (2) if and only if   n  exp  zj Nj  F ∈ D for Im(zj )  0. (1) j =1

(iii) The inverse image of H on x is a PLH if and only if (σ, Z) is a nilpotent orbit. Proof. Note that (iii) follows from (i) and (ii). By Lemma 2.5.4 (ii), we may assume that X is the fs logarithmic point x. Take y ∈ x log . log log We prove (i). Let M =
(x log , Ox ⊗Z HZ ) = τ∗ (Ox ⊗C M), and let log log log Mp =
(x log , F p ) ⊂ M. Then Ox ⊗Z HZ = Ox ⊗C M and F p = Ox ⊗C

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Mp (2.4.4). Hence (d ⊗ 1)(F p ) ⊂ ωx1,log ⊗ F p−1 (∀ p) ⇐⇒ ∇(Mp ) ⊂ ωx1 ⊗C Mp−1 (∀ p), where ∇ : M → ωx1 ⊗C M is the C-linear map induced by d ⊗ 1. Take s ∈ sp(y). s ∼ log Then the composition M → Ox ⊗Z HZ − → HC,y is bijective and induces Mp − → F p (s) for all p. Furthermore, by 2.3.4 (ii), via this composite map, ∇ : M → ωx1 ⊗C M corresponds to N
: HC,y → ωx1 ⊗C HC,y . Hence we have ∇(Mp ) ⊂ ωx1 ⊗C Mp−1 (∀p) ⇐⇒ N
(F p (s)) ⊂ ωx1 ⊗C F p−1 (s) (∀ p) ⇐⇒ N (F p (s)) ⊂ (Mxgp /Ox× ) ⊗ F p−1 (s) (∀ p) ⇐⇒ N (F p (s)) ⊂ F p−1 (s) (∀N ∈ σ, ∀ p). ∼

→ ωx1 (2.1.9). Here the second ⇐⇒ follows from C ⊗Z (Mx /Ox× ) − We prove (ii). Fix a finite subset {q1 , . . . , qm } of Mx − Ox× whose image in Mx /Ox× generates Mx /Ox× . Fix s0 ∈ sp(y). Then, for s ∈ sp(y) varying, we have gp

(˜s : Mx → C converges to α for the topology of simple convergence) ⇐⇒ (˜s (qj )˜s0 (qj )−1 → 0 for any j ) ⇐⇒ (Im((2πi)−1 (s(log(qj )) − s0 (log(qj )))) tends to ∞ for any j ). On the other hand, take a finite family (hj )1≤j ≤n of elements of Hom(Mx /Ox× , N) which generates the monoid Hom(Mx /Ox× , N). Then σ = nj=1 (R≥0 )Nj where Nj is the logarithm of the local monodromy of the element of π1+ (x log ) corresponding to hj . By 2.5.1, we have F (s) = exp( nj=1 zj Nj )F (s0 ) for the element s

of sp(y) characterized by (2π i)−1 (s(log(qj )) − s0 (log(qj ))) = nj=1 zj hj (qj ) for 1 ≤ j ≤ n. Hence the proof of (ii) is reduced to the following lemma which we gp apply by taking Mx /Ox× as S and by taking the homomorphism Mx /Ox× → R, f  → Im((2πi)−1 (s(log(f )) − s0 (log(f )))) as ϕ. 2 Lemma 2.5.6 Let S be an fs monoid such that S × = {1}, let qj (1 ≤ j ≤ m) be eleadd ments of S − {1} which generate S, and let h1 , . . . , hn be elements of Hom(S, R≥0 ) add which generate the cone Hom(S, R≥0 ). For c > 0, define subsets Ac and Bc of add Hom(S, R≥0 ) by   n Ac = yj hj | yj ≥ c (∀ j ) , Bc = {ϕ ∈ Hom(S gp , R add ) | ϕ(qj ) ≥ c (∀ j )}. j =1

Then we have the following. (i) Fix c > 0. Then Ad ⊂ Bc for some d > 0. (ii) Fix c > 0. Then Bd ⊂ Ac for some d > 0. add Proof. We prove (i). Since h1 , . . . , hn generate the cone Hom(S, R
0 ) and since qj  = 1 for any j , (h1 + · · · + hn )(qj ) > 0 for any j . Hence there exist d > 0 such that (dh1 + · · · + dhn )(qj ) ≥ c for any j . If yj ≥ d for any j (1 ≤ j ≤ n), (y1 h1 + · · · + yn hn )(qj ) ≥ (dh1 + · · · + dhn )(qj ) ≥ c for any j .

LOGARITHMIC HODGE STRUCTURES

101

| 1 ≤ j ≤ m}. Let ϕ ∈ Bd . We prove (ii). Let d = max{(ch1 + · · · + chn )(qj ) add Since (ϕ − nj=1 chj )(qj ) ≥ 0 for any j , we have ϕ − nj=1 chj ∈ Hom(S, R≥0 ).

n Hence ϕ − j =1 chj = r1 h1 + · · · + rn hn for some rj ≥ 0. We have ϕ =

n 2 j =1 (c + rj )hj ∈ Ac . Proposition 2.5.7 Let x be an fs logarithmic point. Fix y ∈ x log and s ∈ sp(y). Let (w, (hp,q )p,q , H0 ,  , 0 ) be as in Section 0.7, fix a unipotent action ρ of π1 (x log ) on H0 preserving  , 0 , and let σ be the cone in gR generated by log(ρ(π1+ (x log ))). Let (HZ ,  , ) be the local system on x log whose stalk at y is (H0 ,  , 0 ) with monodromy ρ, and let F+ (resp. F) be the set of all decreasing filtrations F on log Ox ⊗ HZ such that (HZ ,  , , F ) is a PLH (resp. pre-PLH) of weight w and of Hodge type (hp,q ). ˇ F  → F (s) is bijective. (i) The map F → D, (ii) The map in (i) induces a bijection from F+ to the set of all F ∈ Dˇ such that (σ, exp(σC )F ) is a nilpotent orbit. Proof. Take (qj )1≤j ≤n as in the proof of 2.5.1, fix a branch ξy,0 of ξy , and consider the log log associated isomorphism ν : Ox ⊗ HZ  Ox ⊗ H0 . Then we have a bijection F  log Dˇ in which F0 ∈ Dˇ corresponds to F = ν −1 (Ox ⊗C F0 ) ∈ F. We have F (s) = ˇ we see that F → D, ˇ F  → F (s) is ξy,0 (s)F0 . Since ξy,0 (s) is an automorphism of D, a bijection. Let
= ρ(π1 (x log )) ⊂ GZ . The period map associated with the pre-PLH H = (HZ ,  , , F ) with the evident
-level structure sends x to (σ, exp(σC )F (s)). By 2.5.5, H is a PLH if and only if (σ, exp(σC )F (s)) is a nilpotent orbit. 2 Definition 2.5.8 Let  be a fan in gQ and let
be a subgroup of GZ which is strongly compatible with  (1.3.10). Denote by   := w, (hp,q )p,q∈Z , H0 ,  , 0 ,
,   the 6-tuple consisting of the 4-tuple w, (hp,q )p,q∈Z , H0 ,  , 0 as in Section 0.7 and of the above  and
. Let X be an object of A1 (log). A polarized logarithmic Hodge structure on X of type  (PLH on X of type , or PLH on X endowed with a
-level structure whose local monodromies are in the directions in ) is a pre-PLH (HZ ,  , , F ) on X of weight w and of Hodge type (hp,q ), which is endowed with a
-level structure µ, satisfying the following condition (1). Let ϕˇ : X →
\Dˇ orb be the period map associated to H . (1) For each x ∈ X, if we denote ϕ(x) ˇ = ((σ, Z) mod
) with (σ, Z) ∈ Dˇ orb , then there is τ ∈  such that σ ⊂ τ . Furthermore, if we take the smallest such τ , (τ, exp(τC )Z) is a nilpotent orbit. Proposition 2.5.9 Let X be an object of A1 (log). Let  be as in 2.5.8 and let (HZ ,  , , F, µ) be a PLH of type  on X. Then (HZ ,  , , F ) is a PLH of weight w and of Hodge type (hp,q ). Proof. Let ϕˇ : X →
\Dˇ orb be the period map associated with H , let x ∈ X, and write ϕ(x) ˇ = ((σ, Z) mod
). Take the smallest τ ∈  such that σ ⊂ τ . Then

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(τ, exp(τC )Z) is a nilpotent orbit. As σ contains a point of the interior (0.7.7) of τ , this implies that (σ, Z) is a nilpotent orbit. 2 2.5.10 Let H = (HZ ,  , , F, µ) be a PLH of type  on X (2.5.8). We define maps ϕ : X →
\D , ϕ

log

:X

log

→

(1)


\D

(2)

associated with H as below. Let ϕˇ : X →
\Dˇ orb be the period map associated with H in 2.5.3. Let x ∈ X, let ϕ(x) ˇ = ((σ, Z) mod
), and let τ be the smallest element of  such that σ ⊂ τ . We define ϕ(x) := ((τ, exp(τC )Z) mod
) ∈
\D .

(3)

This map ϕ is also called the period map associated with H . If there is a possibility of confusion with the period map ϕ, ˇ we call ϕ the period map of H with respect to . Later, in Chapter 3, we will endow
\D with the structure of a logarithmic local ringed space over C, and ϕ will become the underlying map of a morphism of logarithmic local ringed spaces over C. On the other hand, we consider the map ϕˇ always just as a set-theoretic map. We define the map ϕ log . Let y ∈ Xlog and let x be the image of y in X. Let s be an element of sp(y) satisfying the following condition: The composite map log s → C → C/R  iR coincides with the composite map Ly → Ly → Ox,y − Cont(x, iR) = iR (2.2.4). We define 

ϕ log (y) := ((τ, Z) mod
) ∈
\D ,

(4)

where Z is the τ -nilpotent i-orbit exp(iτR )µ˜ y (F (s)). Then ϕ log (y) is independent of the choices of s and µ˜ y . If ϕ log (y) = ((τ, Z) mod
), then ϕ(x) = ((τ, exp(τC )Z) mod
). 2.5.11 Let X be an object of A1 (log) (2.2.1), let
be a subgroup of GZ , and let H be a pre-PLH on X of weight w and of Hodge type (hp,q ) endowed with a
-level structure. By the set of local monodromy cones of H in gR , we mean the set of all nilpotent cones σ in gR such that ((σ, Z) mod
) belongs to the image of the period ˇ map X →
\Dˇ orb (2.5.3) for some exp(σC )-orbit Z in D. gp

Proposition 2.5.12 Let X be an object of A1 (log) and assume rank Z (MX / × OX )x ≤ 1 for all x ∈ X. Let H be a PLH on X of weight w and of Hodge type (hp,q ) endowed with a
-level structure. Let 
be the set of local monodromy cones of H in gR (2.5.11) and let  = 
∪ {{0}}. Then  ⊂  (1.3.11),  is a fan in gQ and is strongly compatible with
, the period map ϕˇ : X →
\Dˇ orb of H in 2.5.3 takes values in
\D , H is a PLH of type (w, (hp,q ), H0 ,  , 0 ,
, ), and the period map ϕ : X →
\D in 2.5.10 (1) coincides with ϕ. ˇ

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LOGARITHMIC HODGE STRUCTURES

2

Proof. Straightforward. 2.5.13

Schmid [Sc] proved the nilpotent orbit theorem which, roughly speaking, says the following: If X is an analytic manifold and Y is a divisor on X with normal crossings, then a variation of polarized Hodge structure on X − Y yields a nilpotent orbit at each point of Y . By using the relationship between a PLH and a nilpotent orbit given in proposition 2.5.5, we can interpret the nilpotent orbit theorem as Theorem 2.5.14 Let X be a logarithmically smooth, fs logarithmic analytic space, and let U = Xtriv be the open set of X consisting of all points at which the logarith× }). Let (HZ ,  , , F ) mic structure is trivial (that is, U = {x ∈ X | MX,x = OX,x be a variation of polarized Hodge structure on U of weight w and of Hodge type (hp,q ), and assume that the local monodromy of HZ along X − U is unipotent. Then, (HZ ,  , , F ) extends uniquely to a PLH on X of weight w and of Hodge type (hp,q ). This PLH satisfies the big Griffiths transversality (2.4.9), i.e., this is an LVPH in 2.4.9. Here “the local monodromy of HZ along X − U is unipotent” means that the local monodromy of the unique extension of HZ to Xlog as a locally constant sheaf is unipotent. The reason why the nilpotent orbit theorem of Schmid implies theorem 2.5.14 is shown in [KMN]. Here we explain it in the following special case for simplicity. 2.5.15 Recalling the nilpotent orbit theorem of Schmid [Sc, Chapter 4], we explain the above Theorem 2.5.14 in the case X = n and U = (∗ )n . Assume that a variation of polarized Hodge structure H = (HZ ,  , , F ) on (∗ )n is given. Fix p ∈ (∗ )n , let H0 = HZ,p , and let  , 0 : H0 × H0 → Q be the pairing induced by  , . Let hn → (∗ )n ,

(zj )j  → (exp(2π izj ))j .

Denote the pullback of HZ to hn via this map by the same letter HZ . Fix u ∈ hn whose ∼ → H0 on hn (H0 is image in (∗ )n is p. We have a unique isomorphism β : HZ − regarded as a constant sheaf) whose germ at u is the identity map from HZ,u = HZ,p to H0 = HZ,p . For z ∈ hn with image q in (∗ )n , let F (q) be the filtration on HC,q ˜ ˜ : hn → D is the associated defined by F and let (z) = β(F (q)) ∈ D. Then  period map, which is holomorphic. Now assume that the monodromy γj : H0 → H0 is unipotent for every 1 ≤ j ≤ n. (γj is the j th standard generator of π1 ((∗ )n )). Let Nj = log(γj ) : H0,Q → H0,Q . Then ˜ + 1j ) = γj (z) ˜ ˜ (z = exp(Nj )(z)

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for any z ∈ hn and 1 ≤ j ≤ n, where z + 1j is the element of hn defined by (z + 1j )k = zk for any 1 ≤ k ≤ n such that k  = j , and (z + 1j )j = zj + 1. This shows that, if we define   n  ˇ ˜ : hn → D, ˜  z  → exp − zj Nj  (z), j =1

˜ descends to (∗ )n , that is, there is a unique holomorphic map  : (∗ )n → then  Dˇ for which the following diagram is commutative: hn   

˜

 −−−−→ Dˇ



 ˇ (∗ )n −−−−→ D.

Then the nilpotent orbit theorem of Schmid asserts the following (i) and (ii). ˇ (i)  extends to a holomorphic map n → D. n (ii) For any q = (qj )1≤j ≤n ∈  , (σ, exp(σC )(q))

with σ =



(R≥0 )Nj

1≤j ≤n,qj =0

is a nilpotent orbit. We explain why the nilpotent orbit theorem implies that a VPH on (∗ )n extends to an LVPH. Let X = n with the logarithmic structure associated with the normal crossing divisor X − U where U = (∗ )n . Then HZ extends to X log as a locally constant sheaf by 2.3.5. The morphism  : X → Dˇ defines a filtration F on OX ⊗ H0 . Let log

log

ν : OX ⊗ HZ  OX ⊗ H0 be the isomorphism of OX -modules given in 2.3.8. Let F := ν −1 (OX ⊗OX F ) log be the filtration on OX ⊗ HZ . Then, the restriction of F to (∗ )n coincides with the original Hodge filtration on OU ⊗ HZ . This shows that the VPH in question extends to a pre-PLH on X = n . By (ii) of the theorem of Schmid and by 2.5.5, this prePLH is a PLH. It satisfies the big Griffiths transversality because its restriction to (∗ )n satisfies the big Griffiths transversality. Hence this PLH is an LVPH. log

log

2.5.16 Correction. In [KU1, 5.5], there is a mistake in the definition of a PLH of type . The correct definition is the one given in this Section 2.5. The mistake in [KU1, 5.5] is corrected as follows. In the last line of condition (i) in [KU1, 5.5], form a σ -nilpotent orbit must be corrected to are contained in a σ -nilpotent orbit, or, equivalently, to together with their translations by exp(σC ) form a σ -nilpotent orbit.

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LOGARITHMIC HODGE STRUCTURES

2.6 LOGARITHMIC MIXED HODGE STRUCTURES We give here a definition of logarithmic mixed Hodge structure, and define a logarithmic Hodge structure of weight w (w ∈ Z) as a special case of logarithmic mixed Hodge structure. In [D5, 1.8.15], Deligne illustrated how to formulate “good degenerations" of mixed Hodge structures, and Steenbrink and Zucker gave the precise definition in [SZ], [Z3] following the philosophy of Deligne. Our definition of logarithmic mixed Hodge structure below also follows the philosophy of Deligne and is similar to the definition of good degeneration of a mixed Hodge structure in [SZ], [Z3]. This section is not used in the rest of this book. In this section, X denotes an object of A1 (log) (2.2.1). Definition 2.6.1 A prelogarithmic mixed Hodge structure (pre-LMH) on X is a triple (HZ , W, F ), consisting of a locally constant sheaf of free Z-modules HZ of finite rank on X log , a Q-rational increasing filtration W of the sheaf HR of Rlog log modules, and a decreasing filtration F of the OX -module OX ⊗ HZ , which satisfies the following two conditions. (1) For each k ∈ Z, Wk is locally constant. (2) There exist an OX -module M on X and a decreasing filtration (Mp )p∈Z of M by OX -submodules such that M, Mp , M/Mp are locally free of finite log rank, OX ⊗Z HZ = τ ∗ (M) and F p = τ ∗ (Mp ). Definition 2.6.2 Let x be an fs logarithmic point (2.1.9). A pre-LMH (HZ , W, F ) on x is called a logarithmic mixed Hodge structure (LMH) on x if it satisfies the following conditions (1) and (2). (1) (HZ , F ) satisfies the Griffiths transversality (2.4.5). (2) There exists a family (W (S))S of Q-rational increasing filtration W (S) of the sheaf HR given for each face S of the fs monoid Mx /Ox× , satisfying the following conditions (2.1)–(2.3). (2.1) If S = Mx /Ox× , then W (S) = W . add add , where R≥0 means R≥0 (2.2) Let h be a homomorphism Mx /Ox× → R≥0 regarded as a monoid with respect to the addition. Let y ∈ x log and let Nh be the image of h under the composite map add Hom(Mx /Ox× , R≥0 ) ⊂ Hom(Mxgp /Ox× , R) log

 R ⊗Z π1 (x log ) −→ End R (HR,y ), where log is the logarithm of the action of π1 (x log ) on HR,y . Then, if S is a face of Mx /Ox× and S ⊂ Ker(h), we have Nh (W (S)k,y ) ⊂ W (S)k−2,y


for all k ∈ Z. If S is a face of Mx /Ox× and S = S
∩ Ker(h), then, for any integer k and any integer l ≥ 0, we have an isomorphism


∼




(S) W (S ) (S) W (S ) Nhl : gr W )y − → gr W )y . k+l (gr k k−l (gr k

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(2.3) Fix y ∈ x log and let s ∈ sp(y). If s˜ : Mx → C (see 2.4.6) is sufficiently near to the structure morphism of the logarithmic structure α : Mx → C in the topology of simple convergence of C-valued functions, then for any face S of Mx /Ox× , (HZ,y , W (S)y , F (s)) is a mixed Hodge structure in the usual sense. 2.6.3 Remark. The family of filtrations (W (σ ))σ satisfying 2.6.2 (2.1) and 2.6.2 (2.2) is unique if it exists, according to Deligne [D5, 1.6.3]. Definition 2.6.4 A pre-LMH (HZ , W, F ) on X is a logarithmic mixed Hodge structure (LMH) on X if, for any x ∈ X, the inverse image of (HZ , W, F ) on x is an LMH. Here we regard x as an fs logarithmic point with the inverse image of the logarithmic structure of X. Definition 2.6.5 Let w ∈ Z. A logarithmic Hodge structure (LH) on X of weight w is an LMH (HZ , W, F ) on X such that Ww = HR and Ww−1 = 0. 2.6.6 Let w ∈ Z. By Cattani and Kaplan [CK2], a PLH on X of weight w is an LH on X of weight w.

Chapter Three Strong Topology and Logarithmic Manifolds

In this chapter, we define a structure of a logarithmic local ringed space over C on
\D (see Section 3.4), where  is a fan in gQ and
a subgroup of GZ that is strongly compatible with . For this, in Section 3.3, we define for each σ ∈  a subset Eσ of an fs logarithmic analytic space Eˇ σ (a logarithmic version of the subset ˇ and a map Eσ →
\D whose image covers
\D when σ ranges in . D of D) We endow Eσ with the so-called “strong topology” introduced in Section 3.1, and with the inverse images of O and M of Eˇ σ , and then import these structures to
\D . The spaces Eσ and
\D are not necessarily analytic spaces. In Section 3.2, we introduce various categories of various kinds of (logarithmic) generalized analytic spaces and consider relations between them. The spaces Eσ and the spaces
\D , with
neat belong to a category B∗ (log) that appears in Section 3.2, and belong in fact to a much smaller category, the category of “logarithmic manifolds” introduced in Section 3.5. There, as a preparation for the infinitesimal extended period maps, we extend the infinitesimal calculus on analytic spaces to the category B(log) that appears in Section 3.2 (see also 0.4.16) and that contains B ∗ (log). In Section 3.6, we consider “logarithmic modifications” of objects of B(log) (a generalization of the modification in the toric geometry arising from subdivision of a fan) which we will need in Section 4.3 to extend period maps to the boundary.

3.1 STRONG TOPOLOGY Definition 3.1.1 Let X be an analytic space and S a subset of X. (i) The weak topology of S in X is the topology as a subspace of X. We denote this topological space by Sweak/X . (ii) The strong topology of S in X is the strongest topology on S for which the map λ : Y → S is continuous for any analytic space Y and for any analytic morphism λ : Y → X with λ(Y ) ⊂ S. We denote this topological space by Sstr/X . By definition, a subset U of Sstr/X is open if and only if λ−1 (U ) is open in Y for any (Y, λ) as above. We have clearly (weak topology) ≤ (strong topology). By the resolution of singularity of Hironaka [H], [AHV], for any Hausdorff analytic space Y that is countable at infinity there are a smooth analytic space Y
and

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a proper morphism Y
→ Y . Hence we may assume the analytic spaces Y to be smooth in the above definition of strong topology. Note also that if S is a locally closed analytic subspace of X then the strong topology on S coincides with the weak topology on S. We give two examples 3.1.2 and 3.1.3 concerning the strong topology. 3.1.2 Example. Let S be a subset of the complex line X := C. Then the strong topology of S is the one described as follows. Let S
be the interior of S in the usual topology of X, that is, an element s of S belongs to S
if and only if S contains some neighborhood of s in X in the usual topology of X. Then Sstr/X is the disjoint union of the subspace S
of X and the discrete set S − S
. Proof. It is sufficient to prove that, for a smooth analytic space Y and an analytic morphism λ : Y → X with λ(Y ) ⊂ S, if y ∈ Y and λ(y) ∈ S − S
, then λ(y
) = λ(y) for any y
in some neighborhood of y. Take an open neighborhood U of y in Y such that, for any y
∈ U , there exists a connected complex smooth curve C on Y joining y and y
. If the restriction of λ to C is not a constant function, λ : C → X = C is an open map and hence λ(C) must be an open set in X. Since λ(y) ∈ λ(C) ⊂ S, this contradicts the assumption λ(y) ∈ S − S
. Hence the restriction of λ to C is constant and hence λ(y
) = λ(y) for any y
∈ U . 2 3.1.3 Example. Let X := C2 and let S be the subset S := (C2 − ({0} × C)) ∪ {(0, 0)} ⊂ X. (i) The strong topology of S in X is described as follows. (1) The topology on S − {(0, 0)} as a subspace of Sstr/X coincides with the one as a subspace of X. (2) Let ε = (εn )n≥1 be a sequence in R>0 . Define Un (εn ) := {(x, y) ∈ S | |x| < εn , |y| < εn , |y|n < |x|}, and

" U (ε) :=

$

# Un (εn ) ∪ {(0, 0)}.

n≥1

Then the U (ε), where ε runs over all sequences in R>0 , form a fundamental system of neighborhoods of (0, 0) in Sstr/X . (ii) If k ∈ Z≥1 and z ∈ C converges to 0, then (zk , z) converges to (0, 0) in Sstr/X . However, if f : R>0 → R>0 is a map such that, for each n ≥ 1, there is εn > 0 for which f (s) ≤ s n if 0 < s < εn , then (f (s), s) (s → 0) converges to (0, 0) for the weak topology of S but diverges for the strong topology of S. Examples of f are f (s) = e−1/s , f (s) = s 1/s , etc. (iii) Sstr/X is not locally compact.

109

STRONG TOPOLOGY

Proof. (i) (1) follows easily from the definition. For (i) (2), we will prove a more general result, Proposition 3.1.5 (ii), and deduce (i) (2) from it just after the end of the proof of 3.1.5. We prove (ii). The convergence of (zk , z) follows easily from (i) (2). The divergence of (f (s), s) is deduced from (i) (2) as follows. For each n ≥ 1, take εn > 0 such that f (s) ≤ s n if 0 < s < εn . Then (f (s), s) does not belong to U (ε) for any s ∈ R>0 . Otherwise, f (s) < εn , s < εn , s n < f (s) for some n ≥ 1, but this is impossible. We prove (iii). Assume it is locally compact and let K be a compact neighborhood of (0, 0) in Sstr/X . Then we have a set U (ε) in (i) (2) such that U (ε) ⊂ K. Replacing εn by min( 12 , ε1 , . . . , εn ), we may assume 1 > ε1 ≥ ε2 ≥ · · · . Let xn = εnn , yn = εn2 , pn = (xn , yn ) ∈ Un (εn ) ⊂ U (ε) (n ≥ 1). Then pn does not converge to (0, 0) in the strong topology of S. Moreover, any cofinal subsequence of (pn )n does not converge 2n to (0, 0) in the strong topology of S. In fact, let εn
= ε2n (n ≥ 1). We show that

2m m pn ∈ / U (ε ) for any n. If pn ∈ Um (εm ), then εn = yn < xn = εnn . Hence 2m > n 2m


= ε2m < εnn = xn . But this conclusion εm < xn contradicts pn ∈ Um (εm ). and εm On the other hand, since K is compact, some cofinal subsequence of (pn )n must converge to some point p of K. Since it converges to p also in the weak topology, we have p = (0, 0), a contradiction. 2 3.1.4 In Proposition 3.1.5 (ii) below, we generalize Example 3.1.3 (i) to the strong topology of an “analytically constructible” subset in a complex analytic space. A subset S of a complex analytic space X is said to be analytically constructible if the following condition (1) holds locally on X. (1) There exist a finite family (Aj )j ∈J of closed analytic subspaces Aj of X and a closed analytic subspace Bj of Aj for each j such that S = {x ∈ X | x ∈ Bj for any j ∈ J such that x ∈ Aj }. To state Proposition 3.1.5, we fix notation. Let X be a complex analytic space endowed with a metric d : X × X → R≥0 which is compatible with the analytic structure of X. That is, there are an open covering (Uλ )λ∈ of X, analytic immersions ιλ : Uλ → Ck(λ) for some k(λ) ≥ 0, and constants cλ , cλ
> 0 such that cλ |ιλ (x) − ιλ (y)| ≤ d(x, y) ≤ cλ
|ιλ (x) − ιλ (y)|

(∀λ ∈ , ∀x, ∀y ∈ Uλ ).

Here | | denotes the usual Euclidean metric on Ck(λ) . For x ∈ X and for a subset E of X, let d(x, E) = inf {d(x, y) | y ∈ E}. Let S be a subset of X and let (Aj )j ∈J be a finite family of closed analytic subspaces of X. For s ∈ S, for a function n : J → Z>0 , and for δ ∈ R>0 , let Un (s, δ) be the set of all x ∈ S satisfying the following conditions (2) and (3). (2) d(x, s) < δ. / Aj , then d(x, S ∩ Aj )n(j ) < d(x, Aj ). (3) If j ∈ J and if s ∈ Aj and x ∈

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Let E be the set of all families (εn )n of real numbers εn > 0 given for each function n : J → Z>0 . For s ∈ S and for ε = (εn )n ∈ E, define $ U (s, ε) = Un (s, εn ) ⊂ S. n

Proposition 3.1.5 Let X be a complex analytic space endowed with a metric d that is compatible with the analytic structure of X, let S be a subset of X, and let (Aj )j ∈J be a finite family of closed analytic subspaces of X. Let E and U (s, ε) ⊂ S (s ∈ S, ε ∈ E) be as above. (i) If s ∈ S and if ε ∈ E, U (s, ε) is an open neighborhood of s in S in the strong topology. (ii) Assume that there exists a closed analytic subspace Bj of Aj for each j ∈ J such that S = {s ∈ X | s ∈ Bj if j ∈ J and s ∈ Aj }. Then, if s ∈ S and if ε ranges over all elements of E, U (s, ε) form a fundamental system of neighborhoods of s in S in the strong topology. Proof. We prove (i). As is easily seen, if s, s
∈ S, ε ∈ E and s
∈ U (s, ε), then there exists ε
∈ E such that U (s
, ε
) ⊂ U (s, ε). Hence, for the proof of (i), it is sufficient to prove that, if U is a subset of S satisfying the following condition (1), then U is open in the strong topology of S. (1) For any s ∈ U , there exists ε ∈ E such that U (s, ε) ⊂ U . Assume U ⊂ S satisfies (1). Let Y be an analytic space, and λ : Y → X be a morphism satisfying λ(Y ) ⊂ S. We have to show that λ−1 (U ) is an open set on Y . For an analytic space Y
and for a surjective morphism Y
→ Y which is open or closed, we can replace Y by Y
, because the topology on Y is the quotient of the topology on Y
. By the Hironaka resolution of singularities [H], [AHV, Theorem 5.3.1] together with the above remark, we may assume the following conditions (2) and (3). (2) Y is smooth. (3) For each j ∈ J , either λ−1 (Aj ) = Y , or λ−1 (Aj ) is a divisor on Y with normal crossings. Let y ∈ Y and let s = λ(y). It is enough to show that, if z ∈ Y is sufficiently near y, then λ(z) ∈ U . Let J
be the subset of J consisting of all j such that s ∈ Aj and λ−1 (Aj ) is a divisor on Y with normal crossings. For each j ∈ J
, let fj be a generator of the ideal of OY,y whose zero coincides with λ−1 (Aj ) on some e(j,1) e(j,l(j )) neighborhood of y, let fj = fj,1 · · · fj,l(j ) be a prime decomposition in OY,y , and let e(j ) := 1≤k≤l(j ) e(j, k). Since the metric d is compatible with the analytic structure, if the values of n : J → Z>0 are sufficiently large, then for any j ∈ J
, |fj (z)|n(j ) d(λ(z), Aj )−e(j ) converges to 0 when z ∈ Y − λ−1 (Aj ) converges to y. Take such a function n and take δ > 0 such that Un (s, δ) ⊂ U . It is enough to prove that, if z ∈ Y is sufficiently near y, then λ(z) ∈ Un (s, δ). Hence it is sufficient to prove that, if j ∈ J
and if z ∈ Y − λ−1 (Aj ) is sufficiently near y, then d(λ(z), S ∩ Aj )n(j ) < d(λ(z), Aj ).

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Choose local coordinates on Y at y in the form (fj,1 , . . . , fj,l(j ) , tj,1 , . . . , tj,m(j ) ), let |fj,k (z)| be the minimal one among |fj,1 (z)|, . . . , |fj,l(j ) (z)|, and let w be the point on Y with coordinates (fj,1 (z), . . . , fj,k−1 (z), 0, fj,k+1 (z), . . . , fj,l(j ) (z), tj,1 (z), . . . , tj,m(j ) (z)). Note λ(w) ∈ S ∩ Aj . There exists a constant c > 0 which is independent of z and satisfies d(λ(z), λ(w))n(j ) ≤ c|fj,k (z)|n(j ) ≤ c|fj (z)|n(j )/e(j ) < d(λ(z), Aj ). Hence d(λ(z), S ∩ Aj )n(j ) < d(λ(z), Aj ). We prove (ii) of Proposition 3.1.5. For the same reason given at the beginning of the proof of (i), it is sufficient to prove that, if U is an open set of S in the strong topology, then U satisfies the condition (1). Fix s ∈ U and n : J → Z>0 . It is sufficient to prove Un (s, δ) ⊂ U for some δ > 0. Working locally on X, we may assume that, for each j ∈ J , there exist gj,1 , . . . , gj,a(j ) , hj,1 , . . . , hj,b(j ) ∈ OX (X) such that Aj = {x ∈ X | gj,k (x) = 0 (1 ≤ k ≤ a(j ))}, Bj = {x ∈ X | hj,l (x) = 0 (1 ≤ l ≤ b(j ))}. Since the metric d is compatible with the analytic structure, we may further assume that there exist integers e, e
≥ 1 such that max{|hj,l (x)|e | 1 ≤ l ≤ b(j )} ≤ d(x, Bj ),


d(x, Aj )e ≤ max{|gj,k (x)| | 1 ≤ k ≤ a(j )} for any j ∈ J and any x ∈ X. Define a closed analytic subspace Y of X ×

a(j )b(j ) by j ∈J C %   x ∈ X, z = (zj,k,l )j ∈J,1≤k≤a(j ),1≤l≤b(j ) ∈ Ca(j )b(j ) ,      j ∈J   Y := (x, z)  .
e(e n(j )+1)   = zj,k,l gj,k (x) (∀ j, ∀ l)  hj,l (x)     1≤k≤a(j )    

(4)

Let λ : Y → X be the morphism (x, z)  → x. Then λ(Y ) ⊂ S. As is easily seen,

there exists α = (αj )j ∈J ∈ j ∈J Ca(j )b(j ) (αj ∈ Ca(j )b(j ) ) such that (s, α) ∈ Y and such that, if j ∈ J and s ∈ Aj , then αj = 0. Since U is open in the strong topology of S, λ−1 (U ) is a neighborhood of (s, α). Hence there exists δ > 0 such that, if (x, z) ∈ Y and if d(x, s) < δ and |z − α| < δ, then x ∈ U . Thus it is sufficient to prove that, for a given j ∈ J and c > 0, there exists δ such that 0 < δ ≤ c with the following property: If x ∈ Un (s, δ), then there exists zj ∈ Ca(j )b(j ) such that |zj − αj | < c and satisfies 
hj,l (x)e(e n(j )+1) = zk,l gj,k (x) (∀ l). (5) 1≤k≤a(j )

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The existence of such a δ is seen easily if s ∈ / Aj . Assume s ∈ Aj . We show that we can take δ = c. Note that αj = 0 in this case. If x ∈ Aj ∩ Un (s, δ), we can take zj = 0. Asume x ∈ Un (s, δ) but x ∈ / Aj . Then we have d(x, S ∩ Aj )n(j ) < d(x, Aj ) and d(x, S ∩ Aj ) ≤ d(x, s) < δ, and hence


max{|hj,l (x)|e(e n(j )+1) | 1 ≤ l ≤ b(j )}








≤ d(x, Bj )e n(j )+1 ≤ d(x, S ∩ Aj )e n(j )+1 < δd(x, S ∩ Aj )e n(j ) < δd(x, Aj )e ≤ δ max{|gj,l (x)| | 1 ≤ l ≤ a(j )}. Thus there exists zj ∈ Ca(j )b(j ) satisfying (5) and |zj | < δ.




2

Proof of 3.1.3 (i) (2). We deduce 3.1.3 (i) (2) from Proposition 3.1.5 (ii). In 3.1.5, let X = C2 , S = (C2 − ({0} × C)) ∪ {(0, 0)}, J be a one-point set {j }, Aj = {0} × C, and Bj = {(0, 0)}, s = (0, 0). Then if we define Un (s, δ) (δ > 0) in 3.1.4 with respect to the usual metric d of C2 defined by d((x1 , y1 ), (x2 , y2 )) = (|x1 − x2 |2 + |y1 − y2 |2 )1/2 , then Un (s, δ) is contained in the Un (δ) of 3.1.3. If we define Un (s, δ) in 3.1.4 with respect to the metric d((x1 , y1 ), (x2 , y2 )) = (|x1 − x2 | + |y1 − y2 |)/2, then Un (δ) of 3.1.3 is contained in Un+1 (s, δ) when δ < 1/2. Hence 3.1.3 (i) (2) follows from 3.1.5 (ii). 2 The following result will be used in Chapters 6 and 7. Proposition 3.1.6 Let X, d, S be as in 3.1.5, and let A be a closed analytic subspace of X. Let s ∈ S ∩ A and let (sλ )λ be a directed family of elements of S ∩ (X − A) that converges to s in the strong topology of S. Let yλ := | log(d(sλ , A))|. (Note yλ → ∞.) Then, for any k ≥ 0, we have yλk d(sλ , S ∩ A) → 0. Proof. Note that, for each n ≥ 1, there exists εn > 0 such that t < | log(t)|−(k+1)n if 0 < t < εn . Let ε = (εn )n≥1 . By Proposition 3.1.5 (i), U (s, ε) is a neighborhood of s in the strong topology of S. Hence, if λ is large, there exists n ≥ 1 such that d(sλ , s) < εn and d(sλ , S ∩ A)n < d(sλ , A). Since d(sλ , A) ≤ d(sλ , s) < εn , we have d(sλ , A) < yλ−(k+1)n . Hence d(sλ , S ∩ A) < d(sλ , A)1/n < yλ−(k+1) and this shows that yλk d(sλ , S ∩ A) < yλ−1 → 0. 2 3.1.7 We give basic remarks on strong topologies. (i) Let X be an analytic space, let Y be a locally closed analytic subspace of X, and let S be a subset of Y . Then we have Sstr/Y = Sstr/X . (ii) Let X be an analytic space, let S be a subset of X, and let T be a locally closed subset of X contained in S. Then Tstr/X is a topological subspace of Sstr/X . These assertions are proved easily.

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Proposition 3.1.8 Let X and Y be analytic spaces and let S ⊂ X, T ⊂ Y be subsets. Then, (S × T )str/X×Y coincides with Sstr/X × Tstr/Y if either one of the following conditions (1) and (2) is satisfied. (1) S and T are analytically constructible subsets of X and of Y , respectively. (2) T = Y . We do not know whether (S × T )str/X×Y = Sstr/X × Tstr/Y holds in general. Proof of Proposition 3.1.8. If (1) is satisfied, noticing that S × T is a constructible subset of X × Y , we can deduce the conclusion of Proposition 3.1.8 easily from Proposition 3.1.5 (ii). Next assume that (2) is satisfied. By working locally on Y , by the resolution of singularities of Hironaka [H], [AHV], we may assume that Y is an open set of Cn for some n ≥ 0. Then, using an open immersion Y ⊂ PnC , we are reduced to the case Y = PnC . Thus we may assume that Y is compact. Then, since Sstr/X × Y is Hausdorff, it is sufficient to prove that the composite map (S × Y )str/X×Y → Sstr/X × Y → Sstr/X is proper. The fiber of (S × Y )str/X×Y on each s ∈ S is Y and hence is compact. Thus (by [Bn, Ch. I, §10, Theorem 1]), it is sufficient to prove that (S × Y )str/X×Y → Sstr/X is a closed map. Let C be a closed subset of (S × Y )str/X×Y , and let C
be the image of C in S. Let A be an analytic space and let λ : A → X be a morphism such that λ(A) ⊂ S. It is sufficient to prove that λ−1 (C
) is closed in A. Let λY : A × Y → X × Y be the morphism induced by λ. Then λY (A × Y ) ⊂ S × Y . Hence −1 λ−1 Y (C) is closed in A × Y . Since A × Y → A is proper, the image of λY (C) in A, −1
which coincides with λ (C ), is closed in A. 2 Proposition 3.1.9 Let f : X → Y be a morphism of analytic spaces and let S be a subset of Y . Then the topological space f −1 (S)str/X coincides with the fiber product X ×Y Sstr/Y . Proof. Via the embedding
f : X → X × Y,

x  → (x, f (x)),

(1)

consider X as a locally closed analytic subspace of X × Y . By 3.1.7 (i), f −1 (S)str/X 
f (f −1 (S))str/X×Y . Since
f (f −1 (S)) is the intersection of X × S and
f (X) in X × Y ,
f (f −1 (S)) is locally closed in X × S in the weak topology of X × S in X × Y , and hence is locally closed in X × S in the strong topology of X × S in X × Y . Hence, by 3.1.7 (ii),
f (f −1 (S))str/X×Y is a topological subspace of (X × S)str/X×Y and the last space is X × Sstr/Y by Proposition 3.1.8 (2). This proves Proposition 3.1.9. 2 Proposition 3.1.10 Let X be an analytic space, let (Aj )j ∈J be a finite family of closed analytic subspaces of X, and, for each j ∈ J , let Bj be a closed analytic subspace of Aj . Let S = {x ∈ X | x ∈ Bj for any j ∈ J such that x ∈ Aj },

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and let S
be an open set of Sstr/X . Then (i) Assume that X − Aj is dense in X for any j ∈ J . Let x ∈ S
. Then the following conditions (1)–(4) are equivalent. (1) x has a compact neighborhood in (S
)str/X . (2) There exists an open neighborhood U of x in X such that U ∩ Aj = U ∩ Bj for any j ∈ J . (3) There exists an open neighborhood U of x in X such that U ∩ S is a locally closed analytic subset of X. (4) There exists an open neighborhood U of x in X such that U ∩ S
is a locally closed analytic subset of X. (ii) (S
)str/X is locally compact if and only if S
is a locally closed analytic subset of X. Proof. We prove (i). The implications (2) ⇒ (3) ⇒ (4) ⇒ (1) are clear. It is sufficient to prove that (1) ⇒ (2).   We first show that we may assume X is smooth and Aj , Bj , j ∈J Aj , and j ∈J Bj are divisors with normal crossings on X. In fact, by Hironaka’s resolution of singularities [H], [AHV], we may assume that there exists a smooth analytic space −1 −1 Y with  a proper surjective morphism f : Y → X such that f (Ak ), f (Bk ), −1 −1 f ( j ∈J Aj ), and f ( j ∈J Bj ) are divisors with normal crossings on Y . If (X, (Aj )j , (Bj )j , S
, x) satisfies (1), then by Proposition 3.1.9, (Y, (f −1 (Aj ))j , (f −1 (Bj ))j , f −1 (S
), y) satisfies (1) for any y ∈ f −1 (x). We show that, if (Y, (f −1 (Aj ))j , (f −1 (Bj ))j , f −1 (S
), y) satisfies (2) for any y ∈ f −1 (x), then (X, (Aj )j , (Bj )j , S
, x) satisfies (2). In fact, we have an open neighborhood V of f −1 (x) in Y such that V ∩ f −1 (Aj ) = V ∩ f −1 (Bj ) for any j ∈ J , and since C := f (Y − V ) is closed in X by the properness of f , U := X − C is an open neighborhood of x in X and satisfies U ∩ Aj =  U ∩ Bj for any j ∈ J . Now we assume that X is smooth and Aj , Bj , j ∈J Aj , and j ∈J Bj are divisors with normal crossings on X. By replacing X by an  open neighborhood of x in n n X, we may assume that X = C , x = 0 ∈ C , and j ∈J Aj is the set of zeros of

z for some r where the z are the standard coordinates of Cn . Assume (2) is j 1≤j ≤r j not satisfied, that is, for some k ∈ J , U ∩ Ak  = U ∩ Bk for any open neighborhood U of x in X. We may assume that there exist 1 ≤ r1

< r2 ≤ r such that Ak is the set of zeros of 1≤j ≤r1 zj and Bk is the set of zeros of 1≤j ≤r2 zj . Consider Y := {z ∈ Cn | z1 = · · · = zr1 , zr1 +1 = · · · = zr2 , zj = 0 for j > r2 }, and identify Y with C2 via (zr1 , zr2 ). Then T := S ∩ Y is either one of the following two cases: (C2 − ({0} × C)) ∪ {(0, 0)},

(a)

(C − (({0} × C) ∪ (C × {0}))) ∪ {(0, 0)}.

(b)

2

If (1) is satisfied, then there should be a compact neighborhood of (0, 0) in T for the strong topology of T in C2 . This is a contradiction in case (a) by 3.1.3 (iii). By

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blowing up the origin of C2 , Case (b) is reduced to Case (a). Thus, we have proved (1) ⇒ (2) and hence (i). We prove (ii). Let Y be the normalization of the reduced part of X. Since Y → X is proper and surjective, we are reduced to the case where X is normal. We may assume that X is connected. In this case, if X − Aj is not dense, then Aj = X. By this argument, we are reduced to the case (i) where X − Aj are dense for all j . 2

3.2 GENERALIZATIONS OF ANALYTIC SPACES In this section, we consider generalizations of the notion of analytic spaces, and compare the categories of various kinds of these generalized analytic spaces. As we mentioned at the beginning of Section 2.2, we think that the category B(log) is the best one to work with in the theory of moduli of polarized logarithmic Hodge structures. For the roles of other categories, see 3.2.6. 3.2.1 As in Section 0.7, let A (resp. A(log)) be the category of analytic spaces (resp. fs logarithmic analytic spaces). We have considered bigger categories A1 ⊃ A,

A1 (log) ⊃ A(log)

(2.2.1).

Expanding these, we will consider full subcategories A1

⊃

A2

⊃

A

∪ B

∪

(1)

B∗

⊃

of the category of local ringed spaces over C, all of which, except B ∗ , contain A, and consider full subcategories A1 (log) ⊃

A2 (log)

⊃

A(log)

∪ B(log)

⊃

A(log)

∪ ⊃

∗

B (log)

∪ =

(2)

∗

B (log)

of the category of logarithmic local ringed spaces over C, all of which, except B∗ (log), contain A(log). If Ared (resp. Ared (log)) denotes the full subcategory of A (resp. A(log)) consisting of all objects whose structure sheaf O is reduced (i.e., has no nonzero nilpotent local sections), then B∗ (resp. B ∗ (log)) contains Ared (resp. Ared (log)). Among these categories, the categories with the letters B are defined in some concrete way, but the categories with the letters A, other than the categories A and A(log), are defined in some abstract way. We will see that the spaces Eσ and the important spaces
\D belong to B∗ (log) and that, if
is neat, they belong in fact to a much smaller category, the category of “logarithmic manifolds” that will be introduced in Section 3.5. First we define A and A(log). We begin with general settings.

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Definition 3.2.2 Let C be a category and let D be a full subcategory of C. For an object S of C, let hSD be the contravariant functor D → (Sets),

Y  → hSD (Y ) := Mor(Y, S).

Define the categorical closure D of D in C to be the full subcategory of C consisting of all objects S for which Mor(S, Z) → Mor(hSD , hZD ) is bijective for any object Z of C. Then we have D ⊂ D ⊂ C. By definition, an object S of D is determined, up to canonical isomorphisms, by the functor hSD (i.e., by morphisms from objects of D). For example, if C is the category of abelian groups and D is the category of finite abelian groups, then D coincides with the category of torsion abelian groups. Definition 3.2.3 (i) In the case C := (category of local ringed spaces over C) ⊃ D := A in 3.2.2, we denote D by A. (ii) In the case C := (category of fs logarithmic local ringed spaces over C) ⊃ D := A(log), we denote D by A(log). (iii) In the case C := (category of logarithmic local ringed spaces over C) ⊃ D := A(log),








we denote D by A(log) , and we denote by A(log)fs the full subcategory of A(log) consisting of objects whose logarithmic structures are fs.


We will show that A(log)fs = A(log) (see Theorem 3.2.5 below). 3.2.4 We define the other categories in the diagrams (1) and (2) in 3.2.1. Recall that A1 is the category of local ringed spaces over C satisfying the condition 2.2.1 (A1 ) Let A2 be the category of local ringed spaces X over C satisfying the condition 2.2.1 (A1 ) and also the following condition (A2 ). (A2 ) A subset U of X is open if and only if, for any analytic space A and any morphism λ : A → X, λ−1 (U ) is open in A. Let B be the category of local ringed spaces X over C which have an open covering (Uλ )λ satisfying the following condition (B). (B) For each λ, there exist an analytic space Zλ and a subset Sλ of Zλ such that, as local ringed spaces over C, Uλ is isomorphic to Sλ which is endowed with the strong topology in Zλ and the inverse image of OZλ .

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In [KU1], we called an object of A a “generalized analytic space.” Since an object of the category B is also a kind of a “generalized analytic space” and is also important in this book, we now propose to call an object of A a categorical generalized analytic space and an object of B a geometrical generalized analytic space. Let B ∗ be the category of local ringed spaces X over C which have an open covering (Uλ )λ satisfying the following condition (B∗ ). (B∗ ) For each λ, there exist a reduced analytic space Zλ , a closed analytic subspace Aλ of Zλ whose complement Zλ − Aλ is dense in Zλ , and an injection Uλ → Zλ such that, when we regard Uλ as a subset of Zλ , the topology of Uλ coincides with the strong topology of Uλ in Zλ , Uλ is open in Uλ ∪ (Zλ − Aλ ) in the strong topology of Uλ ∪ (Zλ − Aλ ) in Zλ , and OUλ is isomorphic over C to the inverse image of OZλ . For C = A1 , A2 , A, B, B ∗ , let C(log) be the category of objects of C endowed with an fs logarithmic structure. Theorem 3.2.5 We have inclusions in the diagrams (1) and (2) in 3.2.1. Furthermore, we have


A(log)fs = A(log). In theorem 3.2.5, the inclusions A2 ⊂ A1 ,

B∗ ⊂ B

and their “log versions” are clear. We prove B ⊂ A2 (and hence B(log) ⊂ A2 (log)) in 3.2.10 below. The remaining inclusions B ∗ ⊂ A ⊂ A2 ,

B∗ (log) ⊂ A(log) ⊂ A(log)




and the coincidence A(log)fs = A(log) will be proved later in Section 8.3. 3.2.6 Among the above many categories, the categories A1 (log), A2 (log), B(log), B∗ (log), and A(log) play the following special roles in this book. First, as in Chapter 2, A1 (log) is the category of spaces on which we can define a PLH. Next, A2 (log) is the category for which we have the theorem on moduli of PLHs PLH,A2 (log)  Mor(

,
\D )

(theorem B in Section 4.2),

which is more general than Theorem 0.4.27 (i) in Chapter 0. However, the definition of the category A2 (log) is too abstract, and we do not see clearly how big it is. The full subcategory B(log) of A2 (log) can be understood better, and this is the reason we have chosen to state Theorem 0.4.27 (i) in this category. Also, as in Section 3.5 below, B(log) has a good theory of infinitesimal calculus. It has fiber products (3.5.1). We mainly work with B(log) in this book. Finally, it will be proved that our space
\D belongs to B ∗ (log) and hence, by


Theorem 3.2.5, to A(log) = A(log)fs ⊂ A(log) . The facts that
\D ∈ A(log)

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and the isomorphism PLH,A(log)  Mor( ,
\D )|A(log) will show that
\D has the universal property stated in Theorem 0.4.27 (ii) (see 4.2.2). 3.2.7 We describe the differences among the above categories. (1) Let Z be an analytic space and let X be a subset of Z. Endow X with a topology T such that (weak topology) ≤ T ≤ (strong topology), and define OX to be the inverse image of OZ . Then X belongs to A1 . It belongs to A2 if and only if the topology of X is the strong topology. Since A ⊂ A2 by Theorem 3.2.5, this shows that if X in this (1) is a categorically generalized analytic space, then X is a geometrically generalized analytic space (cf. 3.2.4), and this shows that the strong topology occurs naturally when we consider categorically generalized analytic spaces. For example, if Z = C and X is the interval [0, 1] ⊂ R with the weak topology in Z, or if Z = C2 and X = (C2 − ({0} × C)) ∪ {(0, 0)} with the weak topology in Z (not as in 3.1.3 where we considered the strong topology), then the topology of X is different from the strong topology (see 3.1.2 and 3.1.3), and hence X belongs to A1 but not to A2 . (2) Let X be a one-point set and endow X with the ring C{{T }} (resp. C[[T ]]) of convergent power series (resp. formal power series). Then X belongs to B (resp. A) but not to A (resp. B). (3) The authors do not know any example that gives the difference between A(log) and A(log). These statements in 3.2.7 will be proved in 3.2.11 below. We give preliminary lemmas for the proof of B ⊂ A2 . Lemma 3.2.8 Let Z be an analytic space and let X be a subset of Z. Let T be a topology on X satisfying (weak topology) ≤ T ≤ (strong topology). Regard X as a local ringed space over C with the topology T and with the inverse image of OZ . (i) If Y is a local ringed space over C satisfying the condition (A2 ) in 3.2.4, then the canonical map Mor(Y, X) → {λ ∈ Mor(Y, Z) | λ(Y ) ⊂ X} is bijective. (ii) X satisfies the condition (A2 ) in 3.2.4 if and only if T coincides with the strong topology. Proof. (i) follows easily from the definitions, and (ii) follows from (i).

2

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Lemma 3.2.9 Let X be a local ringed space over C. Assume that, for any x ∈ X, the local ring OX,x is Noetherian and its residue field is C. Then, for any open set U of X, the canonical morphism Mor(U, Cn ) → O(U )n is injective. (Here Cn is regarded as an analytic space.) Proof. Let ϕ, ψ ∈ Mor(U, Cn ) and assume zj ◦ ϕ = zj ◦ ψ (1 ≤ j ≤ n), where the zj are the coordinate functions on Cn . We prove ϕ = ψ. We have ϕ(x) = ψ(x) for any x ∈ U . Put y = ϕ(x) = ψ(x) and Y = Cn . It is sufficient to prove that the local ring homomorphisms ϕx∗ , ψx∗ : OY,y → OX,x coincide for any x ∈ U . Since OX,x is Noetherian, OX,x is embedded into its completion Oˆ X,x . Write y = (a1 , . . . , an ) ∈ Cn . Then ϕx∗ and ψx∗ induce local ring homomorphisms ϕˆx∗ , ψˆ x∗ : Oˆ Y,y = C[[T1 , . . . , Tn ]] → Oˆ X,x ,

(1)

where Tj denotes the image of zj − aj . Since C[[T1 , . . . , Tn ]] is generated topologically by T1 , . . . , Tn as a ring over C in the (T1 , . . . , Tn )-adic topology, the homomorphisms ϕˆ x∗ and ψˆ x∗ in (1) are determined by the images of Tj (1 ≤ j ≤ n). Hence ϕˆx∗ and ψˆ x∗ are determined by the images of zj ◦ ϕ and zj ◦ ψ in Oˆ X,x (1 ≤ j ≤ n), respectively. Hence ϕx∗ = ψx∗ . 2 3.2.10 Proof of B ⊂ A2 . Let X be an object of B. Let Mor( , Cn ) be the sheaf on X defined by U  → Mor(U, Cn ) for open sets U of X. By Lemma 3.2.8 (ii), the condition (A2 ) in 3.2.4 is satisfied. Hence, to prove that X belongs to A2 , it is sufficient to show that the condition (A1 ) in 2.2.1 is satisfied, that is, it is sufficient to show ∼ n → OX,x for any x ∈ X. Since the question is local, we may that Mor( , Cn )x − assume that X is a subset of an analytic space Z and X is endowed with the strong topology in Z and with the inverse image of OZ . Then OX,x = OZ,x , and hence OX,x n is Noetherian. By Lemma 3.2.9, Mor( , Cn )x → OX,x is injective. It remains to n n prove the surjectivity. But an element f of OX,x = OZ,x defines a morphism from an open neighborhood of x in Z to Cn , and it induces a morphism from an open neighborhood of x in X to Cn which induces f . 2 3.2.11 Proofs of the statements in 3.2.7. Let X be as in 3.2.7 (1). Then the fact that X belongs to A1 is proved in the same way as the proof of B ⊂ A2 in 3.2.10. The statement concerning A2 follows from Lemma 3.2.8 (ii). Next, as in 3.2.7 (2), let X be a one-point set with the ring C{{T }} (resp. C[[T ]]). For n ≥ 1, let Sn = Spec(C[T ]/(T n )). Let S be the one-point set endowed with the ring C[[T ]]. For an analytic space Y and a morphism Y → X of local ringed spaces, since T ∈ O(X) is contained in the maximal ideal of the local ring of X at the unique point of X, the image of T in OY,y for any y ∈ Y is contained in the maximal ideal of OY,y . This shows that the image of T in OY is locally nilpotent. Hence the functor hX hSn . Hence, for any A is the sheaf on the category A associated to the presheaf lim − → A n

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local ringed space Z over C, we have    Z hSAn , hZA = lim Mor hSAn , hZA Mor hX A , hA = Mor lim − → ← − n n = lim Mor(Sn , Z) = Mor(lim Sn , Z) = Mor(S, Z). − → ← − n n This shows that X does not belong (resp. belongs) to A. On the other hand, clearly X belongs (resp. does not belong) to B. 2 3.3 SETS Eσ AND Eσ In this section, let  be a fan in gQ and let
be a subgroup of GZ that is strongly compatible with  (cf. 1.3.10 (ii)). In the next section, we will endow
\D with the structure of a logarithmic local ringed space over C. For this, we use a surjective map , Eσ →
\D σ ∈

where Eσ is a subset of a certain fs logarithmic analytic space Eˇ σ , and we transport the structure on Eˇ σ of a logarithmic local ringed space to Eσ and then to
\D via this surjection. In this section, we define Eˇ σ , Eσ , and the above surjection. 3.3.1 For an fs monoid S, let S ∨ := Hom(S, N), where N := Z≥0 is regarded as an additive monoid. Then S ∨ is an fs monoid. We have ∼

→ (S ∨ )∨ . S/S × − For an fs monoid S, a face of S is a submonoid S
of S having the following property: If a, b ∈ S and ab ∈ S
, then a, b ∈ S
. There is a bijection ∼

→ {face of S ∨ }, δ : {face of S} −

S
 → {h ∈ S ∨ | h(S
) = {0}}.

(1)

Let  and
be as at the beginning of this section. For σ ∈ ,
(σ ) =
∩ exp(σ ) (1.3.10 (2)) is an fs monoid with
(σ )× = {1} and there is a bijection ∼

→ {face of
(σ )}, {face of σ } −

τ  →
(τ ).

(2)

3.3.2 We define the spaces toricσ , torusσ , and Eˇ σ . mult ) denotes the set C (resp. R≥0 ) regarded as a In the following, Cmult (resp. R≥0 monoid with respect to the multiplication.

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For σ ∈ , denote toricσ = Spec(C[
(σ )∨ ])an = Hom(
(σ )∨ , Cmult ), torusσ = Spec(C[
(σ )∨ gp ])an = Hom(
(σ )∨ gp , C× ) = C× ⊗
(σ )gp ⊂ toricσ , We denote ˇ Eˇ σ = toricσ ×D. (toricσ , torusσ , and Eˇ σ depend on
, not only on σ , and they should be written as toric
(σ ) , torus
(σ ) , and Eˇ
(σ ) , respectively. But we abbreviate them as above since
is usually clear.) We endow toricσ with the canonical logarithmic structure in 2.1.6 (ii). We endow Eˇ σ with the inverse image of the canonical logarithmic structure of the toric variety toricσ . Since Dˇ is smooth, Eˇ σ is a logarithmically smooth fs logarithmic analytic space. We have gp π1 (toriclog σ )  π1 (torusσ ) 
(σ ) .

We often identify these three groups. Let q ∈ toricσ . Then we have a face σ (q) of σ defined as follows. We have a gp log canonical injection π1 (q log ) → π1 (toriclog σ ) =
(σ ) , and, if we regard π1 (q ) as gp a subgroup of
(σ ) via this injection, we have π1+ (q log ) = π1 (q log ) ∩
(σ ) and π1+ (q log ) is a face of
(σ ). Let σ (q) be the face of σ corresponding to the face π1+ (q log ) of
(σ ) via the bijection (2) in 3.3.1. The face of
(σ )∨ corresponding to π1+ (q log ) via the bijection (1) in 3.3.1 is {f ∈
(σ )∨ | f (q)  = 0}. Here we regard f ∈
(σ )∨ as a holomorphic function on Spec(C[
(σ )∨ ])an and so the value f (q) ∈ C of f at q is defined.

3.3.3 We define a canonical pre-PLH Hσ = (Hσ,Z ,  , σ , Fσ ) on Eˇ σ as follows. Let (Hσ,Z ,  , σ ) be the local system on toriclog σ whose stalk at the unit point 1 ∈ torusσ ⊂ toriclog is (H ,  ,  ) with the action of π1 (toriclog 0 0 σ σ ) given by the canonical log gp injection π1 (toricσ ) =
(σ ) → GZ . By 2.3.7, we have a canonical isomorphism log of Otoricσ -modules log

log

ν : Otoricσ ⊗Z Hσ,Z  Otoricσ ⊗Z H0 , log

via which we identify these two Otoricσ -modules. We also denote by (Hσ,Z ,  , σ ) the log local system on Eˇ σ which is the inverse image of (Hσ,Z ,  , σ ) via the projection

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log Eˇ σ → toriclog σ . We have log

log

σ

σ

OEˇ ⊗Z Hσ,Z = OEˇ ⊗Z H0 via the above identification. On the other hand, ODˇ ⊗Z H0 has the universal Hodge ˇ it defines a filtration on O ˇ ⊗Z H0 and hence filtration. Via the projection Eˇ σ → D, Eσ log

log

a filtration Fσ on OEˇ ⊗Z H0 = OEˇ ⊗Z Hσ,Z with which Hσ := (Hσ,Z ,  , σ , Fσ ) σ σ becomes a pre-PLH on Eˇ σ . This pre-PLH has the unique
(σ )gp -level structure µσ whose germ at 1 ∈ torusσ is the class of the identity map of H0 . With the identification 1 ⊗ Hσ,Z = ξ −1 (1 ⊗ H0 ) where ξ is as in 2.3.7 (we take
(σ )∨ as S in 2.3.7), this canonical
(σ )gp -level structure is expressed as ξ(1 ⊗ v) = 1 ⊗ µσ (v)

for v ∈ Hσ,Z .

(1)

3.3.4 We define subsets Eσ and E˜ σ of Eˇ σ . Let Hσ be the canonical pre-PLH on Eˇ σ defined above. We define Eσ to be the set of all points x of Eˇ σ such that the inverse image of Hσ to the fs logarithmic point x is a PLH. We define E˜ σ to be the set of all points x of Eˇ such that the inverse image of Hσ to x satisfies Griffiths transversality (the small Griffiths transversality). We have Eσ ⊂ E˜ σ ⊂ Eˇ σ . 3.3.5 We have a canonical surjective homomorphism e : σC → torusσ = C× ⊗
(σ )gp , z log(γ )  → exp(2π iz) ⊗ γ (z ∈ C, γ ∈
(σ )gp ),

(1)

whose kernel coincides with log(
(σ )gp ). Let q ∈ toricσ , and let S =
(σ )∨ , S
= {f ∈ S | f (q)  = 0} where f ∈ S is regarded as a holomorphic function on toricσ . By passing to the quotient, e induces a commutative diagram with surjective vertical arrows σC / log(
(σ )gp )   

e

−−−−→ torusσ = Hom(S gp , C× ) ∼

∼

σC /(σ (q)C + log(
(σ )gp )) −−−−→

  

(2)

Hom((S
)gp , C× ).

We define the class of q in σC /(σ (q)C + log(
(σ )gp )) to be the element that corresponds to the element f  → f (q) of Hom((S
)gp , C× ) via the lower isomorphism of this diagram (see 3.3.2 for σ (q)).

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Proposition 3.3.6 The period map ϕˇ : Eˇ σ →
(σ )gp \Dˇ orb (2.5.3) of the canonical pre-PLH Hσ on Eˇ σ with the canonical
(σ )gp -level structure µσ (3.3.3) is given by ϕ(q, ˇ F ) = ((σ (q), exp(σ (q)C ) exp(z)F ) mod
(σ )gp ) ˇ where z is any fixed element of σC whose image in σC /(σ (q)C + (q ∈ toricσ , F ∈ D) log(
(σ )gp )) is the class of q in the sense of 3.3.5. Proof. Take y ∈ Eˇ σ lying over (q, F ). Since the
(σ )gp -level structure µσ is given by µσ (v) = ξ(1 ⊗ v) (3.3.3 (2)), we have µ˜ σ,y (Fσ (s)) = ξ(s)F . Let S =
(σ )∨ , and let S
= {f ∈ S | f (q)  = 0} as in 3.3.5. For s ∈ sp(y), let s˜ : S gp → C× be the homomorphism f  → exp(s(log(f ))). When s ranges over sp(y), s˜ ranges over all homomorphisms S gp → C× whose restriction to (S
)gp coincides with f  → f (q). On the other hand, when we regard s˜ as an element of torusσ , we have s˜ = e( nj=1 (2π i)−1 s(log(qj ))Nj ) where (qj )1≤j ≤n and ∨ (Nj )1≤j ≤n are as in 2.3.7 (we take
(σ ) as S in 2.3.7). Hence, as s ranges over n −1 sp(y), ( j =1 (2πi) s(log(qj ))Nj mod log(
(σ )gp )) ranges over all elements of σC / log(
(σ )gp ) whose images in σC /(σ (q)C + log(
(σ )gp )) coincide with the class of q. This proves the result. 2 log

Corollary 3.3.7 Let (q, F ) ∈ Eˇ σ and let z be an element of σC whose image in σC /(σ (q)C + log(
(σ )gp )) coincides with the class of q. (i) (q, F ) ∈ Eσ if and only if exp(σ (q)C ) exp(z)F is a σ (q)-nilpotent orbit. (ii) (q, F ) ∈ E˜ σ if and only if N (F p ) ⊂ F p−1 for all N ∈ σ (q) and for all p. 2

Proof. This follows from 3.3.6 and 2.5.5. 3.3.8

The inverse image of Hσ on Eσ is a PLH. By 3.3.6, it is a PLH of type (w, (hp,q ), H0 ,  , 0 ,
(σ )gp , {face of σ }). The period map ϕˇ : Eσ →
(σ )gp \Dˇ orb takes values in
(σ )gp \Dσ and coincides with the period map ϕ : Eσ →
(σ )gp \Dσ in 2.5.10. 3.3.9 For σ ∈ , denote mult ) ⊂ toricσ , |toric|σ = Hom(
(σ )∨ , R≥0

|torus|σ = Hom(
(σ )∨ gp , R>0 ) = R>0 ⊗
(σ )gp ⊂ torusσ . Define Eˇ σ = |toric|σ ×Dˇ ⊂ Eˇ σ ,

Eσ = Eσ ∩ Eˇ σ .

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3.3.10 We define a kind of a period map ϕ  : Eσ → Dσ which makes the following diagram commutative. ϕ

Eσ −−−−→   

Dσ   

ϕ

Eσ −−−−→
(σ )gp \Dσ . The injection mult |toric|σ = Hom(
(σ )∨ , R≥0 ) ∨ mult ∨ gp → toriclog , S1 ) σ = Hom(
(σ ) , R≥0 ) × Hom(
(σ ) mult ) to (h, 1), where 1 denotes the trivial homomorsending h ∈ Hom(
(σ )∨ , R≥0 ∨ gp 1 phism
(σ ) → S , induces an injection

ˇ ˇ log Eˇ σ = |toric|σ ×Dˇ → toriclog σ ×D = Eσ .

The restriction of ξ = exp( nj=1 (2π i)−1 log(qj ) ⊗ Nj ) ((qj )1≤j ≤n and (Nj )1≤j ≤n are as in 2.3.7 where we take
(σ )∨ as S of 2.3.7) to |toric|σ has one global branch defined by taking the branch of log(qj ) whose restriction to |torus|σ has only real values for each j . This branch of ξ on |toric|σ is independent of the choice of the ∼ → H0 basis (qj )1≤j ≤n of
(σ )∨ gp . By using this ξ , define an isomorphism Hσ,Z − on |toric|σ by v  → ξ(1 ⊗ v) ∈ 1 ⊗ H0 = H0 (v ∈ Hσ,Z ). log Define ϕ  : Eσ → Dσ following the definition of ϕ log : Eˇ σ → Dˇ σ as in 2.5.10 ∼ → H0 on |toric|σ as but without modulo
(σ )gp by using this isomorphism Hσ,Z − µ˜ y in 2.5.10. An explicit description of this map is ϕ  (q, F ) = (σ (q), exp(iσ (q)R ) exp(iy)F ), where y is any element of σR such that the image of iy in σC /(σ (q)C + σR ) coincides with the class of q (3.3.5). 3.3.11 Example. Let σ ∈  with σ  = {0}. Then,
(σ )  N, toricσ = C, |toric|σ = R≥0 , ˇ Eˇ σ = (R≥0 ) × D. ˇ Let γ be the generator of
(σ ) and let N := Eˇ σ = C × D, log(γ ). Then we have     exp((2π i)−1 log(q)N )F ∈ D if q  = 0, and Eσ = (q, F ) ∈ C × Dˇ  , exp(CN )F is a σ -nilpotent orbit if q = 0     exp((2π i)−1 log(q)N )F ∈ D if q  = 0, and Eσ = (q, F ) ∈ (R≥0 ) × Dˇ  . exp(iRN )F is a σ -nilpotent i-orbit if q = 0

125

STRONG TOPOLOGY

Here the cases q  = 0, q = 0 correspond to σ (q) = {0}, σ (q) = σ , respectively. The period map ϕ : Eσ →
(σ )gp \Dσ (resp. ϕ  : Eσ → Dσ ) is (q, F )  → exp((2πi)−1 log(q)N )F if q  = 0, (q, F )  → (σ, exp(CN )F ) (resp. (σ, exp(iRN )F ))

if q = 0.

In the case of Example (i) in 0.3.2 where D = h and Dˇ = P1 (C), for     1 Z 0 R≥0 ,
= ⊂ SL(2, Z), σ = 0 0 0 1 we have Eσ = {(q, z) ∈ C × C | |qe2π iz | < 1} and the map ϕˇ : Eσ →
(σ )gp \Dσ =
\Dσ is identified with Eσ → , (q, z)  → qe2π iz if we identify
\Dσ with  via the bijection that sends q ∈ ∗ =  − {0} to (2πi)−1 log(q) ∈ Z\h =
\D and sends 0 ∈  to ((σ, C) mod
) ∈
\Dσ .

3.4 SPACES Eσ ,
\D , Eσ , AND D Let  be a fan in gQ and let
be a subgroup of GZ that is strongly compatible with  (cf. 1.3.10 (ii)). Let σ ∈ . We define structures of logarithmic local ringed spaces over C on Eσ , E˜ σ , and on
\D , and introduce suitable topologies on Eσ  and on D . 3.4.1 First we consider Eσ and E˜ σ . Recall (3.3.2) that we endow Eˇ σ = toricσ ×Dˇ with the inverse image of the canonical fs logarithmic structure on toricσ . We endow the subsets Eσ and E˜ σ of Eˇ σ , with the following structures of logarithmic local ringed spaces over C. The topology is the strong topology in Eˇ σ . The sheaf O of rings and the logarithmic structure M are the inverse images of O and M of Eˇ σ , respectively. Then the logarithmic structure of Eσ and that of E˜ σ coincide with the inverse images of the canonical logarithmic structure of toricσ . Note that the induced homomorphisms α : MEσ → OEσ and α : ME˜ σ → OE˜ σ are injective. 3.4.2 Next we consider
\D . First, we endow
(σ )gp \Dσ with the quotient topology via the period map Eσ →
(σ )gp \Dσ in 3.3.8. We then endow
\D with the strongest topology for which the maps
(σ )gp \Dσ →
\D are continuous for all σ ∈ . We endow
\D with the following sheaf of rings O
\D over C and the following logarithmic structure M
\D . Let πσ : Eσ →
(σ )gp \Dσ →
\D be the composite map. For any open set U of
\D and for any σ ∈ ,

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CHAPTER 3

let Uσ := πσ−1 (U ) and define O
\D (U ) (resp. M
\D (U )) := {map f : U → C | f ◦ πσ ∈ OEσ (Uσ ) (resp. ∈ MEσ (Uσ )) (∀σ ∈ )}. Here, in the definition of M
\D , we identify MEσ with its image in OEσ via the injection α : MEσ → OEσ . 3.4.3 We introduce the topology of Eσ as a subspace of Eσ (cf. 3.3.9). We introduce the quotient topology of Dσ via the surjection Eσ → Dσ in 3.3.10. We introduce on  D the strongest topology for which the inclusion maps Dσ → D (σ ∈ ) are  continuous. Note that the surjection D →
\D (1.3.9) becomes continuous. Proposition 3.4.4

(i) Let (σ, Z) ∈ D , let F ∈ Z, and write σ = 1≤j ≤n R≥0 Nj . Then      zj Nj  F mod
 in
\D . ((σ, Z) mod
) = lim exp  Im(zj )→∞ 1≤j ≤n

1≤j ≤n



(ii) Let (σ, Z) ∈ D , let F ∈ Z, and let Nj be as above. Then     iyj Nj  F in D . (σ, Z) = lim exp  yj →∞ 1≤j ≤n

Proof. Replacing F by exp that

1≤j ≤n

 1≤j ≤n



exp 

iaj Nj F for some aj  0, we may assume



 iyj Nj  F ∈ D

1≤j ≤n

for any yj ≥ 0 (1 ≤ j ≤ n). Let 0 ∈ toricσ = Hom(
(σ )∨ , Cmult ) be the element that sends all nontrivial elements of
(σ )∨ to 0 ∈ C. Then, by the morphism of analytic spaces toricσ → Eˇ σ ,

q  → (q, F ),

a sufficiently small open neighborhood U of 0 in toricσ is sent into Eσ , and hence gives a continuous map U → Eσ in the strong topology of Eσ in Eˇ σ . When Im(zj ) → ∞ (1 ≤ j ≤ n),    zj Nj  → 0 in toricσ , (1) e 1≤j ≤n

127

STRONG TOPOLOGY

where e is as in 3.3.5, and hence      e  zj Nj  , F  → (0, F )

in Eσ

(2)

1≤j ≤n

in the strong toplogy of Eσ . By taking the image of (2) under the continuous map Eσ →
\D , we have      exp  zj Nj  F mod
 → ((σ, Z) mod
) in
\D , 1≤j ≤n

proving (i). Next, when yj → ∞ (1 ≤ j ≤ n), we have, by (2),      e  iyj Nj  , F  → (0, F )

in Eσ .

(3)

1≤j ≤n

By applying the continuous map Eσ → D to (3), we obtain     exp  iyj Nj  F → (σ, Z) in D .

2

1≤j ≤n

3.4.5 Example. In the case of Example (i) in 0.3.2, by 3.4.2, the bijection between
(σ )gp \Dσ and  given in 3.3.11 is in fact an isomorphism
(σ )gp \Dσ   of logarithmic local ringed spaces over C (cf. 0.4.13).

3.5 INFINITESIMAL CALCULUS AND LOGARITHMIC MANIFOLDS We show that the usual infinitesimal calculus on analytic spaces is naturally extended to that on objects of the category B(log) in 3.2.4. We consider “logarithmic manifolds” as a special kind of “logarithmically smooth” objects of B(log). Proposition 3.5.1 (i) The category B(log) has fiber products. (ii) Let X ×Z Y be a fiber product in B(log) and let X ×cl Z Y be the fiber product of X → Z ← Y in the category of topological spaces. (Here cl means classical.) Consider the canonical continuous map f : X ×Z Y → X ×cl Z Y . This map f is bijective if either one of the following conditions (1) and (2) is satisfied. (1) Either X → Z or Y → Z is strict (that is, either the logarithmic structure of X or that of Y coincides with the inverse image of that of Z). (2) The logarithmic structure of Z is trivial.

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CHAPTER 3

The map f is a homeomorphism if either one of the above conditions (1) and (2) is satisfied and furthermore either one of the following conditions (3) and (4) is satisfied. (3) Either one of X and Y is an fs logarithmic analytic space. (4) Locally, X and Y are analytically constructible subsets of fs logarithmic analytic spaces X
and Y
, respectively, X (resp. Y ) is endowed with the strong topology in X
(resp. Y
), and O and M of X (resp. Y ) are the pullbacks of those of X
(resp. Y
). Proof. We prove (i). Let p1 : X → Z, p2 : Y → Z be morphisms in B(log). Working locally, we may assume that there are objects X
, Y
, and Z
of A(log) (0.7.4) such that X (resp. Y , resp. Z) is a subset of X
(resp. Y
, resp. Z
) endowed with the strong topology and with the inverse images of the structure sheaf O and the logarithmic structure M of X
(resp. Y
, resp. Z
), and morphisms p1
: X
→ Z
and p2
: Y
→ Z
in A(log) such that the following diagram is commutative: p1

X −−−−→    p1


p2

Z ←−−−−   

Y   

p2


X
−−−−→ Z
←−−−− Y
. Let X
×Z
Y
be the fiber product in A(log) (2.1.10), and let S ⊂ X
×Z
Y
be the
cl

cl


inverse image of X ×cl Z Y under X ×Z
Y → X ×Z
Y where X ×Z
Y denotes


the fiber product of X → Z ← Y in the category of analytic spaces (which is the fiber product also as a topological space). Endow S with the strong topology in X
×Z
Y
and with the inverse images of O and M of X
×Z
Y
. We prove that S is the fiber product X ×Z Y in B(log). For an object T of B(log), by 3.2.8 and by B(log) ⊂ A2 (log) (3.2.10), we have the identifications {λ : T → S, a morphism} = {λ : T → X
×Z
Y
, a morphism such that λ(T ) ⊂ S}    λ : T → X
, λ2 : T → Y
morphisms, = (λ1 , λ2 ) 1 , λ1 (T ) ⊂ X, λ2 (T ) ⊂ Y , p1
◦ λ1 = p2
◦ λ2 = {(λ1 , λ2 ) | λ1 : T → X, λ2 : T → Y morphisms, p1 ◦ λ1 = p2 ◦ λ2 }. We prove (ii). Working locally, let X
, Y
, etc. be as in the proof of (i). Let X × Y be the product in B(log) and let X ×cl Y be the product in the category of topological spaces. The topology of X × Y is the strong topology in X
× Y
. The canonical bijection X × Y → X ×cl Y is continuous, and, by 3.1.8, it is a homeomorphism if one of the conditions (3) and (4) is satisfied. By Proposition 3.1.9 applied to the morphism X
×Z
Y
→ X
× Y
and to the subset X × Y of X
× Y
, X ×Z Y coincides as a topological space with the fiber product of X
×Z
Y
→ X
× Y
← X × Y in the category of topological spaces. Let P be the fiber product of X
×cl Z
Y
→ X
× Y
← X × Y in the category of topological spaces. Then the canonical

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STRONG TOPOLOGY

map P → X ×cl Z Y is bijective and continuous. If one of the conditions (1) and (2) is satisfied, locally, (p1
, p2
) also satisfies one of the conditions (1) and (2). Hence ∼ ∼
X
×Z
Y
− → X
×cl → P as topological Z
Y by 2.1.10, and this shows X ×Z Y − spaces. Hence X ×Z Y → X ×cl Z Y is bijective. If one of the conditions (3) and cl (4) is satisfied, P → X ×cl Z Y is a homeomorphism since X × Y → X × Y is a homeomorphism. These results prove (ii). 2 In this Section 3.5, we use direct products X × Y (= X ×Spec(C) Y with the trivial logarithmic structure on Spec(C)) in B(log). In the next Section 3.6, we will use more general fiber products in B(log). 3.5.2 1 of logarithmic differential Let X be an object of B(log). We define the sheaf ωX 1-forms on X, imitating its definition in 2.1.7 for an fs logarithmic analytic space, by using the product X × X in 3.5.1. q q,log log q q 1 (q ≥ 0), ωX := OX ⊗τ −1 (OX ) τ −1 (ωX ), d : OX → Define ωX := OX ωX gp

log

1,log

1 1 ωX , d log : MX → ωX , d : OX → ωX •,log • ωX and ωX , just as in 2.1.7.

and the logarithmic de Rham complexes

Lemma 3.5.3 Let Z be an object of A(log), let X be a subset of Z, and endow X with the strong topology and with the inverse images of OZ and MZ . Then q

∼

q

→ ωX , ωZ |X − Proof. These are proved easily.

q,log

ωZ

∼

q,log

→ ωX |Xlog −

,

(q ≥ 0). 2

Definition 3.5.4 (i) Recall (2.1.11) that an object of A(log) is logarithmically smooth if it is locally an open set of a toric variety endowed with the canonical logarithmic structure (2.1.6 (ii)). (ii) An object of B(log) is said to be logarithmically smooth if it is locally isomorphic to a subset of a logarithmically smooth object Z of A(log), endowed with the strong topology in Z and with the inverse images of OZ and MZ . (iii) Let X be a logarithmically smooth object of B(log). We define the sheaf θX 1 on X, called the sheaf of logarithmic vector fields on X, as the OX -dual of ωX . 1 (Note that ωX is locally free of finite rank by 3.5.3.) Proposition 3.5.5 Let X and Y be objects of B(log). Let I be an ideal of OY which is locally of finite type and which satisfies I 2 = 0, and let Y0 be the object of B(log) whose underlying topological space is that of Y , and which is endowed with OY0 = OY /I and with the inverse image of MY . Let f : Y0 → X be a morphism, and let P be the set of all morphisms Y → X that extend f . (i) Assume that P is not empty. Then P is a principal homogeneous space under 1 the group Q := HomOY0 (f ∗ (ωX ), I). (ii) If X is logarithmically smooth, then P is not empty locally on Y .

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Proof. We prove (i). We define an action of Q on the set P as follows. Let δ ∈ Q. Then, for g ∈ P , g
= δg ∈ P is the following extension Y → X of f . The homomorphism g
∗ : f −1 (OX ) → OY is given by a  → g ∗ (a) + δ(da), and the homomorphism g
∗ : f −1 (MX ) → MY is given by a  → g ∗ (a)(1 + δ(d log(a))). We prove that, for any g, g
∈ P , there exists a unique δ ∈ Q that sends g to g
. 1
Let 1X be ωX
where X is the same as X as a local ringed space over C but the logarithmic structure of X
is trivial. Since the restriction of (g
, g) : Y → X × X to Y0 coincides with the composite of f : Y0 → X and the diagonal morphism X → X × X, and since I 2 = 0, there is a unique homomorphism u : f ∗ 1X → I gp such that g
∗ (b) = g ∗ (b) + u(db) for any b ∈ OX . Furthermore, for a ∈ f −1 (MX ), gp gp gp ∗
∗ g (a) ∈ MY and g (a) ∈ MY have the same image in MY0 , and hence there exists gp a unique homomorphism v : f −1 (MX ) → I such that g
∗ (a) = g ∗ (a)(1 + v(a)) gp gp −1 −1 for a ∈ f (MX ). For a ∈ f (MX ), α(g
∗ (a)) = α(g ∗ (a)(1 + v(a))) shows that gp 1 u(dα(a)) = α(a)v(a). Since ωX is the quotient of 1X ⊕ (OX ⊗Z MX ) by the OX -submodule generated by {(−dα(a), α(a) ⊗ a) | a ∈ MX }, we have a unique 1 ) → I which induces u on f ∗ 1X and which satisfies homomorphism δ : f ∗ (ωX gp v(a) = δ(d log(a)) for a ∈ f −1 (MX ). We prove (ii). Because Mor(Y, Spec(C[S])an )  Hom(S, MY ) for any object Y of A1 (log) and any fs monoid S, (ii) is reduced to the fact that, for an fs monoid S, a homomorphism S → MY0 lifts locally to a homomorphism gp gp S → MY . Since MY → MY0 is surjective and the kernel ( I) is divisible as an gp gp gp abelian group, S gp → MY0 lifts to S gp → MY locally. Any lifting S gp → MY of S → MY0 gives a homomorphism S → MY . 2 3.5.6 For a logarithmically smooth object X of B(log), we define the logarithmic tangent bundle TX of X as the vector bundle associated with the sheaf θX . Here, for an object X of B(log) and for a locally free OX -module F of finite rank on X, the vector bundle V (F) associated with F is the object of B(log) that represents the functor Y → {(f, a) | f : Y → X, a ∈
(Y, f ∗ F)} from B(log) to the category of sets. The existence of V (F) is shown as follows. We ⊕r for some r ≥ 0. Then may work locally on X and hence we may assume F = OX r since B(log) ⊂ A1 (log) (3.2.10), the product X × C , as in 3.5.1, has the property of V (F). By 3.5.5, for a logarithmically smooth object X of B(log), TX represents the functor Y  → Mor(Y [T ]/(T 2 ), X)

(1)

from B(log) to the category of sets, where Y [T ]/(T 2 ) denotes the object of B(log) whose underlying topological space is the same as Y , whose structure sheaf is

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STRONG TOPOLOGY

OY [T ]/(T 2 ) (T is an indeterminate), and whose logarithmic structure is the inverse image of that of Y . If X is a complex manifold with the trivial logarithmic structure, TX coincides with the usual tangent bundle of X. For a morphism f : X → X
of logarithmically smooth objects of B(log), the 1 1 canonical homomorphism f ∗ (ωX
) → ωX induces a morphism TX → TX
, which we denote by df . This morphism df is also obtained from the above functorial interpretation (1) of the logarithmic tangent bundle. Definition 3.5.7 By a logarithmic manifold, we mean a logarithmic local ringed space over C which has an open covering (Uλ )λ with the following property: For each λ, there exist a logarithmically smooth fs logarithmic analytic space Zλ , a finite subset Iλ of
(Zλ , ωZ1 λ ), and an isomorphism of logarithmic local ringed spaces over C between Uλ and an open set of Sλ = {z ∈ Zλ | the image of Iλ in ωz1 is zero},

(1)

where Sλ is endowed with the strong topology in Zλ and with the inverse images of OZλ and MZλ . We will see (Theorem A in Section 4.1 below) that Eσ and
\D , for
neat and strongly compatible with , are logarithmic manifolds. The following 3.5.8 is seen easily. Proposition 3.5.8 (i) A logarithmic manifold is a logarithmically smooth object of B(log). (ii) A logarithmic manifold is an object of the category B ∗ (log) in 3.2.4. Proposition 3.5.9 Let Z be a logarithmically smooth fs logarithmic analytic space, let I be a finite subset of
(Z, ωZ1 ), and let S be the set of all z ∈ Z such that the image of I in the sheaf ωz1 of logarithmic differential 1-forms on the fs logarithmic point z is zero. Then S is an analytically constructible subset (3.1.4) of Z. Note that, by 3.5.9 and 3.1.5, the topology of a logarithmic manifold is well understood. Proof of Proposition 3.5.9. We may assume that Z is an open set of the toric variety Spec(C[S])an for an fs monoid S and that the logarithmic structure of Z is induced ∼ → from the canonical logarithmic structure of Spec(C[S])an . We have OZ ⊗Z S gp − 1

ωZ , f ⊗ g  → f d log(g) (2.1.8 (ii)). For each face S of S, let A(S ) be the closed analytic subspace of Z defined by the ideal of OZ generated by the image of the complement S − S
of S
. Let JS
be the coherent ideal of OA(S
) generated by the images of I under h

→ OA(S
) , ωZ1  OZ ⊗Z S gp → OA(S
) ⊗Z (S gp /S
gp ) −

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where h ranges over all homomorphisms h : S gp /S
gp → Z. Let B(S
) be the closed analytic subspace of A(S
) defined by JS
. It is sufficient to prove that S = {z ∈ Z | z ∈ B(S
) for any face S
of S such that z ∈ A(S
)}.

(1)

For z ∈ Z, let S(z) := {f ∈ S | f (z)  = 0} be the face of S corresponding to z, then, as is easily seen, we have (1) For a face S
of S, z ∈ A(S
) if and only if S(z) ⊂ S
. ∼ → ωz1 , 1 ⊗ g  → d log(g). (2) C ⊗Z (S gp /S(z)gp ) − 2

These results (2) and (3) yield (1).

We now prove that the space E˜ σ ⊂ Eˇ σ (3.3.4, 3.4.1) is a logarithmic manifold. Note that Eˇ σ is a logarithmically smooth fs logarithmic analytic space. Proposition 3.5.10 Let the notation be as in Section 3.3, and denote Eˇ σ (resp. E˜ σ ) by Z (resp. there exist an open covering (Uλ )λ of Z and a  S). Then, finite subset Iλ of
Uλ , ωU1 λ for each λ such that S ∩ Uλ = {z ∈ Uλ | the image of Iλ in ωz1 is zero}. In particular, E˜ σ is a logarithmic manifold. Proof. Let Y = toricσ . Since Y = Spec(C[
(σ )∨ ])an , ωY1 is identified with OY ⊗Z (
(σ )∨ )gp and, via log :
(σ )gp → σC , it is identified with the OY -dual of OY ⊗C σC . Hence, for each q ∈ Y , the fiber ωY1 (q) of ωY1 at q is identified with the dual Cvector space of σC . Furthermore, this identification induces an identification of the logarithmic differential module ωq1 of the fs logarithmic point q, which is a quotient of ωY1 (q), with the dual C-vector space of σ (q)C (for σ (q), see 3.3.2). On the other hand, let G := ODˇ ⊗C σC ⊃ G −1 := {X ∈ G | XFuniv ⊂ Funiv (∀p ∈ Z)}, p

p−1

ˇ Then, G, G −1 , and G/G −1 where Funiv is the universal filtration of ODˇ ⊗Z H0 on D. ˇ the fiber G −1 (F ) of G −1 at F are locally free ODˇ -modules of finite rank. For F ∈ D, p p−1 (∀p ∈ Z)}. By 3.3.7 (ii), for z = (q, F ) ∈ coincides with {X ∈ σC | XF ⊂ F Eˇ σ , z ∈ S if and only if the map σ (q)C → σC /G −1 (F ) = G(F )/G −1 (F ) is the zero map. Let p1 : Z → Y and p2 : Z → Dˇ be the projections, and define a locally free OZ -module F of finite rank by F := p2∗ (G/G −1 ). Then, the canonical projection OZ ⊗C σC → F defines a canonical global section of p1∗ (ωY1 ) ⊗OZ F. Hence we obtain a global section of ωZ1 ⊗OZ F, which we denote by η. For z = (q, F ) ∈ Z, since ωq1  ωz1 , the canonical map σ (q)C → σC /G −1 (F ) is identified with the image of η in ωz1 ⊗C F(z). Hence z ∈ S if and only if the image of η in ωz1 ⊗C F(z) is zero.

STRONG TOPOLOGY

133

an open set U of Z and if we write η|U =

If we have a basis (ej )j ∈J of F on 1 η ⊗ e with η a section of ω | j j j U j ∈J Z , then S ∩ U = {z ∈ U | the images of ηj in ωz1 are zero for all j ∈ J }. This proves 3.5.10.

2

3.6 LOGARITHMIC MODIFICATIONS The subject of this section is as follows. Let S be an fs monoid, let X be the toric variety Spec(C[S])an , and let U be the dense open set Spec(C[S gp ])an of X. Then in the toric geometry [KKMS], for a finite add rational subdivision  of the cone Hom(S, R≥0 ), we have a modification X() of X associated with , with a proper morphism f : X() → X which induces an ∼ → U . If I is an ideal of OX generated by some elements of isomorphism f −1 (U ) − S, the normalization of the blow-up of X with respect to I is isomorphic to X() for some . In this section, we consider such a modification X() (“logarithmic modification” 3.6.12) of an object X of B(log). (See [Kk2, 9] for a similar theory for schemes.) This is important when we try to extend period maps to the boundary in Section 4.3. We also consider a space Xval defined by using projective limits of logarithmic modifications (3.6.18, 3.6.23). This space Xval will play important roles in later sections from Chapter 5, when we consider the “valuative spaces” in the fundamental diagram in Introduction. Proposition 3.6.1 Let X be an object of B(log) and J be a subset of gp gp × ×
(X, MX /OX ) such that the submonoid of
(X, MX /OX ) generated by J has gp × finite generators. (For example, any finite subset J of
(X, MX /OX ) satisfies this condition.) Then (i) There is an object X[J ] of B(log) over X having the following property. For any object Y of B(log) over X, the set Mor X (Y, X[J ]) of morphisms Y → X[J ] over X is either a one-point set or an empty set, and it is nonempty if and only if the gp pullbacks of all elements of J on Y in
(Y, MY /OY× ) belong to
(Y, MY /OY× ). × −1 (ii) If a ∈
(X, MX /OX ) for all a ∈ J , X[J ] is an open subobject of X in B(log). Here we say an object U of B(log) is an open subobject of an object X of B(log) if U is an open set of X and OU = OX |U and MU = MX |U . The characterizing property of X[J ] in proposition 3.6.1 (i) shows that, if J  gp × denotes the submonoid of
(X, MX /OX ) generated by J , then X[J ] = X[J ]. 3.6.2 Before we prove proposition 3.6.1, we consider a special case of it. Let S be an fs monoid, let X = Spec(C[S])an with the canonical logarithmic structure, and let J˜ be a subset of S gp that generates a finitely generated submonoid of S gp . Let J be

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gp × ). Then X[J ] = Spec(C[S
])an , where S
is the the image of J˜ in
(X, MX /OX gp smallest fs submonoid of S containing S and J˜, endowed with the canonical logarithmic structure. If S

denotes the submonoid of S gp generated by S and J˜, then S
= {a ∈ S gp | a n ∈ S

for some n ≥ 1}, and Spec(C[S
])an coincides with the normalization of Spec(C[S

])an .

3.6.3 Example. If X = Spec(C[T1 , T2 ])an with the logarithmic structure associated with × N2 → OX , (m, n)  → T1m T2n , and if J is the one-point set {T1 /T2 mod OX }, then X[J ] = Spec(C[T1 /T2 , T2 ])an with the logarithmic structure associated with N2 → OX[J ] , (m, n)  → (T1 /T2 )m T2n . 3.6.4 Proof of Proposition 3.6.1. We may assume that J is a finite set. (i) It is sufficient to prove the existence of X[J ] locally on X. Locally on X, there are an fs monoid S, a homomorphism h : S → MX , and a finite subset J˜ × of S gp such that J = h(J˜) mod OX . Since B(log) ⊂ A1 (log) (3.2.10), we have a morphism X → Spec(C[S])an corresponding to h (2.2.2). Let S
be the smallest fs submonoid of S gp containing S and J˜ and let X
be the fiber product X ×Spec(C[S])an Spec(C[S
])an in the category B(log) (3.5.1). Then this X
has the property of X[J ] stated in Proposition 3.6.1. (ii) Under the assumption of (ii), in the proof of (i), we can take J˜ such that (J˜)−1 ⊂ S. In this situation, Spec(C[S
])an is an open subobject of Spec(C[S])an . Hence the fiber product X
becomes also an open subobject of X.
3.6.5 Let X, J , and X[J ] be as in 3.6.1. Then the map of sheaves Mor( , X[J ]) → Mor( , X) on the category B(log) is injective by the characterizing property of X[J ] in Proposition 3.6.1 (i). However the map of the spaces X[J ] → X is not necessarily injective. For example, in the example in 3.6.3, the map X[J ] → X is described as C2 → C2 , (x, y)  → (xy, y) and hence not injective. Proposition 3.6.6 Let X be an object of B(log) and I be a nonempty finite subset gp × of
(X, MX /OX ). Then (i) There is an object BI (X) of B(log) over X (which we call the logarithmic blow-up of X with respect to I ) having the following property. For any object Y of B(log) over X, the set Mor X (Y, BI (X)) is either a one-point set or an empty set, and it is nonempty if and only if, locally on Y , there is an element a of I such that gp the pullbacks of a −1 b in MY /OY× for all b ∈ I belong to MY /OY× . (ii) There is an object BI∗ (X) of B(log) over X having the following property. For any object Y of B(log) over X, the set Mor X (Y, BI∗ (X)) is either a one-point set or an empty set, and it is nonempty if and only if for any a, b ∈ I , locally on Y , gp at least one of the pullbacks of a −1 b and ab−1 in MY /OY× belong to MY /OY× .

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STRONG TOPOLOGY

(iii) There is a morphism BI∗ (X) → BI (X) over X. Furthermore, BI∗ (X) = gp × BI
(X) for some nonempty finite subset I
of
(X, MX /OX ). gp ×
(iv) Let I be a nonempty subset of
(X, MX /OX ). Then Mor( , BI∗
(X)) ⊂ Mor( , BI∗ (X)) in Mor( , X) if I
⊃ I, Mor( , BI (X)) ∩ Mor( , BI
(X)) = Mor( , BI I
(X)) in Mor( , X) where I I
= {aa
| a ∈ I, a
∈ I
}. (v) Assume that X = Spec(C[S])an for an fs monoid S and that I ⊂ gp ×
(X, MX /OX ) is the image of a nonempty finite subset I˜ of S. Then, as an analytic space, BI (X) coincides with the normalization of the blow-up of X along the closed analytic subspace of X defined by the ideal of OX generated by I˜. Proof. We prove (i). For a ∈ I , let a −1 I = {a −1 b | b ∈ I } and X[a −1 I ] be as in 3.6.1. By 3.6.5, Mor( , X[a −1 I ]) is  a subsheaf of the sheaf Mor( , X) on the category B(log). Define the sheaf a∈I Mor( , X[a −1 I ]) as the union of the sheaves Mor( , X[a −1 I ]) in Mor( , X). For a, b ∈ I , Mor( , X[a −1 I ]) ∩ Mor( , X[b−1 I ]) is represented by the open subobject X[a −1 I ][(a −1 b)−1 ] of X[a −1 I ] in B(log) (Proposition 3.6.1 (ii)), which is identified with the open subobject X[b−1 I ][(b−1 a)−1 ] of X[b−1 I ] in B(log). Hence the sheaf  Mor( , X[a −1 I ]) is represented by an object BI (X) := ∪a∈I X[a −1 I ], the a∈I union of all X[a −1 I ], which we glue as open subobjects. The assertion in (iv) concerning the product I I
follows from the characterizing property of BI (X) in (i) easily. By this, if I
is the product of all sets {a, b} with a, b ∈ I in this sense, BI
(X) has the property of BI∗ (X) stated in (ii). Hence we have (ii). The assertions in (iii) and the remaining assertion in (iv) follow from the characterizing properties of BI (X) and BI∗ (X) given in (i) and (ii). The assertion (v) follows from 3.6.2. (We take the normalization, since Spec(C[S
])an is the normalization of Spec(C[S

])an in 3.6.2.) 2 3.6.7 Let X be an object of B(log), let NQ be a finite-dimensional Q-vector space, and assume that we are given an element × s ∈
(X, MX /OX ) ⊗ NQ . gp

We will consider an object X() of B(log) over X associated with a rational fan  in NR = R ⊗Q NQ . For an object Y of B(log) over X and for a point y ∈ Y , we have a homomorphism sy : π1 (y log ) → NQ ⊂ NR which is the germ in (MY /OY× )y ⊗ NQ = Hom(π1 (y log ), NQ ) of the pullback of s gp in
(Y, MY /OY× ) ⊗ NQ . gp

Lemma 3.6.8 Let σ be a finitely generated rational cone in NR . Then there is an object X(σ ) of B(log) over X having the following property. For any object Y of

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B(log) over X, the set Mor X (Y, X(σ )) is either a one-point set or an empty set, and it is nonempty if and only if sy (π1+ (y log )) ⊂ σ for any y ∈ Y . Proof. Take a finitely generated Z-submodule NZ of NQ such that NQ = Q ⊗ NZ gp × and such that s ∈
(X, MX /OX ) ⊗ NZ . Let × ), J (σ ) = {h(s) | h ∈ HomZ (NZ , Z), h(σ ) ⊂ R≥0 } ⊂
(X, MX /OX gp

× ) ⊗ NZ → where h(s) denotes the image of s under 1 ⊗ h :
(X, MX /OX gp ×
(X, MX /OX ). Since J (σ ) is a finitely generated monoid, by 3.6.1 (i), we have X[J (σ )]. This X[J (σ )] has the property of X(σ ) stated in Lemma 3.6.8. 2 gp

Lemma 3.6.9 Let σ and τ be finitely generated rational cones in NR . (i) If τ ⊂ σ , then Mor( , X(τ )) ⊂ Mor( , X(σ )) in Mor( , X) as sheaves on B(log). (ii) Mor( , X(σ )) ∩ Mor( , X(τ )) = Mor( , X(σ ∩ τ )) in Mor( , X). (iii) If τ is a face of σ , then X(τ ) is an open subobject (3.6.1) of X(σ ). Proof. The assertions (i) and (ii) follow from 3.6.8. We prove (iii). Take NZ as in the proof of 3.6.8. Then J (τ ) is generated by J (σ ) and the h(s)−1 for the elements h of HomZ (NZ , Z) satisfying h(σ ) ⊂ R≥0 and h(τ ) = 0. Hence X(τ ) is an open subobject of X(σ ) by 3.6.1 (ii). 2 Proposition 3.6.10 Let X be an object of B(log), let (NQ , s) be as in 3.6.7, and let  be a rational fan in NR . Then there is an object X() of B(log) over X having the following property. For any object Y of B(log) over X, the set Mor X (Y, X()) is either a one-point set or an empty set, and it is nonempty if and only if, for each y ∈ Y , there is an element σ of  such that sy (π1+ (y log )) ⊂ σ .  Each X(σ ) (σ ∈ ) is an open subobject of X(), and X() = σ ∈ X(σ ). We have X(σ ) = X({face of σ }). Proof. This follows from 3.6.8 and 3.6.9.

2

3.6.11 We describe the relation of X() to toric varieties associated with fans ([KKMS]). (i) Assume first that we are given a chart h : S → MX with S a sharp fs monoid, gp × ) ⊗ NQ be the element correlet NQ = Hom(S gp , Q), and let s ∈
(X, MX /OX sponding to h. Assume that we are given a rational fan  in NR = Hom(S, R add ) add ) for any σ ∈ . Let V () be the toric variety consuch that σ ⊂ Hom(S, R≥0 structed in [KKMS]. It is covered by open subspaces V (σ ) = Spec(C[S(σ )])an where S(σ ) is a submonoid of S gp consisting of all f ∈ S gp that are sent to R≥0 by all elements of σ . We endow V () with a unique logarithmic structure whose restriction to each V (σ ) is the canonical logarithmic structure 2.1.6 (ii). The inclusions

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S ⊂ S(σ ) induce a canonical morphism V () → Spec(C[S])an . We have X() = X ×Spec(C[S])an V (),

X(σ ) = X ×Spec(C[S])an V (σ )

(σ ∈ )

as objects of B(log). These fiber products are taken here in B(log) (3.5.1 (i)) but as topological spaces, they are the fiber products in the category of topological spaces by 3.5.1 (ii) (condition (1) there is satisfied because X → Spec(C[S])an is strict, and condition (3) there is satisfied because V () and V (σ ) are analytic spaces). add If  here is a finite subdivision of Hom(S, R≥0 ) (this means that  is finite and  add σ ∈ σ = Hom(S, R≥0 )), the map V () → Spec(C[S])an is proper surjective with connected fibers, and hence X() → X is proper surjective with connected fibers. (ii) Next we consider the general situation. Let (NQ , s) be as in 3.6.7 and let  be a rational fan in NR . Locally on X, there is a chart h : S → MX with S a sharp gp × fs monoid (2.1.5 (iii)). Since S gp → MX /OX is a surjective homomorphism of sheaves, locally on X, there is a homomorphism a : Hom(NQ , Q) → Q ⊗ S gp such gp × that the composition h ◦ a : Hom(NQ , Q) → Q ⊗ (MX /OX ) is the map induced by s. Let b : Hom(S gp , R) → NR be the R-homomorphism induced by a. Let 
be add ) ∩ b−1 (σ ) (σ ∈ ) and the rational fan in Hom(S gp , R) consisting of Hom(S, R≥0 their faces. Then X() = X(
) = X ×Spec(C[S])an V (
). Here X(
) is defined with respect to h. From this, we see that the map X() → X is separated (0.7.5) in general. In fact, to show this we may work locally, and we are reduced to the well-known fact that V (
) is Hausdorff. Definition 3.6.12 Let X be an object of B(log) and let Y be an object of B(log) over X. We say Y is a logarithmic modification of X if, locally on X, there are a chart h : S → MX with S a sharp fs monoid and a finite rational subdivision  of add Hom(S, R≥0 ) such that Y  X() over X. Here X() is defined with respect to the chart h as in 3.6.11 (i). By 3.6.11, a logarithmic modification is proper and surjective with connected fibers as a map of topological spaces. Lemma 3.6.13 A logarithmic blow-up is a logarithmic modification. × Proof. Let I be a nonempty finite subset of
(X, MX /OX ). Locally on X, we have a chart S → MX with S a sharp fs monoid and a finite subset I˜ of S gp gp × whose image in
(X, MX /OX ) coincides with I . Let NQ = Hom(S gp , Q) and gp × let s ∈
(X, MX /OX ) ⊗ NQ be the induced element. For a ∈ I˜, let σa be the add cone {h ∈ Hom(S, R≥0 ) | h(b) ≥ h(a) (∀ b ∈ I˜)}. Then for a, b ∈ I˜, σa ∩ σb is a face of σa and also a face of σb . Hence we have a finite subdivision  of the add ˜ cone  Hom(S, R≥0 ) consisting of σa (a ∈ I ) and their faces. We have BI (X) = 2 a∈I X(σa ) = X(). gp

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3.6.14 In 3.6.15 below, we give set-theoretic descriptions of X() and X()log , and also those for logarithmic blow-ups. For an object X of B(log), we define sets Q(X) and Q
(X) as follows. Let Q1 (X) be the set of all pairs (x, P ) where x ∈ X and P is a submonoid of gp × × × )x ⊂ P and P × ∩ (MX /OX )x = {1}. Let Q(X) be (MX /OX )x such that (MX /OX the set of all triples (x, P , h) where (x, P ) ∈ Q1 (X) and h is a homomorphism gp (P˜ )× → C× , where P˜ denotes the inverse image of P in MX,x and (P˜ )× denotes × the group of all invertible elements of P˜ , such that h(f ) = f (x) for f ∈ OX,x .
Let Q1 (X) be the set of all pairs (x, σ ) where x ∈ X and σ is a submonoid of π1+ (x log ) which contains some point in the interior (0.7.7) of π1+ (x log ). We have a map Q
1 (X) → Q1 (X), (x, σ )  → (x, P (σ )), × where P (σ ) denotes the submonoid of (MX /OX )x consisting of all elements f log such that the homomorphism π1 (x ) → Z corresponding to f (2.2.9) sends σ into N. Let Q
(X) be the set of all triples (x, σ, h) such that (x, σ ) ∈ Q
1 (X) and (x, P (σ ), h) ∈ Q(X). We have an evident map gp

Q
(X) → Q(X), (x, σ, h)  → (x, P (σ ), h). For an object Y of B(log) over X, we have maps qY
: Y → Q
(X),

qY : Y → Q(X)

defined as follows. The map qY is induced from qY
via Q
(X) → Q(X), and qY
(y) = (x, σ, h), where x is the image of y in X, σ is the image of π1+ (y log ) in π1 (x log ) (then P := P (σ ) coincides with the inverse image of (MY /OY× )y under the map gp gp × × (MX /OX )x → (MY /OY× )y ), and h is the composition (P˜ )× → OY,y → C× where the last arrow is f  → f (y). Forgetting h, we have maps
qY,1 : Y → Q
1 (X),

qY,1 : Y → Q1 (X).

Lemma 3.6.15 Let X be an object of B(log) and let Y be an object of B(log) over X. Assume that we have one of the following four cases (1)–(4). × ) which generates a finitely (1) Y = X[J ] for some subset J of
(X, MX /OX generated monoid (3.6.1 (i)). gp × (2) Y = BI (X) for some nonempty finite subset I of
(X, MX /OX ) (3.6.6 (i)). gp × ∗ (3) Y = BI (X) for some nonempty finite subset I of
(X, MX /OX ) (3.6.6 (ii)). (4) Y = X() for (NQ , s) as in 3.6.7 and for a rational fan  in NR (3.6.10). gp

Then (i) The maps qY
: Y → Q
(X) and qY : Y → Q(X) (3.6.14) are injective. gp gp × )x → (MY /OY× )y is (ii) For any y ∈ Y with image x in X, the map (MX /OX surjective, and qY
(y) = (x, σ, h) ∈ Q
(X) is recovered from qY (y) = (x, P , h) ∈ Q(X) as σ = {γ ∈ π1 (x log ) | hγ (P ) ⊂ N} where hγ denotes the homomorphism gp × (MX /OX )x → Z corresponding to γ .

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In the following, by (i), we regard Y as a subset of Q
(X) via the injection qY
and also as a subset of Q(X) via the injection qY . (iii) In the case (4), as a subset of Q
(X), we have Y = {(x, σ, h) ∈ Q
(X) | σ is a face of π1+ (x log ) ∩ sx−1 (τ ) for some τ ∈ }. (iv) In the case (1), as a subset of Q(X), we have × Y = {(x, P , h) ∈ Q(X) | P = (MX /OX )x , Jx , (Jx
)−1 sat } × for some finite subset J
of J . Here (MX /OX )x , Jx , (Jx
)−1  denotes the submonoid gp × × of (MX /OX )x generated by (MX /OX )x , the stalk Jx of J , and the inverses of gp × elements of the stalk Jx
, and −sat means the saturation of − in (MX /OX )x , i.e., × −sat = {a ∈ (MX /OX )x | a n ∈ − for some n ≥ 1}. gp

In the case (2), as a subset of Q(X), we have     P = (MX /O× )x , {a −1 b | a ∈ I
, b ∈ I }sat X Y = (x, P , h) ∈ Q(X)  . for some subset I
of I In the case (3), as a subset of Q(X), we have     P = (MX /O× )x , {a −1 b | a, b ∈ I, f (a) ≤ f (b)}sat X Y = (x, P , h) ∈ Q(X)  . for some map f : I → N (v) The map Y log → Y ×X X log is injective, and the image consists of all elements ((x, P , h), (x, h
)) ((x, P , h) ∈ Y ⊂ Q(X), (x, h
) ∈ Xlog ) such that h
(a) = h(a)/|h(a)| for any a ∈ (P˜ )× . The topology of Y log coincides with the topology as a subset of Y ×X X log . (vi) Let Z be an object of B(log) over X. Then there is a morphism Z → Y over X if and only if, for any z ∈ Z with qZ
(z) = (x, τ, f ) (resp. qZ (z) = (x, Q, f )),
there is (x, σ ) ∈ qY,1 (Y ) (resp. (x, P ) ∈ qY,1 (Y )) such that τ ⊂ σ (resp. P ⊂ Q). If there is a morphism Z → Y over X, and z, x, τ , f , σ , P are as above, the image of z in Y ⊂ Q
(X) and in Y ⊂ Q(X) are described as (x, σ
, h) and (x, P
, h), respectively, where σ
is the smallest face of σ such that τ ⊂ σ , P
= P , (P ∩ Q× )−1 , and h is the restriction of f to ((P
)∼ )× . Proof. The assertions about the case (1) follow from the explicit construction of X[J ] in 3.6.4. The assertions in (ii) for the cases (2), (3), and (4) are reduced to the case (1). By (ii), the injectivity of Y → Q(X) is reduced to the injectivity of Y → Q
(X). We prove the injectivity of Y → Q
(X) for the case (4). If y1 , y2 ∈ Y have the log same image (x, σ, h) in Q
(X), syj (π1+ (yj )) = sx (σ ) for j = 1, 2, and hence there is τ ∈  such that syj (π1+ (yj )) ⊂ τ for j = 1, 2. We have y1 , y2 ∈ X(τ ) and hence, by 3.6.8, we are reduced to the case (1). The injectivity of Y → Q
(X) for the cases (2) and (3) follows from that for the case (4), since locally on X the cases (2) and (3) are special cases of (4) by 3.6.6 (iii) and 3.6.13. The other assertions in 3.6.15 are reduced to the case (1). 2 log

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Definition 3.6.16 For an abelian group L and for a submonoid V of L, we say V is valuative if V ∪ V −1 = L. 3.6.17 Example. A valuative submonoid of Z2 is one of the following types (i)–(iii). Let  ,  : Z2 × R 2 → R be the standard pairing ((x1 , x2 ), (y1 , y2 ))  → x1 y1 + x2 y2 . (i) Z2 . (ii) V for a half line  = (R≥0 )v (v ∈ R 2 − {(0, 0)}), defined by V = {x ∈ Z2 | x,  ⊂ R≥0 }. (iii) V,
for a rational half line  = (R≥0 )v (v ∈ Q2 − {(0, 0)}) and for a half line  = (R≥0 )v
in the one-dimensional quotient vector space R 2 /R (v
∈ R 2 /R − {0}), defined by


V,
= {x ∈ Z2 | x,  ⊂ R≥0 . If x,  = 0, then x, 
 ≥ 0}. Definition 3.6.18 Let X be an object of B(log). As a set, Xval is the set of all elements (x, V , h) of Q(X) (3.6.14) such that V gp × )x . That is, Xval is the set of all triples (x, V , h) where is valuative in (MX /OX gp × × x ∈ X, V is a valuative submonoid of (MX /OX )x containing (MX /OX )x such that × × × × V ∩ (MX /OX )x = {1}, and h is a homomorphism (V˜ ) → C , where V˜ denotes gp the inverse image of V in MX,x and (V˜ )× denotes the group of all invertible elements × of V˜ , such that h(f ) = f (x) for all f ∈ OX,x . In 3.6.23 below, we will define a structure on Xval of a logarithmic local ringed space over X. 3.6.19 × ), we define a canonical map Xval → BI∗ (X) For a finite subset I of
(X, MX /OX
as (x, V , h)  → (x, P , h ) (3.6.6 (ii)), where gp

× P = (MX /OX )x , V ∩ {ax−1 bx | a, b ∈ I }sat ,

and h
is the restriction of h to (P˜ )× (3.6.15 (iv) for the case (3)). gp × If I
is another nonempty finite subset of
(X, MX /OX ) such that I
⊃ I , the map Xval → BI∗ (X) coincides with the composition Xval → BI∗
(X) → BI∗ (X). This follows from 3.6.15 (vi) for the case (3). × × ) → (MX /OX )x is surjective for Proposition 3.6.20 Assume that
(X, MX /OX ∗ any x ∈ X. Then the maps Xval → BI (X) in 3.6.19 induce a bijection gp

gp

∼

Xval − → lim BI∗ (X) ← − I

× where I ranges over all non-empty finite subset of
(X, MX /OX ). gp

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2

Proof. This follows from 3.6.15 (iv) and (vi), for the case (3).

Lemma 3.6.21 Let (NQ , s) be as in 3.6.7. Assume we are given a chart S → MX . (i) Let  be a finite rational fan in NR . Then, locally on X, there is a nonempty finite subset I˜ of S gp such that the fiber product X() ×X BI (X) in the category B(log) is an open subobject (3.6.1)of BI (X). Here I denotes the image of I˜ in gp ×
(X, MX /OX ). (ii) Let Y be a logarithmic modification of X. Then, locally on X, there are a nonempty finite subset I˜ of S gp and a morphism BI (X) → Y over X. Here I denotes gp × the image of I˜ in
(X, MX /OX ). Proof. (i) It is sufficient to treat the case  = {face of σ } for some finitely generated sharp rational cone σ in NR . Take NZ ⊂ NQ as in 3.6.7, and let I be a finite subset gp × ) which generates J (σ ) (3.6.8) such that 1 ∈ I . Locally on X, we of
(X, MX /OX gp × ) coincides with I . We have a finite subset I˜ of S gp whose image in
(X, MX /OX −1 have X(σ ) = X[I ] = X[1 I ] and this is an open subobject of BI (X). (ii) By (i), locally on X, we have I˜ such that the fiber product Y ×X BI (X) in B(log) is an open subobject of BI (X). It is sufficient to prove that Y ×X BI (X) = BI (X). Since Y ×X BI (X) → BI (X) is proper (3.6.12), for each x ∈ X, the fiber F1 in Y ×X BI (X) over x is a nonempty compact open subset of the fiber F2 in BI (X) over x. Since F2 is a Hausdorff space, F1 is an open and closed subset of F2 . 2 Since F2 is connected by 3.6.12 and 3.6.13, we have F1 = F2 . 3.6.22 In general, for a directed ordered set  and for a projective system (Xλ )λ∈ of logarithmic local ringed spaces, the projective limit limλ Xλ in the category of local ← − ringed spaces exist. As a topological space, this projective limit P is the projective limit of the topological spaces Xλ . The structural sheaf OP and the logarithmic structure MP of P are given by OP = lim pλ−1 (OXλ ), − → λ

MP = lim pλ−1 (MXλ ) − → λ

where pλ is projection P → Xλ . 3.6.23 Let X be an object of B(log). We endow Xval with the structure of a logarithmic local ringed space over C as follows. First assume that there is a chart S → MX . Then, by 3.6.6 (iii), 3.6.13, and 3.6.21 (ii), we have isomorphisms of logarithmic local ringed spaces lim (logarithmic modifications of X)  lim BI∗ (X)  lim BI∗ (X), ← − ← − ← − I

I˜

(1)

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where the transition morphisms in the first projective system are all morphisms over gp × X, I in the second ranges over all nonempty finite subsets of
(X, MX /OX ), I˜ in gp the third ranges over all nonempty finite subsets of S , and I in the third projective gp × ). By the existence of the chart, system denotes the image of I˜ in
(X, MX /OX the assumption of Proposition 3.6.20 is satisfied. Hence, by Proposition 3.6.20, the underlying set of the projective limit in (1) above is identified with Xval . We endow Xval with the structure of a logarithmic local ringed space over X as the projective limit in (1). In the case where X has a chart, by the presentation of X as the third projective limit in (1), we see that the logarithmic local ringed space Uval over X for an open set U of X is identified with the open subobject of Xval whose underlying set is the inverse image of U in Xval . Hence for a general object X of B(log), globally on Xval , we have a structure of a logarithmic local ringed space over X which induces the logarithmic local ringed structure of Uval for any open subset U of X which has a chart. Lemma 3.6.24 The map Xval → X is proper and surjective, and the fibers are connected. Proof. Since the problem is local on X, we may assume that X has a chart. In this case, since Xval is the projective limit of logarithmic modifications of X, 3.6.24 follows from 3.6.12. 2 3.6.25 Example. Let X = C2 with the logarithmic structure given by the divisor C2 − (C× )2 with normal crossings. Then Xval coincides with the space (C2 )val in 0.5.21. Let f : × (C2 )val → C2 be the projection. Let x = (0, 0) ∈ C2 and identify (MX /OX )x with −1 2 N in the natural way. Then the point (0, 0)s of f (x) (s ∈ R>0 − Q>0 ) in 0.5.21 is the point (x, V , h) of Xval (3.6.18) where V = V with  = (R≥0 )(1, s) in the nota× tion of 3.6.17 and h : (V˜ )× → C× is the evident one (note that (V˜ )× = OX,x in this case). The point (0, 0)0 (resp. (0, 0)∞ , resp. (0, 0)s,0 with s ∈ Q>0 , resp. (0, 0)s,∞ with s ∈ Q>0 ) of f −1 (x) in 0.5.21 is the point (x, V , h) where V = V,
with  = (R≥0 )(1, 0) (resp. (R≥0 )(0, 1), resp. (R≥0 )(1, s), resp. (R≥0 )(1, s)) and 
is the image of (R≥0 )(0, 1) (resp. (R≥0 )(1, 0), resp. (R≥0 )(1, 0), resp. (R≥0 )(0, 1)), × and h : (V˜ )× → C× is the evident one (note that (V˜ )× = OX,x in this case). Finally let s ∈ Q>0 and write s = m/n with m, n ∈ Z, m, n > 0, and GCD(m, n) = 1. Then the point (0, 0)s,z with s ∈ Q>0 and z ∈ C× of f −1 (x) in 0.5.21 is the point (x, V , h) × of Xval , where V = V with  = (R≥0 )(1, s), V˜ in this case is generated by OX,x m n 2 × × and q1 /q2 with (q1 , q2 ) the standard coordinate of C , and h : (V˜ ) → C is the × homomorphism which sends f ∈ OX,x to f (x) and q1m /q2n to z. 3.6.26 log

Let Xval be the set of all pairs (x, h) where x ∈ Xval and h is a homomorphism gp × MX,x → S1 such that h(u) = u(x)/|u(x)| for any u ∈ OX . Then we have a val ,x

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bijection log ∼

→ {((x, V , h), (x, h
)) ∈ Xval ×X X log | Xval − h
(f ) = h(f )/|h(f )| for any f ∈ (V˜ )× }. log

We endow Xval with the topology as a subspace of Xval ×X X log . Since this is a log closed subspace of Xval ×X X log , Xval is proper over Xval . In the case X has a chart, we have a homeomorphism log ∼

→ lim Y log Xval − ← − Y

where Y ranges over all logarithmic modifications of X. The bijectivity of this map follows from 3.6.15 (v), and the fact that it is a homeomorphism follows from the log properness of Xval and of limY Y log over X. ← − Lemma 3.6.27 Let X be an object of B(log), let (NQ , s) be as in 3.6.7, and let  be a rational fan in NR . (i) Via the morphism X()val → Xval induced by X() → X, X()val is an open subobject (3.6.1) of Xval . (ii) As a subset of Xval , X()val is the set of all (x, V , h) ∈ Xval such that P ⊂ V for some (x, P , h) ∈ X() ⊂ Q(X). Proof. (i) This is reduced to the case where  is finite and X has a chart, and then to 3.6.21 (i). (ii) Write Y = X(). The inclusion map Yval → Xval sends (y, V1 , h1 ) ∈ Yval (3.6.18) with y = (x, P , h2 ) ∈ Y ⊂ Q(X) (3.6.14) (P is the inverse image of gp gp × )x → (MY /OY× )y and h2 : (P˜ )× → C× is induced (MY /OY× )y under (MX /OX from h1 : (V˜1 )× → C× ) to (x, V , h), where V is the inverse image of V1 ∼ gp gp gp × × under (MX /OX )x → (MX /OX )x /P × − → (MY /OY× )y , and h : (V˜ )× → C× is the homomorphism induced by h1 . Hence, if (x, V , h) ∈ Xval is the image of (y, V1 , h1 ) ∈ Yval with y = (x, P , h2 ) ∈ Y ⊂ Q(X), then P ⊂ V as desired. Conversely, let (x, V , h) ∈ Xval and assume that there is (x, P , h2 ) ∈ Y such that P ⊂ V . gp × Let P
be the submonoid of (MX /OX )x generated by P and the inverses of elements × of P contained in V . Let y := (x, P
, h
2 ) ∈ Y ⊂ Q(X), where h
2 : (P˜
)× → C× ∼ gp gp × is the homomorphism induced by h2 . We have (MX /OX )x /(P
)× − → (MY /OY× )y .
× Hence (x, V , h) is the image of (y, V1 , h1 ) ∈ Yval , where V1 = V /(P ) and where × h1 : (V˜1 )× → C× is the homomorphism induced by h and OY,y → C× , f  → f (y), × by the fact that (V˜1 )× is the push-out of (V˜ )× ← (P
)× → OY,y . 2 Theorem 3.6.28 Let X be an object of B(log), let (NQ , s) be as in 3.6.7, and let  be a rational fan in NR . Then the following six conditions (1)–(6) are equivalent. (1) X() is a logarithmic modification of X. (2) The map X() → X is proper and surjective.

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(3) Locally on X, there is a logarithmic modification Y of X and a morphism Y → X() over X.  (4) For any x ∈ X, π1+ (x log ) = σ ∈ π1+ (x log ) ∩ sx−1 (σ ), and the set of submonoids π1+ (x log ) ∩ sx−1 (σ ) of π1+ (x log ), where σ ranges over , is finite. (5) X()val = Xval . gp × (6) For any x ∈ X and any valuative submonoid V of (MX /OX )x with × × × (MX /OX )x ⊂ V and V ∩ (MX /OX )x = {1}, there is (x, P , h) ∈ X() ⊂ Q(X) such that P ⊂ V . Proof. The implication (1) ⇒ (2) is already shown in 3.6.12. We prove the implication (2) ⇒ (3). Assume that X() → X is proper and surjective. Since {X(σ )}σ ∈ is an open covering of X() and X() → X is proper, by working locally on X, we may assume that there is a finite subset 1 of  such  that σ ∈1 X(σ ) = X(). Hence we may assume that there is a finite subfan 2 of  such that X(2 ) = X(). Since 2 is finite, locally on X, by 3.6.21 (i), 3.6.6 (i), and 3.6.13, there is a logarithmic modification Y → X such that the fiber product X() ×X Y is an open subobject of Y . It is sufficient to prove X() ×X Y = Y . Since X() → X is proper and surjective by the assumption and since X() ×X Y → X() is proper and surjective (3.6.12), the composition X() ×X Y → X is proper and surjective. Hence, for each x ∈ X, the fiber F1 in X() ×X Y over x is a nonempty compact open subset of the fiber F2 in Y over x, and F2 is Hausdorff and connected by 3.6.12. Hence F1 = F2 , and thus we have X() ×X Y = Y , and (3) is proved. We prove the implication (3) ⇒ (4). Let x ∈ X. By 3.6.15 (iii), there are only
finitely many submonoids σ of π1+ (x log ) such that (x, σ ) is in the image of qY,1 :  n +
log Y → Q1 (X). Let σ1 , . . . , σn be all such submonoids. We have π1 (x ) = j =1 σj . Since there is a morphism Y → X(), we have by 3.6.15 (vi) that for each j , there
is a submonoid τj of π1+ (x log ) such that (x, τj ) is in the image of qX(),1 : X() → n + log
Q1 (X) and σj ⊂ τj . We have π1 (x ) = j =1 τj . It remains to show that there are only finitely many submonoids τ of π1+ (x log ) such that (x, τ ) belongs to the
: X() → Q
1 (X). If τ is such submonoid, τ = nj=1 τ ∩ τj , and image of qX(),1 by 3.6.15 (iii), τ ∩ τj is a face of τ . Hence τ = τ ∩ τj for some j . This shows that τ is a face of τj . But there are only finitely many faces of τj . We prove the implication (4) ⇒ (1). Let x ∈ X. Working locally at x, take a chart ∼ × → (MX /OX )x (2.1.5 (iii)). Take NZ ⊂ NQ as in the proof of S → MX such that S − gp × is surjective, locally at 3.6.8. Since the homomorphism of sheaves S gp → MX /OX gp x there is a homomorphism a : Hom(NZ , Z) → S such that the homomorphism a gp × coincides with the composition Hom(NZ , Z) − → s : Hom(NZ , Z) → MX /OX gp × add S gp → MX /OX . Let b : Hom(S, R≥0 ) → NR be the homomorphism induced by a. ∼

b

add → Hom(S, N) ⊂ Hom(S, R≥0 )− → Since sx : π1+ (x log ) → NR factors as π1+ (x log ) −  add −1 −1 NR , we have Hom(S, R≥0 ) = σ ∈ b (σ ) and that the set of cones b (σ ) (σ ∈ ) is finite. Hence X() coincides with the logarithmic modification defined add by the finite subdivision of Hom(S, R≥0 ) consisting of all faces of b−1 (σ ) for all σ ∈ . Thus we have shown that the conditions (1)–(4) are equivalent.

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145

The implication (1) ⇒ (5) is clear. We prove the implication (5) ⇒ (2). In general, let A → B and B → C be continuous maps of topological spaces, and assume that the composition A → C is proper, that A → B is surjective, and that B → C is separated (see 0.7.5). Then B → C is proper. We apply this to A = Xval , B = X(), C = X. Note Xval → X is proper by 3.6.24, Xval = X()val → X() is surjective by 3.6.24, and X() → X is separated by 3.6.11 (ii). Finally, the equivalence (5) ⇔ (6) follows from 3.6.27 (ii). 2

Chapter Four Main Results

In this chapter, we state the main results of this book: Theorem A in Section 4.1, and Theorem B in Section 4.2. We extend Griffiths’ period maps in Section 4.3 and the infinitesimal period maps in Section 4.4.

4.1 THEOREM A: THE SPACES E σ ,
\D  , AND
\D   4.1.1 We state our first main theorem, whose proof will be given in Chapter 7 below. A subgroup
of GZ is said to be neat if, for each γ ∈
, the subgroup of C× generated by all the eigenvalues of γ is torsion-free. It is known that there exists a neat subgroup of GZ of finite index. Theorem A Let  be a fan in gQ and let
be a subgroup of GZ that is strongly compatible with  (1.3.10). Then we have: (i) For σ ∈ , Eσ is open in E˜ σ in the strong topology of E˜ σ in Eˇ σ (3.3.4). In particular, by 3.5.10, Eσ is a logarithmic manifold (Definition 3.5.7). (ii) If
is neat, then
\D is also a logarithmic manifold. (iii) Let σ ∈  and define the action of σC on Eσ over
(σ )gp \Dσ by a · (q, F ) := (e(a)q, exp(−a)F ) (a ∈ σC , (q, F ) ∈ Eσ ), where e(a) ∈ torusσ is as in 3.3.5 and e(a)q is defined by the natural action of torusσ on toricσ . Then, Eσ →
(σ )gp \Dσ is a σC -torsor in the category of logarithmic manifolds. That is, locally on the base
(σ )gp \Dσ , Eσ is isomorphic as a logarithmic manifold to the product of σC and the base endowed with the evident action of σC . (iv) If
is neat, then, for any σ ∈ , the map
(σ )gp \Dσ →
\D is open and locally an isomorphism of logarithmic manifolds. (v) The topological space
\D is Hausdorff. (vi) If
is neat, then there is a homeomorphism of topological spaces (
\D )log 
\D , 

that is compatible with τ : (
\D )log →
\D and the projection
\D → 
\D induced by D → D in 1.3.9.

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4.2 THEOREM B: THE FUNCTOR PLH In this section, we state the second main result of this book, Theorem B, whose proof will be given in Chapter 8 below. 4.2.1 Let  be a fan in gQ and let
be a subgroup of GZ which is strongly compatible with  (1.3.10). As in 2.5.8, we denote by  = (w, (hp,q )p,q∈Z , H0 ,  , 0 ,
, )

(1)

the 6-tuple consisting of the 4-tuple (w, (hp,q )p,q∈Z , H0 ,  , 0 ) as in Section 0.7 and of the above  and
. Theorem B We assume
is neat (4.1.1). Define a contravariant functor PLH from the category A2 (log) (3.2.4) to the category of sets by PLH (X) := (isomorphism classes of PLH on X of type ) (2.5.8). Then this functor PLH is represented by
\D , i.e., there exists an isomorphism of functors PLH  Mor( ,
\D ) such that, for any object X of A2 (log) and any element of PLH (X), the  induced maps X →
\D and X log → (
\D )log 
\D coincide with the maps in 2.5.10. 4.2.2 Since A(log) ⊂ B(log) ⊂ A2 (log) by Theorem 3.2.5, Theorem B contains Theorem 0.4.27 (i) in Chapter 0. Proof of Theorem 0.4.27 (ii). We show that Theorem 0.4.27 (ii) follows from Theorem A, Theorem B, and Theorem 3.2.5. Once we know PLH |A(log)  Mor(


\D

 ,
\D )|A(log) = hA(log)

by Theorem B, Theorem 0.4.27 (ii) becomes equivalent to ∼


\D

 → Mor(hA(log) , hZA(log) ) Mor(
\D , Z) −

for logarithmic local ringed spaces Z over C, and hence to the statement that
\D
belongs to A(log) . Since
\D is a logarithmic manifold by Theorem A, it is an
2 object of B ∗ (log) and, due to Theorem 3.2.5, it is an object of A(log) . If we restrict our attention to nilpotent cones of rank one, Theorem B shows the following. × )x ≤ 1 Corollary 4.2.3 Let X be an object of A2 (log) such that rank Z (MX /OX for any x ∈ X. Let
be a neat subgroup of GZ of finite index. Then, isomorphism gp

148

CHAPTER 4

classes of PLH on X endowed with a
-level structure correspond bijectively to morphisms from X to
\D of logarithmic local ringed spaces over C.

4.3 EXTENSIONS OF PERIOD MAPS Assuming the results in Sections 4.1 and 4.2, in this section we extend the period map associated with a variation of polarized Hodge structure over a boundary. We will give other extension theorems in Sections 8.4, 9.4, and 12.6. These extension theorems represent one of the main motivations for Griffiths’ hope to enlarge D. The authors are grateful to Professor Chikara Nakayama for his essential contribution to the proof of the following theorem. Theorem 4.3.1 Let X be a connected, logarithmically smooth, fs logarithmic analytic space and let U = Xtriv be the open subspace of X consisting of all points at which the logarithmic structure is trivial. Let H = (HZ ,  , , F ) be a variation of polarized Hodge structure on U with unipotent local monodromy along X − U . Fix a base point u ∈ U and let (H0 ,  , 0 ) = (HZ,u ,  , u ). Let
be a subgroup of GZ that contains the global monodromy group Image(π1 (U, u) → GZ ), and assume that
is neat. Let ϕ : U →
\D be the associated period map. (i) Assume that X − U is a smooth divisor. Then there exists a fan  in gQ which is strongly compatible with
such that the period map ϕ : U →
\D extends to a morphism X →
\D of logarithmic manifolds. U

⊂

↓
\D

X ↓

⊂


\D .

(A natural choice of  is given in 4.3.2 below.) In the case where
is of finite index in GZ , we can take the fan  in (0.4.5, 1.3.11) as . (ii) For any point x ∈ X, there exist an open neighborhood W of x, a logarithmic modification W
of W (3.6.12), a commutative subgroup

of
, and a fan  in gQ that is strongly compatible with

such that the period map ϕ|U ∩W lifts to a morphism U ∩ W →

\D which extends to a morphism W
→

\D of logarithmic manifolds: U

⊃

↓
\D

U ∩W

⊂

W


⊂




↓ ←





\D

↓
\D .

(iii) Assume
is commutative. Then we can take

=
in (ii). Assume furthermore that the following condition (1) is satisfied. (1) There is a finite family (Xj ) 1≤j ≤n of connected locally closed analytic subspaces of X such that X = nj=1 Xj as a set and such that, for each j , the × on Xj is locally constant. inverse image of the sheaf MX /OX

149

MAIN RESULTS

Then there is a logarithmic modification X
of X and a fan  in gQ that is strongly compatible with
such that the period map ϕ extends to a morphism X
→
\D of logarithmic manifolds. U

⊂

↓
\D

X
↓

⊂


\D .

Note that a logarithmic modification X
→ X (or W
→ W ) is proper and surjective, and the induced morphism U ×X X
→ U (or (U ∩ W ) ×W W
→ U ∩ W ) is an isomorphism. The assertions in 4.3.1 (iii) may be true without the assumption that
is commutative, but the authors need this assumption for the proof. The condition (1) in 4.3.1 (iii) is satisfied, for example, if X is an open set of the toric variety Spec(C[S])an for some fs monoid S, or if X − U is compact. (If X − U is compact, X has a finite open covering with one member U and with all other members isomorphic to open sets of toric varieties, and hence X satisfies (1) in 4.3.1 (iii).) In the following, assuming results in Sections 4.1 and 4.2, we prove 4.3.1 (i) in 4.3.2, and prove 4.3.1 (ii) and (iii) in 4.3.9 at the end of this section after preparation. 4.3.2 Proof of 4.3.1 (i). By the nilpotent orbit theorem of Schmid interpreted as Theorem 2.5.14, H extends to a PLH H
on X. Denote by  the set of local monodromy cones of H
in gR (2.5.11). Then
is strongly compatible with  (2.5.12). We show that ϕ : U →
\D extends to a morphism X →
\D of logarithmic manifolds. Since gp × )x ≤ 1 for any x ∈ X, H
is of type (w, (hp,q ), H0 ,  , ,
, ) rank Z (MX /OX where w is the weight of H and (hp,q ) is the Hodge type of H (2.5.12). Hence we have the morphism X →
\D associated with H
whose restriction to U is the original period map U →
\D. If
is of finite index in GZ , since
is compatible with  and  ⊂ , we have the composite morphism X →
\D →
\D . 2 Corollary 4.3.3 Let H be a variation of polarized Hodge structure on a punctured disc ∗ =  − {0} with unipotent local monodromy at 0 ∈ . Fix a base point u ∈ ∗ and let (H0 ,  , 0 ) = (HZ,u ,  , u ). Let
be the image of π1 (∗ , u)  Z → GZ , and let σ be the cone of gR generated by the logarithm of the image in GZ of positive generator (appendix A1) of π1 (∗ ). Then the period map ϕ : ∗ →
\D extends to a morphism  →
\Dσ of logarithmic manifolds. 4.3.4 The reason that we need a logarithmic modification in 4.3.1 (ii) and (iii) is the following. We still have a PLH H
on X that extends H on U . However, if the gp × )x ≤ 1 is not satisfied, it can happen that there is no fan condition rank Z (MX /OX in gQ which contains the set of local monodromy cones of H
in gR (2.5.11) as a subset. We will give such examples (Examples (i) and (ii) below).

150

CHAPTER 4

Let X = n with the logarithmic structure associated to the normal crossing divisor n − (∗ )n , let x = (0, . . . , 0) ∈ n , let ρ : π1 (x log ) → GZ be the monodromy action, let (γj )1≤j ≤n be the standard generator of π1+ (x log )  Nn , and let Nj = ˇ
) log(ρ(γj )) ∈ gQ . Then, if x
= (xj
)1≤j ≤n is the point of X = n , the image ϕ(x

ˇ under the period

map ϕˇ : X →
\Dorb (2.5.3) has the form ((σ (x ), Z(x )) mod
)
with σ (x ) = j ; x
=0 (R≥0 )Nj . j

Examples. (i) It can happen that σ (x) is not a sharp cone. In fact, it can happen that n = 2 and N2 = −N1  = 0. In this case, σ (x)  Z and σ (x) is not sharp. (ii) It can happen that n = 3, N1 and N2 are linearly independent over R, and N3 = N1 + N2 . If x
= (xj
)1≤j ≤3 is a point of 3 such that x3
= 0 and x1
and

x2
are not zero, then σ (x
) = (R≥0 )N3 is not a face of σ (x) = 3j =1 (R≥0 )Nj = (R≥0 )N1 + (R≥0 )N2 . Hence there is no fan in gQ which contains both σ (x) and σ (x
). Assume that
is commutative so that the adjoint action of
on σ (x) is trivial. In the above examples, our method for extending the period map, which is explained below in detail, is as follows. In Example (i), we subdivide σ (x) into sharp cones (R≥0 )N1 and (R≥0 )N2 . In Example (ii), we subdivide σ (x) into the cones (R≥0 )N1 + (R≥0 )N3 and (R≥0 )N2 + (R≥0 )N3 . Then we have a fan  in gQ that is strongly compatible with
and that subdivides the monodromy cones σ (x
) of any point x
of X. As is shown below, our subdivision yields a logarithmic modification X() of X to which we can extend the period morphism. 4.3.5 Let X be an object of B(log), let
be a subgroup of GZ , and let H be a pre-PLH on X of weight w and of Hodge type (hp,q ) endowed with a
-level structure µ. Let  be a fan in gQ which is compatible with
(here
and  need not be strongly compatible). We define an object X() of B(log) over X. Consider the homomorphism × ) ⊗ HQ N : HQ → (MX /OX gp

(1)

defined in 2.3.4. Via the
-level structure, N induces × ) ⊗ gQ )), N ∈
(X,
\((MX /OX gp

gp × ) ⊗ gQ )
\((MX /OX

(2)

on X, where γ ∈
acts a global section of the sheaf of sets gp × on (MX /OX ) ⊗ gQ as a ⊗ b  → a ⊗ Ad(γ )b. gp × ) ⊗ gQ . By 3.6.10, using this Locally on X, lift N to an element of
(X, MX /OX lifting of N as s there, we have an object X() of B(log) over X. Since  is stable under the adjoint action of
, these local constructions are patched up to yield a global object X() of B(log) over X. Let x ∈ X and let y be a point of X log lying over x. Take a representation µ˜ y : ∼ (HZ,y , ( , )y ) − → (H0 ,  , 0 ) of the stalk µy of the
-level structure µ. Then we

151

MAIN RESULTS

× )x ⊗ gQ associated to µ˜ y . The homomorphism sx : have a lifting N ∈ (MX /OX log π1 (x ) → gQ in 3.6.7 (we take this lifting of N as s in 3.6.7) is nothing but the logarithm of the local monodromy action of π1 (x log ) on H0 via µ˜ y . In the following, for an object X
of B(log) over X and for x
∈ X
, let σ (x
) be the cone in gR generated by the image of π1+ ((x
)log ) under the logarithm π1 ((x
)log ) → gQ of the local monodromy action on H0 given by the inverse image of H on X
and by the
-level structure of H . Then σ (x
) is determined modulo the adjoint action of
. The object X() over X is characterized as follows (3.6.10). For any object X
of B(log) over X, Mor X (X
, X()) is either empty or a one point set, and it is nonempty if and only if, for any x
∈ X
, there exists τ ∈  such that σ (x
) ⊂ τ . gp

Proposition 4.3.6 Let X be an object of B(log), let
be a subgroup of GZ , and let H be a PLH on X of weight w and of Hodge type (hp,q ) endowed with a
-level structure. Let C be the set of local monodromy cones of H in gQ (2.5.11). Assume that we are given a fan  in gQ satisfying the following conditions (1)–(3).   (1) σ ∈ σ = σ ∈C σ . (2)  is compatible with 
. (3) For any σ ∈ C, σ = nj=1 τj for some n ≥ 1 and for some τj ∈ . Then (i)  is strongly compatible with
. (ii) The inverse image of H on X() is of type  = (w, (hp,q ), H0 ,  , 0 ,
, ). Consequently, when
is neat, we have the corresponding period morphism X() →
\D . (iii) X() is a logarithmic modification of X. Proof. The assertion (i) follows from the condition (1) and (2). We prove (ii). Let x
∈ X() and let x ∈ X be the image of x
in X. By the characterizing property of X() (3.6.10 restated at the end of 4.3.5), there is τ ∈  such that σ (x
) ⊂ τ . Take the smallest such τ . Our task is to prove that the pullback of H
to x
induces a τ -nilpotent orbit, not only a σ (x
)-nilpotent orbit. Since H induces a σ (x)-nilpotent orbit, for the proof of the fact that the pullback of H
to x
induces a τ -nilpotent orbit, it is sufficient to prove τ ⊂ σ (x). Let p be an element in the interior (0.7.7) of σ (x
). By the minimality of τ , p belongs to the interior of τ . On the other hand, σ (x
) ⊂ σ (x) and σ (x) is the union of some elements τj of  such that τj ⊂ σ (x) by the condition (3). Hence we have p ∈ τj for some j . Since τ ∩ τj is a face of τ and contains a point in the interior of τ , we have τ ∩ τj = τ , that is, τ ⊂ τj . Hence τ ⊂ σ (x). We prove (iii). It is sufficient to prove that  satisfiesthe condition (4) in 3.6.28. Let x ∈ X. By the condition (1), we have π1+ (x log ) = σ ∈ π1+ (x log ) ∩ sx−1 (σ ). It remains to prove that the set of submonoids π1+ (x log ) ∩ sx−1 (σ ) of π1+ (x log ), where σ ranges over , is finite. For this it is sufficient to prove that the set of  cones σ (x) ∩ τ , where τ ranges over , is finite. By the condition (3), σ (x) = nj=1 τj  for some τj ∈ . If τ ∈ , σ (x) ∩ τ = nj=1 τj ∩ τ . Since τj ∩ τ is a face of τj and

152

CHAPTER 4

since the number of faces of each τj is finite, the set of cones σ (x) ∩ τ , where τ ranges over , is finite as desired. 2 Lemma 4.3.7 Let X be an object of B(log),
a subgroup of GZ , H a pre-PLH of weight w and of Hodge type (hp,q ) endowed with a
-level structure, and let C be the set of local monodromy cones of H in gR (2.5.11). (i) Let x ∈ X. Fix a cone σ (x) which is a representative of the class (σ (x) mod
). Then, if we replace X by a sufficiently small open neighborhood of x, the
-level structure of H comes from a
(σ (x))gp -level structure of H , σ (x
) ⊂ σ (x) for any x
∈ X for some choice of a representative σ (x
) of (σ (x
) mod
), and the quotient set
\C of C by the adjoint action of
is finite. × (ii) If MX /OX is a locally constant sheaf, the map x
 → (σ (x
) mod
) is a locally constant function on X. (iii) Assume that the condition (1) in 4.3.1 (iii) is satisfied (here
need not be commutative). Then the quotient set
\C is finite. (iv) Assume that
is commutative. Then Ad(γ )v = v for any σ ∈ C, any v ∈ σ , and any γ ∈
. Proof. (i) We first prove the assertion concerning the
-level structure. Take y ∈ Xlog ∼ lying over x, and let µ˜ y : (HZ,y ,  , y ) − → (H0 ,  , 0 ) be a representative of the germ µy of the
-level structure µ of H . Via µ˜ y , the local monodromy action of π1 (x log ) on (HZ,y ,  , y ) induces a homomorphism π1 (x log ) →
. If we replace µ˜ y by γ µ˜ y for some suitable γ ∈
, the image of π1 (x log ) in
is contained in
(σ (x))gp . For this new µ˜ y , the inverse image H |x of H on the fs logarithmic point x has a unique
(σ (x))gp -level structure µ
whose germ at y is the class of µ˜ y . The
-level structure µ of H |x is induced from µ
. By the proper base change theorem (Appendix A2) applied to the proper map Xlog → X and to the sheaves of sets
\Isom((HZ ,  , ), (H0 ,  , 0 )) and
(σ (x))gp \Isom((HZ ,  , ), (H0 ,  , 0 )) on Xlog , for some open neighborhood W of x, µ
extends to a
(σ (x))gp -level structure of H |W which induces µ of H |W . ∼

We prove the rest of (i). Locally at x, take a chart S → MX such that S − → gp × × (MX /OX )x . Locally at x, N ∈
(X,
\((MX /OX ) ⊗ gQ )) comes from an element N˜ ∈ S gp ⊗ gQ . In this situation, for x
∈ X, (σ (x
) mod
) is described as follows. × )x
, N) → Hom(S, N), Let σ1 (x
) be the image of π1+ ((x
)log )  Hom((MX /OX where the first isomorphism is by the duality 2.2.9 and the next arrow is induced by × S → (MX /OX )x
. Let σ (x
) be the cone in gR generated by the images of N˜ under gp the maps S ⊗ gQ → gQ induced by all elements of σ1 (x
). Then σ (x
) is a representative of (σ (x
) mod
). Since σ1 (x) = Hom(S, N), we have σ1 (x
) ⊂ σ1 (x) and hence σ (x
) ⊂ σ (x). Furthermore, σ1 (x
) is a face of Hom(S, N). This is shown × as follows. Let S(x
) ⊂ S be the kernel of S → (MX /OX )x
. Then S(x
) is a face of ∼ × → (MX /OX )x
. Hence σ1 (x
) S and we have an isomorphism (S(x
)gp S)/S(x
)gp − coincides with the set of all homomorphisms S → N which kill S(x
). Hence σ1 (x
) is a face of Hom(S, N). Since there are only finitely many faces of Hom(S, N), there are only finitely many (σ (x
) mod
).

153

MAIN RESULTS

× ) ⊗ HQ (2.3.4). Since (ii) Consider the homomorphism N : HQ → (MX /OX × MX /OX is locally constant, the map N is locally constant. Hence (σ (x
) mod
) is locally constant when x
varies. gp

The assertion (iii) follows from (ii). The assertion (iv) follows from the fact that each σ ∈ C is generated as a cone by logarithms of some unipotent elements of
. 2 Proposition 4.3.8 Let X be an object of B(log), let
be a subgroup of GZ , and let H be a PLH on X of weight w and of Hodge type (hp,q ) endowed with a
-level structure. (i) For any point x ∈ X, there exist an open neighborhood W of x, a logarithmic modification W
of W , a commutative subgroup

of
, and a fan  in gQ which is strongly compatible with

such that the
-level structure of H comes from a

-level structure of H , with which and with  the inverse image of H on X
is of type (w, (hp,q ), H0 ,  , 0 ,

, ). (ii) Assume that
is commutative and assume that the set C of local monodromy cones of H in gR (2.5.11) is finite. Then there exist a logarithmic modification X
of X and a fan  in gQ which is strongly compatible with
such that the inverse image of H on X
is of type (w, (hp,q ), H0 ,  , 0 ,
, ). Proof. By 4.3.7, it is sufficient to prove (ii). By 4.3.6, for the proof of 4.3.8, it is sufficient to prove that there exists a fan  in gQ satisfying the conditions (1)–(3) in 4.3.6. For each τ ∈ C, subdivide τ into sharp cones. Let B be the set of all cones that appear as a result. For each τ ∈ B, take a finite fan τ ingQ such that ∪τ
∈τ τ
= gQ
and τ ∈ τ . Let  be the set of all cones of the form τ ∈B c(τ ) where c(τ ) is an element of τ for each τ ∈ B. Then 
is a fan. Let  be the subset of 
consisting of all σ ∈ 
such that σ ⊂ τ for some τ ∈ C. Then  is a fan satisfying the conditions (1)–(3) in 4.3.6. In fact, (1) and (3) are checked easily, and (2) follows from the fact that, by the commutativity of
, the adjoint action of
on ∪σ ∈ σ is trivial. 2 4.3.9 Proofs of 4.3.1 (ii) and (iii). By the nilpotent orbit theorem of Schmid interpreted as Theorem 2.5.14, we have a PLH on X which extends H on U . Hence 4.3.1 (ii) is reduced to 4.3.8 (i), and 4.3.1 (iii) is reduced to 4.3.7 (iii) and (iv) and 4.3.8 (ii). 2

4.4 INFINITESIMAL PERIOD MAPS The purpose of this section is to generalize Griffiths’ formulation of infinitesimal period maps to our extended period maps. Let the notation be as in 4.2.1.

154

CHAPTER 4

4.4.1 Let X be an object of A1 (log) (2.2.1) and let (HZ ,  , , F ) be a PLH on X (2.4.8). log Then, as in 2.4.1 and 2.3.3, M = τ∗ (OX ⊗Z HZ ) is a locally free OX -module of rank = rank Z HZ and Mp = τ∗ (F p ) is locally a direct summand of M. Let End  ,  (M) := {ν ∈ HomOX (M, M) | ν(x), y + x, ν(y) = 0 (∀ x, ∀ y ∈ M)}, F p End  ,  (M) := {ν ∈ End  ,  (M) | ν(Mq ) ⊂ Mp+q (∀q ∈ Z)}. Then, as an OX -module, End  ,  (M) is locally free of finite rank and F p End  ,  (M) is locally a direct summand of it. 4.4.2 Let the notation be as in 4.4.1 and assume that X is a logarithmically smooth object 1,log log of B(log) (3.5.4, 3.2.4). Then, d ⊗ 1HC : OX ⊗C HC → ωX ⊗C HC induces a connection 1 ∇ : M → ωX ⊗OX M.

(1)

Let θX be the sheaf of logarithmic vector fields on X (3.5.4 (iii)). We define a map  (2) HomOX (Mp , M/Mp ) θX → p

by sending δ ∈ θX to the element of ⊕p HomOX (Mp , M/Mp ) induced by the map ∇

δ⊗1M

1 M− → ωX ⊗OX M −−−→ M.

(3)

Note that the map (3) is not OX -linear but it induces an OX -linear map M → M/Mp . The image of the map (2) is contained in the image of the injection  HomOX (Mp , M/Mp ). End  ,  (M)/F 0 End  ,  (M) → p

p

Thus, we have a homomorphism of OX -modules θX → End  ,  (M)/F 0 End  ,  (M).

(4)

Proposition 4.4.3 Let X =
\D with
neat (4.1.1), let (HZ ,  , , F, µ) be the universal PLH of type  on X (Theorem B), and define End  ,  (M) as in 4.4.1. Then the homomorphism 4.4.2 (4) is an isomorphism ∼

θX − → End  ,  (M)/F 0 End  ,  (M). Proof. Let U be an open set of X and let U
:= U [T ]/(T 2 ) (3.5.6). Define the sets P , Q, and R as follows. Let P be the set of all morphisms U
→ X that extend the inclusion morphism U → X, let Q = HomOU (ωU1 , OU ) =
(U, θX ), and let R be the set of all isomorphism classes of PLH (HZ
,  , 
, F
, µ
) on U
of type

155

MAIN RESULTS

 whose pullbacks to U coincide with the restriction of (HZ ,  , , F, µ) to U . Since X =
\D is the fine moduli space of PLH of type , the map P → R, obtained by pulling back the universal object, is bijective. On the other hand, by 3.5.5, the trivial extension ι : U
→ U → X of the inclusion morphism U → X ∼ → P , δ  → δι. Furthermore, we have a bijection between induces a bijection Q − R and
U, End  ,  (M)/F 0 End  ,  (M) given as follows. For an element ν of the last set, the corresponding (HZ
,  , 
, F
, µ
) is given by (HZ
,  , 
, µ
) = log (HZ ,  , , µ)|U and by F
p , whose germ at each point y ∈ U
log is the OU
,y submodule of OU
,y ⊗Z HZ generated by a + T ν(a) (a ∈ τ −1 (My ), where y is log regarded that this correspondence gives a bijection  as a point of U ).0It is easily seen from
U, End  ,  (M)/F End  ,  (M) onto R. It is also easily seen that the com ∼ ∼ ∼ →R− →
U, End  ,  (M)/F 0 End  ,  (M) is →P − posite map
(U, θX ) = Q − nothing but the homomorphism 4.4.2 (4). 2 log

p

4.4.4 Let X =
\D with
neat. We define the horizontal logarithmic tangent bundle TXh of X, which is a subbundle of the logarithmic tangent bundle TX (3.5.6), as follows. Let M be as in 4.4.1 and let θXh be the OX -submodule of θX whose image in End  ,  (M)/F 0 End  ,  (M) under the isomorphism of 4.4.3 coincides h with gr −1 F End  ,  (M). Then, θX is locally a direct summand of θX , and is called the horizontal submodule of θX . We define TXh to be the vector bundle associated with the sheaf θXh (cf. 3.5.6). We have θXh  gr −1 F End  ,  (M)   p p−1  (hp )p ∈ HomOX (gr M , gr M ) p

    h2w+1−p = −t hp (∀p ∈ Z) . 

(1)

Here t ( ) means the transposed with respect to  , . 4.4.5 Assume that
is neat. Denote
\D by Z. Let X be a logarithmically smooth object of B(log) (3.5.4), let (HZ ,  , , F, µ) be a PLH on X of type  and let ϕ : X → Z be the corresponding period map. Then M, Mp and F p End  ,  (M) on X (4.4.1) are the pullbacks of M, Mp and F p End  ,  (M) of the universal PLH on Z by ϕ, respectively. If we regard the isomorphism in 4.4.3 as identification, the map θX → End  ,  (M)/F 0 End  ,  (M) in 4.4.2 (4) is identified with the ∗ h canonical map θX → ϕ ∗ (θZ ) and gr −1 F End  ,  (M) is identified with ϕ (θZ ). Hence we have Proposition 4.4.6 In the notation in 4.4.5, the following four conditions are equivalent.

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CHAPTER 4

(1) The PLH (HZ ,  , , F ) satisfies the big Griffiths transversality 1 ∇(Mp ) ⊂ ωX ⊗OX Mp−1 (∀p ∈ Z) 1,log

(or equivalently, (d ⊗ 1HC )(F p ) ⊂ ωX

⊗Olog F p−1 (∀p ∈ Z)). X

gr −1 F

End  ,  (M). (2) The map 4.4.2 (4) factors through (3) The map θX → ϕ ∗ (θZ ) factors through ϕ ∗ (θZh ). (4) The map dϕ : TX → TZ factors through TZh . 4.4.7 Relation to the logarithmic Kodaira-Spencer map. Let the notation be as in 4.4.5, and assume that X is a logarithmically smooth fs logarithmic analytic space and that (HZ ,  , , F, µ) is obtained from “geometry" as in 0.2.21. In this case, M = R w f∗ (ωY• /X ),

≥p

Mp = R w f∗ (ωY /X ),

p

p

gr M = R w−p f∗ (ωY /X ),

and this PLH satisfies the big Griffiths transversality, i.e., the LVPH (2.4.9, 4.4.6). We define θY/X := Ker(θY → f ∗ (θX )). The following theorem about the infinitesimal period map dϕ is proved in the same way as in the nonlogarithmic case ([G2, 1.23]). Theorem 4.4.8 In the notation in 4.4.7, the following diagram is commutative: θX   K-S

dϕ

−−−−→

via coupling

R 1 f∗ θY/X −−−−−−→

h = gr −1 ϕ ∗ θ
\D F End  ,  (M)    ∩

 p

p

p−1

HomOX (R m−p f∗ ωY /X , R m−p+1 f∗ ωY /X )

Here K-S means the logarithmic Kodaira-Spencer map (i.e., the connecting map coming from the exact sequence 0 → θY /X → θY → f ∗ (θX ) → 0) and the bottom p p−1 horizontal arrow is induced by the pairing θY /X ⊗ ωY /X → ωY /X .

Chapter Five Fundamental Diagram

The aim of this chapter is to construct the following “fundamental diagram” which gives an illustration of various enlargements of D and of our method to prove Theorems A and B: DSL(2),val

 D,val

→

→

DBS,val

↓

↓

DSL(2)

DBS

→ XBS

(5.0.1)

↓
\D

←



D

 In this diagram, D , D , D,val , DSL(2) , DSL(2),val , DBS,val , and DBS are enlargements of D. The first two appeared in Section 1.3, the last four were defined in our  previous paper [KU2], as reviewed in Sections 5.1 and 5.2 below, and D,val will be defined in Section 5.3. Here, X denotes the space of all maximal compact subgroups of GR and XBS denotes the Borel-Serre space (X in their notation) constructed in [BS] as a nice enlargement of X . In (5.0.1), all arrows are continuous, all the vertical arrows are kinds of projective limits of blow-ups and are proper and surjective, and  the map DBS → XBS and the induced map
\D →
\D are also proper and surjective. To prove that
\D ( is a fan and
⊂ GZ is strongly compatible with ) has the nice properties stated in Theorems A and B (Chapter 4), our method is to transport the nice properties of XBS along this fundamental diagram from the right to the left. For example, we explain roughly how we prove that the quotient space
\D is Hausdorff. It is known that the action of
on XBS is proper. From this, we deduce that the actions of
on DBS , DBS,val , and DSL(2),val are proper. Using the fact that DSL(2),val → DSL(2) is proper and surjective, we obtain that the action of
on DSL(2) is proper. (These results were obtained in [KU2].) From  this, we deduce that the action of
on D,val is proper. By using the fact that    D,val → D is proper and surjective, we obtain that the action of
on D is proper.   Hence the quotient space
\D is Hausdorff. Since
\D →
\D is proper and surjective, this shows that
\D is Hausdorff. (These arguments will be given in Section 7.4.)

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In Section 5.1, we review the spaces XBS , DBS , and DBS,val . In Section 5.2, we  review the spaces DSL(2),val and DSL(2) . In Section 5.3, we define the space D,val .  In Section 5.4, we show how to define the map D,val → DSL(2) in (5.0.1) by using the work of Cattani, Kaplan, and Schmit [CKS]. Theorems 5.4.3 and 5.4.4 state that this map is actually well defined and continuous; the proofs will be given in Chapter 6.

5.1 BOREL-SERRE SPACES (REVIEW) 5.1.1 Summary. Let X be the set of all maximal compact subgroups of GR . Then GR acts on X transitively by inner automorphisms. Since the normalizer of GR at each K ∈ X is K itself, we have a GR -equivariant isomorphism ∼

→ X, GR /K −

g  → Int(g)K = gKg −1 ,

for each fixed K ∈ X . By using this isomorphism, we introduce a topology of X . This topology does not depend on the choice of K. Borel and Serre constructed in [BS] a topological space XBS that contains X as an open dense subset. This Section 5.1 is a review of [BS] and [KU2, §2] for our later use, i.e., we review the Borel-Serre space XBS associated with X , a similar enlargement DBS associated with D, and the projective limit DBS,val of the blow-ups of DBS . These spaces are related by continuous proper surjective maps in the following way: DBS,val    DBS

−−−−→ XBS .

5.1.2 For F ∈ D, define compact groups KF , KF
such that KF
⊂ KF ⊂ GR as follows. Let KF
:= {g ∈ GR | gF = F }. Let KF be the subgroup of GR consisting of all elements that preserve the Hermitian inner product (x, y)  → CF (x), y on H0,C . Here CF is the Weil operator (1.1.2) associated with F . Then KF is a maximal compact subgroup of GR , and we have a canonical continuous map D → X, which is proper and surjective.

F  → KF ,

159

FUNDAMENTAL DIAGRAM

5.1.3 Borel-Serre action. Let P be a parabolic subgroup of GR , Pu its unipotent radical, and C the center of P /Pu . In what follows, we regard P , Pu , and C as the groups of their R-valued points, respectively. Let K ∈ X . Then, for each a ∈ C, there exists a unique element aK of P having the following properties (1) and (2). (1) The image of aK in P /Pu coincides with a. (2) For the Cartan involution θK : GR → GR associated with the maximal compact subgroup K, we have −1 . θK (aK ) = aK

Recall that θK is the unique automorphism of GR such that θK2 = id and K = {g ∈ GR | θK (g) = g}. If K = KF with F ∈ D, θK (g) = CF gCF−1 for g ∈ GR where CF is the Weil operator (1.1.2). We call aK the Borel-Serre lifting of a at K. The map C → P,

a  → aK ,

is a homomorphism of algebraic groups over R. Furthermore, we have an action ◦ of the group C on D (resp. X ) defined by (resp. a ◦ K = Int(aK )K).

a ◦ F = aKF F We call this the Borel-Serre action. This action satisfies a ◦ pF = p(a ◦ F )

(resp. a ◦ Int(p)K = Int(p)(a ◦ K))

(3)

for a ∈ C, p ∈ P , F ∈ D, K ∈ X ([KU2, 2.4]). 5.1.4 Now let P be a Q-parabolic subgroup of GR , SP the maximal Q-split torus in the center C of P /Pu , and AP the connected component of the group of R-valued points of SP which contains the unity. Definition 5.1.5 ([KU2, 2.5]) The Borel-Serre space DBS (resp. XBS ) is defined by     P is a Q-parabolic subgroup of GR ,  . DBS (resp. XBS ) := (P , Z)  Z is an (AP ◦)-orbit in D (resp. X ) Definition 5.1.6 ([KU2, 2.6]) We define the space DBS,val by    T is an R-split torus of GR ,        Z is a (T>0 )-orbit in D, DBS,val := (T , Z, V )  .     V is a valuative submonoid of X(T ),   which satisfy the following conditions (1)–(3)

160

CHAPTER 5

Here, T>0 is the connected component of the group of R-valued points of T containing the unity, and X(T ) denotes the character group of T . A submonoid V of X(T ) is said to be valuative if X(T ) = V ∪ V −1 (cf. 3.6.16). (1) θKF (t) = t −1 (∀ F ∈ Z, ∀ t ∈ T ). (2) V × = {1}.  (3) Let H0,R = χ ∈X(T ) H0,R (χ ) be the decomposition into eigenspaces H0,R (χ ) := {v ∈ H0,R | tv = χ (t)v (∀t ∈ T )}. Then, for α ∈ X(T ), the R-subspace  H0,R (αχ −1 ) Wα := χ ∈V

of H0,R is Q-rational. Note that, since the set {Wα | α ∈ X(T )} of R-subspaces of H0,R is totally ordered with respect to the inclusion, we have a Q-parabolic subgroup PT ,V := {g ∈ (G◦ )R | g preserves (Wα )α∈X(T ) }. (For G◦ , see Section 0.7.) 5.1.7 We have morphisms DBS,val   α DBS

β

−−−−→ XBS ,

α : (T , Z, V )  → (PT ,V , APT ,V ◦ Z), β : the map induced by D → X , F  → KF . 5.1.8 For a Q-parabolic subgroup P of GR , we define DBS (P ) := {(Q, Z) ∈ DBS | Q ⊃ P }, XBS (P ) := {(Q, Z) ∈ XBS | Q ⊃ P }, DBS,val (P ) := {(T , Z, V ) ∈ DBS,val | PT ,V ⊃ P }. In 5.1.9–5.1.12 below, we give preliminaries needed to define the topologies on the spaces DBS , XBS , and DBS,val . 5.1.9 Let P be a Q-parabolic subgroup of GR . A subset P of X(SP ) is defined as follows. Let S˜P ⊂ P be any torus such that the projection P → P /Pu induces an

161

FUNDAMENTAL DIAGRAM ∼

→ SP . Let isomorphism S˜P − 
P := {χ ∈ X(S˜P ) | 0  = ∃v ∈ Lie(Pu ) such that Ad(a)v = χ (a)−1 v (∀ a ∈ S˜P )}. ∼ → SP . Then the Identify X(S˜P ) with X(SP ) via the canonical isomorphism S˜P − subset 
P of X(SP ) is independent of the choice of the liftings S˜P , and it is a finite subset of X(SP ) which generates Q ⊗ X(SP ) over Q. There exists a unique subset P of 
P satisfying the following two conditions:

(1) The number (P ) of the elements of P coincides with rank SP . (2) 
P is contained in the submonoid of X(SP ) generated by P . Let P be as above. Let Q be a Q-parabolic subgroup of GR containing P . Then there are injective maps (3)

SQ → SP , Q → P ,

(4)

obtained as follows, which are regarded as inclusion maps. Note that Q ⊃ P ⊃ Pu ⊃ Qu . We have that SQ ⊂ P /Qu in Q/Qu , that the canonical map SQ → P /Pu is injective, and that the image of this map is contained in SP . This gives the injection (3). Let I := {χ ∈ P | χ annihilates SQ }, and let J ⊂ P be the complement of ∼ → Q . The injection (4) is I in P . Then the restriction to SQ gives a bijection J − ∼ − J → P . It is known that we have a bijection obtained as the composite Q ← ∼

→ {subset of P }, Q  → Q , {Q-parabolic subgroup of GR containing P } − (5) (cf. [BS, §4],[B, §11]). 5.1.10 Identification of DBS (P ) with D ×AP AP . Let P be a Q-parabolic subgroup of GR . Let X(SP )+ be the submonoid of X(SP ) consisting of elements χ such that, for some n ≥ 1, χ n belongs to the submonoid of X(SP ) generated by P . Then X(SP )+ is an fs monoid, and (X(SP )+ )gp = X(SP ). We define mult AP := Hom(X(SP )+ , R≥0 ) ⊃ Hom(X(SP )+ , R>0 ) = Hom(X(SP ), R>0 ) = AP .

Then, AP

∼

− →

Map(P , R≥0 )

 AP



 ∼

− →

Map(P , R>0 )

r R≥0

 

r R>0 ,

where r := (P ) = rank SP . The group AP acts on AP in the natural way. Denote D ×AP AP := (D × AP )/AP under the action a · (F, b) = (a ◦ F, a −1 b)

(a ∈ AP , (F, b) ∈ D × AP ).

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CHAPTER 5

Then we have a bijection DBS (P )  D ×AP AP ,

(Q, Z) ←→ (F, b),

defined as follows. For (Q, Z) ∈ DBS (P ), F is any element of Z, and b ∈ AP is defined by b(χ ) = 0 if χ ∈ Q and b(χ ) = 1 if χ ∈ P − Q . Conversely, for (F, b) ∈ D ×AP AP , Q is the Q-parabolic subgroup of GR containing P such that Q = {χ ∈ P | b(χ ) = 0}, and Z := {a ◦ F | a ∈ AP , χ (a) = b(χ ) for any χ ∈ P − Q }. 5.1.11 Valuative spaces associated with toric varieties. Let S be an fs monoid. We denote the space Spec(C[S])an,val (the space Xval in 3.6.18–3.6.24 for X = Spec(C[S])an = Hom(S, Cmult )) also by Hom(S, Cmult )val . We regard the open subset Hom(S gp , C× ) of Hom(S, Cmult ) as an open subset of Hom(S, Cmult )val via the homeomorohism ∼

→ Hom(S gp , C× ). Hom(S gp , C× ) ×Hom(S,Cmult ) Hom(S, Cmult )val − Let mult )val ⊂ Hom(S, Cmult )val Hom(S, R≥0

be the closure of Hom(S, R>0 ). When S = Nn , Hom(S, Cmult )val and n mult )val are also denoted by (Cn )val and (R≥0 )val , respectively. Hom(S, R≥0 We identify, as sets,    V is a valuative submonoid of S gp        with V ⊃ S,  , (1) Spec(C[S])an,val = (V , h)  mult     h : V → C−1 is× a homomorphism   × such that h (C ) = V    V is a valuative submonoid of S gp        with V ⊃ S, mult  Hom(S, R≥0 )val = (V , h)  . (2) mult   ≥0 is a homomorphism   h : V → R−1   such that h (R>0 ) = V × Here (V , h) as in (1) corresponding to (x, V
, h
) ∈ Spec(C[S])an,val is as follows: V is the inverse image of V
under the canonical surjective homomorphism S gp → gp × )x , and h is the homomorphism induced by h
. The identification (2) is the (MX /OX one induced from (1). In the identification (1) (resp. (2)), a directed family ((Vλ , hλ ))λ converges to mult )val ) if and only if the following (V , h) in Spec(C[S])an,val (resp. Hom(S, R≥0 condition (3) is satisfied. (3) For each χ ∈ V , χ ∈ Vλ for sufficiently large λ and hλ (χ ) converges to h(χ ). Since the logarithmic structure of the open set Spec(C[S gp ])an ⊂ Spec(C[S])an is trivial, we have dense open immersions Spec(C[S gp ])an ⊂ Spec(C[S])an,val , mult )val . Hom(S gp , R>0 ) ⊂ Hom(S, R≥0

163

FUNDAMENTAL DIAGRAM

Furthermore, the natural action of the group Spec(C[S gp ])an (resp. Hom(S gp , R>0 )) mult on Spec(C[S])an (resp. Hom(S, R≥0 )) extends uniquely to a continuous action of mult )val ). In the identification (1) this group on Spec(C[S])an,val (resp. Hom(S, R≥0 (resp. (2)), an element a of this group sends (V , h) to (V , ah), where ah is the homomorphism sending x ∈ V to a(x)h(x). 5.1.12 Identification of DBS,val (P ) with D ×AP (AP )val . Let P be a Q-parabolic subgroup of GR . Define mult )val , (AP )val := Hom(X(SP )+ , R≥0

where X(SP )+ is the submonoid of X(SP ) introduced in 5.1.10. If we identify the submonoid of X(SP )+ generated by P with Nr (r := rank(SP )), the inclusion Nr ⊂ X(SP )+ induces ∼

r → (R≥0 )val . (AP )val −

For the set DBS,val (P ) in 5.1.8, we have a bijection DBS,val (P ) → D ×AP (AP )val , (T , Z, V
)  → (F, (V , h)), which is given by   F ∈ Z (any element), V := (the inverse image of V
under X(SP ) → X(T )),   mult , h(χ ) = 1 (resp. 0) if χ ∈ V × (resp. χ ∈ / V × ). h : V → R≥0

(1)

(2)

Here, in the definition of V , we regard T as a subtorus of SP via the composite of the embeddings T → SPT ,V
→ SP (5.1.6 and 5.1.9). The inverse map D ×AP AP → DBS,val (P ), (F, (V , h))  → (T , Z, V
), is given by    the image of the annihilator of V × in SP   ,  T := under the Borel-Serre lifting SP → GR at KF Z := {a ◦ F | a ∈ AP , χ (a) = h(χ ) (∀χ ∈ V × )},   
V := (the image of V under X(SP ) → X(T )) (so that V
 V /V × ). 5.1.13 Topologies of DBS , XBS and DBS,val . Let P be a Q-parabolic subgroup of GR . We have DBS (P )  D ×AP AP (see 5.1.10), XBS (P )  X ×AP AP (analogously to the above), DBS,val (P )  D ×AP (AP )val (see 5.1.12). Via these isomorphisms, we introduce topologies on DBS (P ), XBS (P ), and DBS,val (P ), respectively. We introduce the strongest topology on DBS (resp. XBS ,

164

CHAPTER 5

DBS,val ) for which the map DBS (P ) → DBS (resp. XBS (P ) → XBS , DBS,val (P ) → DBS,val ) is continuous for every Q-parabolic subgroup P of GR . Then, it can be shown, as in [BS], that all these maps DBS (P ) → DBS , XBS (P ) → XBS and DBS,val (P ) → DBS,val are open embeddings. We have the following results on DBS and DBS,val which are analogous to the similar results on XBS obtained in [BS]. Theorem 5.1.14 ([KU2, 2.17, 5.2, 5.8]) (i) The maps α : DBS,val → DBS , β : DBS → XBS in 5.1.7 are proper and surjective. (ii) The spaces DBS , DBS,val are Hausdorff and moreover locally compact. (iii) For a subgroup
of GZ , the actions of
on DBS and on DBS,val are proper, and the quotient spaces
\DBS and
\DBS,val are Hausdorff. If
is neat, the projections DBS →
\DBS and DBS,val →
\DBS,val are local homeomorphisms. If
is of finite index,
\DBS and
\DBS,val are compact spaces. See Section 7.2 below for the definition of “proper action” and the basic facts concerning this notion. 5.1.15 Relationship to Iwasawa decomposition. The theory of Borel-Serre spaces is closely related to the theory of Iwasawa decomposition. Let P be a minimal parabolic subgroup of GR and K a maximal compact subgroup of GR so that we have the Iwasawa decomposition associated with (P , K), r GR  Pu × R>0 ×K

(homeomorphism),

(1)

where r is the dimension of a maximal R-split torus of GR . This homeomorphism is defined in the following way. Let SPR be the maximal R-split torus of the center of P /Pu , let AR P be the connected component of the group of R-valued points r of SPR containing the unity, and identify AR P with R>0 just as in 5.1.10. Then the homeomorphism (1) is defined by ptK k  → (p, t, k). Take F ∈ D. Then the Iwasawa decomposition (1) associated with (P , KF ) induces r X  Pu × R>0 , Int(ptKF )KF ↔ (p, t), r D  Pu × R>0 × (KF /KF
), ptKF kF ↔ (p, t, k mod KF
).

(2)

Now let Q be a Q-rational parabolic subgroup of GR , and take a minimal parabolic subgroup P of GR such that P ⊂ Q. Then SQ ⊂ P /Qu → P /Pu induces injections r SQ → SPR , AQ → AR P = R>0 . Since a ◦ Int(ptKF )KF = Int(p(at)KF )KF and a ◦ ptKF kF = p(at)KF kF for a ∈ AQ , (2) induces r ×AQ AQ ), XBS (Q)  Pu × (R>0 r DBS (Q)  Pu × (R>0 ×AQ AQ ) × (KF /KF
),

DBS,val (Q) 

r Pu × (R>0

×

AQ

AQ,val ) × (KF /KF
).

(3)

165

FUNDAMENTAL DIAGRAM

In the case P = Q and SPR = SQ , the homeomorphisms in (3) have simpler forms r , XBS (P )  Pu × R≥0 r × (KF /KF
), DBS (P )  Pu × R≥0

DBS,val (P ) 

r Pu × (R≥0 )val

(4)

× (KF /KF
).

5.2 SPACES OF SL(2)-ORBITS (REVIEW) This Section 5.2 is a review of [KU2, §3, §4] for our later use, i.e., we review spaces of SL(2)-orbits DSL(2) and the projective limit DSL(2),val of the blow-ups of DSL(2) . These spaces, together with the spaces in the previous section, will form the following diagram: DSL(2),val

→

DBS,val

↓

↓

DSL(2)

DBS

→ XBS .

In general, there is no direct relation between DSL(2) and DBS (see [KU2, §6]), which is the reason that we introduce DSL(2),val and DBS,val . 5.2.1 SL(2)-orbits in several variables. We review the definition of SL(2)-orbits in several variables ([CKS, 4], [KU2, §3]). Let (ρ, ϕ) be a pair of a homomorphism ρ : SL(2, C)n → GC of algebraic groups, ˇ Throughout this book, such a which is defined over R and a map ϕ : P1 (C)n → D. pair (ρ, ϕ) is called an SL(2)-orbit in n variables if it satisfies the following three conditions (1)–(3): (1) ϕ(gz) = ρ(g)ϕ(z) for all g ∈ SL(2, C)n and all z ∈ P1 (C)n . (2) The Lie algebra homomorphism ρ∗ : sl(2, C)⊕n → gC , associated with ρ, satisfies ρ∗ (F p (z)(sl(2, C)⊕n )) ⊂ F p (ϕ(z))(gC )

(∀ z ∈ P1 (C)n , ∀ p ∈ Z).

Here F p (z)(sl(2, C)⊕n ) := {(Xj )j ∈ sl(2, C)⊕n | Xj F q (zj ) ⊂ F q+p (zj ) (∀ q ∈ Z, 1 ≤ j ≤ n)}, p F (ϕ(z))(gC ) := {X ∈ gC | XF q (ϕ(z)) ⊂ F q+p (ϕ(z)) (∀ q ∈ Z)}, where F (zj ) (resp. F (ϕ(z))) denotes the filtration on C2 (resp. H0,C ) corresponding to zj ∈ P1 (C) (1.2.3) (resp. ϕ(z) ∈ Dˇ (1.2.2)). (3) Let h be the upper-half plane. Then ϕ(hn ) ⊂ D. We can replace the conditions (2) and (3) by

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CHAPTER 5

(4) Let i := (i, . . . , i) ∈ hn . Then ϕ(i) ∈ D, and ρ∗ (F p (i)(sl(2, C)⊕n )) ⊂ F p (ϕ(i))(gC )

(∀ p ∈ Z).

5.2.2 Let (ρ, ϕ) be an SL(2)-orbit in n variables. For 1 ≤ j ≤ n, let   0 1 ∈ gR , Nj := ρ∗j 0 0 where ρ∗j : sl(2, C) → gC denotes the restriction of ρ∗ to the j th factor of sl(2, C)⊕n . n → SL(2, R)n be the homomorphism defined by Let  : Gm,R   −1   −1 tn 0 t1 0 (t1 , . . . , tn )  → ,..., . 0 tn 0 t1 Define a homomorphism n ρ˜ : Gm,R → GR ,

ρ(t ˜ 1 , . . . , tn ) := ρ((t1 · · · tn , t2 · · · tn , . . . , tn−1 tn , tn )). (1)

For 1 ≤ j ≤ n, let ρ˜j : Gm,R → GR

(2)

n be the restriction of ρ˜ to the j -th factor of Gm,R . Then, ρ˜j (t) = ρ( (t, . . . , t , 1 . . . , 1)). . /0 1 j

5.2.3 Rank of an SL(2)-orbit. Let (ρ, ϕ) be an SL(2)-orbit in n variables. Let J be the set of all j (1 ≤ j ≤ n) such that the associated homomorphism ρ∗ of Lie algebras is not the zero map on the j th factor of sl(2, C)⊕n , and let r be the number of elements of the set J . Then there exists a unique SL(2)-orbit (ρ
, ϕ
) in r variables such that (ρ, ϕ) = (ρ
, ϕ
) ◦ πJ where πJ : (SL(2, C) × P1 (C))n → (SL(2, C) × P1 (C))r is the projection to the J -factor. We call r the rank of (ρ, ϕ), and (ρ
, ϕ
) the SL(2)-orbit of rank r associated with (ρ, ϕ). 5.2.4 For a nilpotent element N ∈ gR , the weight filtration associated with N is the increasing filtration W = W (N) of H0,R characterized by the following two conditions ([D5]): (1) N Wk ⊂ Wk−2 for all k ∈ Z. ∼ → gr W (2) N k : gr W k − −k for all k ∈ Z≥0 . The following is proved in [CK2].

167

FUNDAMENTAL DIAGRAM

Theorem ([CK2]) Let (σ, Z) be either a nilpotent orbit or a nilpotent i-orbit (1.3.6). Then, for any elements N , N
of the interior (0.7.7) of σ , the filtrations W (N) and W (N
) of H0,R coincide. This common filtration is denoted by W (σ ).

5.2.5 Family of weight filtrations associated with an SL(2)-orbit. As in [CKS, §4], an SL(2)-orbit (ρ, ϕ) in n variables defines an ordered family of nilpotent i-orbits (σj , Zj )1≤j ≤n consisting of σj := (R≥0 )N1 + · · · + (R≥0 )Nj (Nk is as in 5.2.2), Zj := exp(iσj,R )ϕ(0, . . . , 0, i, . . . , i) = ϕ(iR, . . . , iR , i, . . . , i). . /0 1 . /0 1 j

(1)

j

We call the family W = (W (σj ))1≤j ≤n the family of weight filtrations associated with (ρ, ϕ). We say the weight filtrations of (ρ, ϕ) are rational if the filtrations W (σj ) are defined over Q for 1 ≤ j ≤ n. The set J in 5.2.3 coincides with {j | 1 ≤ j ≤ n, W (σj )  = W (σj −1 )}, where we understand σ0 = {0}. We have W (σj )  = W (σk ) if j, k ∈ J and j  = k. Definition 5.2.6 ([KU2, 3.6]) We define the set DSL(2) as follows. For j = 1, 2, let (ρj , ϕj ) be an SL(2)-orbit in nj variables. We say (ρ1 , ϕ1 ) and (ρ2 , ϕ2 ) are equivalent if the following condition (1) is satisfied. Let rj be the rank of (ρj , ϕj ), and let (ρj
, ϕj
) be the SL(2)-orbit of rank rj associated to (ρj , ϕj ) as in 5.2.3. r such that ρ2
= (1) r1 = r2 (denoted r := r1 = r2 ), and there exists t ∈ R>0




Int(ρ1 ((t))) ◦ ρ1 and ϕ2 = ρ1 ((t)) · ϕ1 .

We define DSL(2) to be the set of all equivalence classes of SL(2)-orbits (ρ, ϕ) of n variables for all n ≥ 0 whose weight filtrations are rational. We denote by [ρ, ϕ] the point of DSL(2) represented by (ρ, ϕ). We denote by DSL(2),n ⊂ DSL(2) the part of DSL(2) consisting of classes of SL(2)orbits of rank n. Then, we have , DSL(2) = DSL(2),n , D = DSL(2),0 . n≥0

Let p ∈ DSL(2),n and let (ρ, ϕ) be a representative of [ρ, ϕ] in n variables. Then, the family W = (W (σj ))1≤j ≤n of weight filtrations of H0,R associated with (ρ, ϕ) (cf. 5.2.5) depends only on p and is independent of the choice of (ρ, ϕ). We call W the family of weight filtrations associated to p. In our previous paper [KU2, §3], we considered only SL(2)-orbits in n variables of rank n, but the space DSL(2) in this book coincides with the one in that paper.

168

CHAPTER 5

Definition 5.2.7 ([KU2, 3.7]) For a non-negative integer n, we define     p ∈ DSL(2),n , Z ⊂ D, V ⊂ X(Gn ) m,R  . DSL(2),val,n := (p, Z, V )  which satisfy the following conditions (1) and (2) n Let X(Gm,R )+ be the inverse image of Nn ⊂ Zn under the canonical isomorphism n X(Gm,R )  Zn . n n ) such that X(Gm,R )+ ⊂ V and (1) V is a valuative submonoid of X(Gm,R n × X(Gm,R )+ ∩ V = {1}. (2) Let n T := {t ∈ Gm,R | χ (t) = 1 (∀χ ∈ V × )},

and let (ρ, ϕ) be a representative of p in n variables. Then Z is a ρ(T ˜ >0 )-orbit n in ϕ(iR>0 ). Here we denote by T>0 the connected component of the group T containing the unity. We define DSL(2),val :=

,

DSL(2),val,n .

n≥0

We have the canonical surjection DSL(2),val → DSL(2) , (p, Z, V )  → p. n Note that ρ(T ˜ >0 ) and ϕ(iR>0 ) in 5.2.7 (2) depend only on p and are independent of the choice of a representative (ρ, ϕ).

Lemma 5.2.8 ([KU2, 3.8]) Let (ρ, ϕ) be an SL(2)-orbit and put r = ϕ(i). Then n θKr (ρ(t)) ˜ = ρ(t) ˜ −1 (∀ t ∈ Gm,R ).

Lemma 5.2.9 ([KU2, 3.9]) Let (ρ, ϕ) and r be as in 5.2.8. For 1 ≤ j ≤ n, let W (j ) = W (σj ) be as in 5.2.5 and let Pj be the Q-parabolic subgroup of GR defined by W (j ) . Then, the Borel-Serre lifting at Kr of the j th weight map (j )

)k Gm,R → Pj /Pj,u , tj  → (tjk on gr W k n coincides with ρ˜j in 5.2.2 (2) (i.e., the restriction of ρ˜ to the j th factor of Gm,R ).

Lemma 5.2.10 ([KU2, 3.10]) Let (ρ, ϕ), r = ϕ(i) and W be as in 5.2.8 and 5.2.5. Then the SL(2)-orbit (ρ, ϕ) is determined by (W, r). Theorem 5.2.11 ([KU2, 3.11]) There is an injective map DSL(2),val → DBS,val ,

([ρ, ϕ], Z, V )  → (ρ(T ˜ ), Z, V
).


(1)

×

Here T is the subtorus of in 5.2.7 (2), and V  V /V , which is regarded as a subset of the character group of ρ(T ˜ ). n Gm,R

5.2.12 A family (W (j ) )1≤j ≤n of increasing filtrations W (j ) of H0,Ris called a compatible family if there exists a direct sum decomposition H0,R = m∈Zn H (m) such that

169

FUNDAMENTAL DIAGRAM (j )



Wk = m∈Zn , mj ≤k H (m) for any j and k. (We used this terminology “compatible family” in our previous paper [KU2] and also use it in this book. But, after writing this book, we realized that this notion is equivalent to the notion “distributive family” of Kashiwara [K].) Note that, for [ρ, ϕ] ∈ DSL(2),n , the family of weight filtrations (W (σj ))1≤j ≤n associated with [ρ, ϕ] in 5.2.5 is a compatible family. Let W = (W ( j ) )1≤j ≤n be a compatible family of Q-rational increasing filtrations (j ) W of H0,R . We denote GW,R = {g ∈ GR | gW (j ) = W (j ) (1 ≤ j ≤ n)}. For W = (W (j ) )1≤j ≤n as above, we define the subset DSL(2) (W ) of DSL(2) by    ∃sj ∈ Z (1 ≤ j ≤ m) such that $   . DSL(2) (W ) := p ∈ DSL(2),m 1 ≤ s1 < · · · < sm ≤ n and   W (p)(j ) = W (sj ) (∀ j ) 0≤m≤n Here (W (p)(j ) )1≤j ≤m is the family of weight filtrations associated with p ∈ DSL(2),m (5.2.6). We define the subset DSL(2),val (W ) of DSL(2),val by the pullback of DSL(2) (W ). Definition 5.2.13 ([KU2, 3.13]) We define the topology of DSL(2),val as the weakest one in which the following two families of subsets are open: (1) The pullbacks on DSL(2),val of open subsets of DBS,val . (2) The subset DSL(2),val (W ) for any n and any compatible family of Q-rational increasing filtrations W = (W (j ) )1≤j ≤n . We induce the quotient topology on DSL(2) of the above one under the projection DSL(2),val → DSL(2) . 5.2.14 The topology of DSL(2) defined in 5.2.13 has the following property (see [KU2, 4.19]). For an SL(2)-orbit (ρ, ϕ) of rank n, [ρ, ϕ] ∈ DSL(2) is the limit of ϕ(iy1 , . . . , iyn ) ∈ D, as yj > 0 and

yj yj +1

→ ∞ for 1 ≤ ∀ j ≤ n (yn+1 denotes 1).

Note that the space DSL(2),val is Hausdorff from 5.1.14 and 5.2.13. We have, moreover, Theorem 5.2.15 ([KU2, 3.14, 4.18, 5.2, 5.8]) (i) The canonical map DSL(2),val → DSL(2) is proper and surjective. (ii) The spaces DSL(2) and DSL(2),val are Hausdorff. (iii) The spaces DSL(2) (W ) and DSL(2),val (W ) are regular spaces (cf. 6.4.6 below). (iv) For a subgroup
of GZ , the actions of
on DSL(2) and on DSL(2),val are proper, and the quotient spaces
\DSL(2) and
\DSL(2),val are Hausdorff. If
is neat, the projections DSL(2) →
\DSL(2) and DSL(2),val →
\DSL(2),val are local homeomorphisms.

170

CHAPTER 5

Proposition 5.2.16 ([KU2, 4.15]) Let (ρ, ϕ) be an SL(2)-orbit in n variables and of rank n whose associated weight filtrations are rational, and let r = ϕ(i). Then the sets n B(U, U
, U

) := {ρ(t)gk ˜ · r | k ∈ U, t ∈ R>0 ∩ U
, ν Int(ρ(t)) ˜ (g) ∈ U

(ν = 0, ±1)}

form a basis of the filter {D ∩ V | V is a neighborhood of [ρ, ϕ] in DSL(2) }, when U n , 1 in GR ). (resp. U
, U

) ranges over all neighborhoods of 1 in Kr (resp. 0 in R≥0 In Chapter 10, we will give new proofs of the results 5.2.15 (i)–(iii) and 5.2.16. 5.2.17 Correction. In [KU2, Lemma 4.7], the definition of B(U, U
, U

) is written wrongly as {g ρ(t)kr ˜ | · · · }. It should be corrected to {ρ(t)gkr ˜ | · · · }, that is, g and ρ(t) ˜ in [KU2, Lemma 4.7] should be interchanged, as in Proposition 5.2.16 above. The proof of [KU2, Lemma 4.7] and the rest of the paper [KU2] concerning this lemma are correct without any change.

5.3 SPACES OF VALUATIVE NILPOTENT ORBITS 



In this section, we introduce the spaces Dval and Dval . Dval contains, as a subspace,   the project limit D,val of the blow-ups of the space of nilpotent i-orbits D , with which we can describe the relationship between the space of SL(2)-orbits DSL(2)  and the space D . Definition 5.3.1 We define    A is a Q-linear subspace of gQ consisting of        mutually commutative nilpotent elements,  V := (A, V )  . ∗   V is a valuative submonoid of A := HomQ (A, Q)    with V ∩ (−V ) = {0} Note that a valuative submonoid V of A∗ as above is stable under the multiplication by elements of Q≥0 . In fact, assume x ∈ A∗ and nx ∈ V for some n ∈ Z>0 . We prove x ∈ V . Assume x  ∈ V . Then −x ∈ V and hence −nx ∈ V ∩ (−V ) = {0}. Therefore x = 0, which contradicts x  ∈ V . 5.3.2 For (A, V ) ∈ V, let F(A, V ) be the set of all rational nilpotent cones σ ⊂ gR satisfying the following conditions (1) and (2). (1) σR = AR . (2) Let (σ ∩ A)∨ := {h ∈ A∗ | h(σ ∩ A) ⊂ Q≥0 }. Then (σ ∩ A)∨ ⊂ V . Note that (σ ∩ A)∨ = {h ∈ A∗ | h(σ ) ⊂ R≥0 }. This set F(A, V ) has the following properties (3)–(5).

171

FUNDAMENTAL DIAGRAM

(3) If σ, τ ∈ F(A, V ), then σ ∩ τ ∈ F(A, V ). (4) If σ ⊂ AR is a rational nilpotent cone containing some τ ∈ F(A, V ), then σ ∈ F(A,  V ). (5) V = σ ∈F(A,V ) (σ ∩ A)∨ . (4) is evident. We prove (3) and (5). For a finitely generated (Q≥0 )-cone C in A (resp. A∗ ) such that CQ = A (resp. ∗ A ) and C ∩ (−C) = {0}, we have C ∨∨ = C, where C ∨ = Hom(C, Qadd ≥0 ). For the proof of (3), it is sufficient to show that (σ ∩ τ ∩ A)∨ = (σ ∩ A)∨ + (τ ∩ A)∨ . But ( )∨ of both sides coincide with σ ∩ τ ∩ A. Next we prove (5). When C ranges  over all finitely generated (Q≥0 )-subcones of V such that CQ = A∗ , then V = C. If σC denotes {a ∈ AR | C(a) ⊂ R≥0 }, then C ∨ = σC ∩ A. We have σC ∈ F(A, V ) since C ∨∨ = C ⊂ V . These prove (5). Definition 5.3.3 (i) We define

    (A, V ) ∈ V, Z is an exp(AC )  Dˇ val (resp. Dˇ val . ) := (A, V , Z)  (resp. exp(iAR ))-orbit in Dˇ

(ii) We define Dval (resp.

 Dval )

 

   (A, V , Z) ∈ Dˇ val (resp. Dˇ  ),   val := (A, V , Z)  ∃σ ∈ F(A, V ) such that Z is a   σ -nilpotent orbit (resp. i-orbit)

(cf. 1.3.7). There is a natural map 

Dval → Dval ,

(A, V , Z)  → (A, V , exp(AC )Z).

 Lemma 5.3.4 Let  be a fan in gQ . Let (A, V , Z) ∈ Dˇ val (resp. Dˇ val ). Assume that there is σ ∈  such that σ ∩ AR ∈ F(A, V ), and let σ0 be the smallest one among such σ . (Note that the smallest one exists by 5.3.2 (3).) Assume that exp(σ0,C )Z (resp. exp(iσ0,R )Z) is a σ0 -nilpotent orbit (resp. i-orbit). Then Z is a (σ0 ∩ AR )-nilpotent orbit (resp. i-orbit).

Proof. It is sufficient to prove that σ0 ∩ AR contains a point of the interior (0.7.7) of σ0 . Assume σ0 ∩ AR = ∪τ τ ∩ AR , where τ ranges over all faces of σ0 that are different from σ0 . Then, since τ ∩ AR is a face of σ0 ∩ AR and its dimension is strictly smaller than the dimension of σ0 ∩ AR unless τ ∩ AR = σ0 ∩ AR , we should have τ ∩ AR = σ0 ∩ AR for some face τ  = σ0 of σ0 . But this contradicts the minimality of σ0 . 2  Definition 5.3.5 For a fan  in gQ , we define D,val ⊂ Dˇ val (resp. D,val ⊂  ˇ Dval ) as     (A, V , Z) ∈ Dˇ val (resp. Dˇ  ),      val   σ ∩ AR ∈ F(A, V ) for some σ ∈ ,    D,val (resp. D,val ) := (A, V , Z)  .    exp(σ0,C )Z (resp. exp(iσ0,R )Z) is      a σ0 -nilpotent orbit (resp. i-orbit)

172

CHAPTER 5

Here σ0 is as in Lemma 5.3.4. Then D,val ⊂ Dval ,





D,val ⊂ Dval .

For a sharp rational nilpotent cone σ , we define Dσ,val := D{face of σ },val , We have D,val =

$

Dσ,val ,







$

Dσ,val := D{face of σ },val . D,val =



Dσ,val .

σ ∈

σ ∈

There is a natural map 

D,val → D,val ,

(A, V , Z)  → (A, V , exp(AC )Z).

We define maps D,val → D , (A, V , Z)  → (σ0 , exp(σ0,C )Z),   D,val → D , (A, V , Z)  → (σ0 , exp(iσ0,R )Z),

where σ0 ∈  is as in Lemma 5.3.4. 5.3.6 Let (
, ) be a strongly compatible pair of a subgroup
of GZ and a fan  in gQ . For σ ∈ , we denote toricσ,val := Spec(C[
(σ )∨ ])an,val , mult )val ⊂ toricσ,val , |toric|σ,val := Hom(
(σ )∨ , R≥0

(1)

(5.1.11). There is a canonical bijection between toricσ,val (resp. |toric|σ,val ) and the set of all triples (A, V , z) (resp. (A, V , y)) where (A, V ) is an element of V such that A ⊂ σR and σ ∩ A ∈ F(A, V ), and z ∈ σC /(AC + log(
(σ )gp ) (resp. y ∈ σR /AR ). The bijection is given as follows. For (A, V , z) (resp. (A, V , y)), the corresponding element of toricσ,val (resp. |toric|σ,val ) is the pair (V
, h) (5.1.11) where V
is the set of all elements χ of
(σ )∨ gp such that the induced homomorphism exp ∼

χ

→Q A → σR ∩ gQ −→ Q ⊗
(σ )gp − mult ) that belongs to V , and h is the homomorphism V
→ Cmult (resp. V
→ R≥0

×
× sends V − (V ) to 0 and an element χ of (V ) to the image of e(z) (resp. e(iy)) (3.3.5) under χ : C× ⊗
(σ )gp → C× (resp. (R>0 ) ⊗
(σ )gp → R>0 ).

5.3.7 Let (
, ) be as in 5.3.6. For σ ∈ , we have Eσ,val = toricσ,val ×toricσ Eσ

173

FUNDAMENTAL DIAGRAM 

as a topological space. Define the topological space Eσ,val by 

Eσ,val := |toric|σ,val ×|toric|σ Eσ .

(1)

Then we have commutative diagrams with surjective horizontal arrows Eσ,val −−−−→
(σ )gp \Dσ,val       Eσ

−−−−→
(σ )gp \Dσ ,





Eσ,val −−−−→ Dσ,val       Eσ

(2)

−−−−→ Dσ ,

where the upper horizontal maps are defined, respectively, as follows: (V , h, F )  → (A, V
, exp(AC ) exp(z)F ), (V , h, F )  → (A, V
, exp(iAR ) exp(iy)F ), ˇ and (A, V
, z) (resp. (A, V
, y)) corresponds to the element (V , h) where F ∈ D, of toricσ,val (resp. |toric|σ,val ) (5.1.11) as in 5.3.6. 

Definition 5.3.8 We introduce the topology on
(σ )gp \Dσ,val (resp. Dσ,val ) as a  quotient of Eσ,val (resp. Eσ,val ) by 5.3.7 (2).  We introduce the strongest topology on
\D,val (resp. D,val ) for which the map  
(σ )gp \Dσ,val →
\D,val (resp. Dσ,val → D,val ) is continuous for every σ ∈ .

5.4 VALUATIVE NILPOTENT i-ORBITS AND SL(2)-ORBITS  The aim of Section 5.4 is to give the map D,val → DSL(2) in the fundamental diagram (5.0.1), by using the work of Cattani, Kaplan, and Schmid [CKS].

5.4.1 ˇ Let N1 , . . . , Nn ∈ gR be mutually commutative nilpotent elements. Let F ∈ D, and assume that (N1 , . . . , Nn , F ) generates a nilpotent orbit. This means that, for

σ = 1≤j ≤n (R≥0 )Nj , exp(σC )F is a σ -nilpotent orbit (1.3.6). In [CKS], Cattani, Kaplan, and Schmid defined an SL(2)-orbit in n variables associated with the family (N1 , . . . , Nn , F ). Here the order of N1 , . . . , Nn is important. The definition of this SL(2)-orbit will be reviewed in Section 6.1. Here we state three theorems 5.4.2, 5.4.3, and 5.4.4, related to this SL(2)-orbit. Their proofs will be the main subjects of Chapter 6.

Theorem 5.4.2 Let (Nj )1≤j ≤n and F be as in 5.4.1. Put σj := 1≤k≤j (R≥0 )Nk (1 ≤ j ≤ n). Assume that the associated weight filtration W (σj ) is Q-rational for any j . Then the class [ρ, ϕ] in DSL(2) of the SL(2)-orbit associated with

174

CHAPTER 5

(N1 , . . . , Nn , F ) by [CKS] coincides with the limit    lim exp  iyj Nj  F yj /yj +1 →∞ 1≤j ≤n

in DSL(2) , where we put yn+1 = 1 (so

1≤j ≤n yn yn+1

→ ∞ means yn → ∞).

 Theorem 5.4.3 Let p = (A, V , Z) ∈ Dval (5.3.3).

(i) There exists a family (Nj )1≤j ≤n of elements of AR satisfying the following two conditions: (1) If F ∈ Z, (N1 , . . . , Nn , F ) generates a nilpotent orbit. (2) Via (Nj )1≤j ≤n : A∗ = HomQ (A, Q) → R n , V coincides with the set of all elements of A∗ whose images in R n are ≥ 0 with respect to the lexicographic order. (ii) Take (Nj )1≤j ≤n as in (i), let F ∈ Z and define ψ(p) ∈ DSL(2) to be the class of the SL(2)-orbit associated with (N1 , . . . , Nn , F ) by [CKS]. Then ψ(p) is independent of the choices of (Nj )1≤j ≤n and F . By Theorem 5.4.3, we obtain a map 

ψ : Dval → DSL(2) ,

(3)

which we call the CKS map. 

Theorem 5.4.4 Let  be a fan in gQ . Then, ψ : D,val → DSL(2) is continuous.  Thus ψ : D,val → DSL(2) is the unique continuous extension of the identity map of D.

Chapter Six 

The Map ψ : Dval → DSL(2) This chapter is devoted to the proofs of Theorems 5.4.2, 5.4.3, and 5.4.4. In Section 6.1, we recall the theory of SL(2)-orbits in several variables in [CKS]. We prove Theorem 5.4.2 in Section 6.2, Theorem 5.4.3 (i) in Section 6.3, and Theorem 5.4.3 (ii) and Theorem 5.4.4 in Section 6.4.

6.1 REVIEW OF [CKS] AND SOME RELATED RESULTS In this section, we review some results in [CKS]. We review in 6.1.1 the theory of the associated SL(2)-orbit in one variable, in 6.1.2 the theory of R-split mixed Hodge stuructures, and in 6.1.3 the theory of the associated SL(2)-orbit in several variables. In the remaining part of this section, we give some supplements. 6.1.1 The associated SL(2)-orbit in one variable. Let N be a nonzero nilpotent eleˇ and assume that (N, F ) generates a nilpotent orbit, that ment of gR , let F ∈ D, is, ((R≥0 )N, exp(CN )F ) is a nilpotent orbit. Then, an SL(2)-orbit (ρ, ϕ) in one variable is associated to (N, F ) canonically in [Sc],[CKS, 3]. This SL(2)-orbit has the following two properties (1), (2).  (1) The homomorphism ρ∗ : sl(2, C) → gC , associated with ρ, sends 00 10 to N . (2) There are an ∈ gR (n ≥ 1) such that n≥1 an t n absolutely converges for t ∈ C with |t| sufficiently small, that     √ exp(iyN )F = exp ˜ y)−1 ϕ(i) an y −n ϕ(iy) = exp an y −n ρ( n≥1

n≥1

for y  0, and that the component of an of (Ad ◦ρ)-weight ˜ e (e ∈ Z) is zero unless e ≤ n − 1 (see 5.2.2 for the notation ρ). ˜ Here the component of (Ad ◦ρ)-weight ˜ e means the component on which Ad(ρ(t)) ˜ acts as the multiplication by t e for t ∈ Gm,R = R × . From (2), we obtain another presentation (3) of exp(iyN )F .

(3) There are bn ∈ gR (n ≥ 1) such that n≥1 bn t n absolutely converges for t ∈ C with |t| sufficiently small,   √ −n √ exp(iyN )F = ρ( ˜ y)−1 exp ϕ(i) bn y n≥1

176

CHAPTER 6

for y  0, and such that the component of bn of (Ad ◦ρ)-weight ˜ e (e ∈ Z) is zero unless |e| ≤ n − 1. In fact, (3) is deduced from (2) as      √ √ √ −(2n−e) ˜ y)−1 = ρ( an y −n ρ( ˜ y)−1 exp an (e) y exp n≥1

n≥1,e∈Z

√ = ρ( ˜ y)−1 exp



 √ −n , bn y

n≥1

where an (e) denotes the component of an of (Ad ◦ρ)-weight ˜ e and bn = k ak (2k − n). Note that ak (2k − n) = 0 unless |2k − n| < n, for the conditions |2k − n| < n and 2k − n < k are equivalent (since k > 0). The associated SL(2)-orbit (ρ, ϕ) has the following characterization in terms of the Borel-Serre action. Let P be the parabolic subgroup (G◦ )W (N),R = (G◦ )R ∩ GW (N),R of GR . Let wN : Gm,R → P /Pu be the weight map characterized by the property that, for t ∈ (N) Gm,R , wN (t) acts on gr W as the multiplication by t k for any k. Then (ρ, ϕ) is k characterized by the following (4) and (5). (4) The weight filtration associated with (ρ, ϕ) is W (N ). √ (5) ϕ(i) = limy→∞ wN ( y) ◦ exp(iyN )F , where ◦ is the Borel-Serre action with respect to P (5.1.3). This characterization is essentially obtained in [Sc], but not so explicitly. We give a proof of this characterization in 6.1.12 below. We will use the following three facts concerning the associated SL(2)-orbit in one variable. (6) For y ∈ R, (N, exp(iyN )F ) also generates a nilpotent orbit, and the SL(2)orbit associated with this pair is also (ρ, ϕ). (7) For g ∈ GR , (Ad(g)N, gF ) also generates a nilpotent orbit, and the SL(2)orbit associated with this pair is (Int(g) ◦ ρ, gϕ).  (8) Let (ρ, ϕ) be an SL(2)-orbit in one variable of rank 1 and let N = ρ∗ 00 10 . Then, the SL(2)-orbit associated with (N, ϕ(0)) is (ρ, ϕ) itself. 6.1.2 R-split mixed Hodge structure. The point ϕ(0) ∈ Dˇ of the SL(2)-orbit (ρ, ϕ) associated with (N, F ) as above depends, in fact, only on the pair (W (N ), F ). As is explained in [CKS], ϕ(0) is the R-split mixed Hodge structure associated with the mixed Hodge structure (W (N )[−w], F ). Here, for an increasing filtration W and l ∈ Z, we define the increasing filtration W [l] by W [l]k := Wk+l . We review the theory of the associated R-split mixed Hodge structure in [CKS, 2, 3]. Let V be a finite-dimensional R-vector space. Let (W, F ) be a mixed Hodge structure on V , i.e., W is the weight filtration, which is an increasing filtration on V , and F is the Hodge filtration, which is a decreasing filtration on VC := C ⊗R V such that, for each integer k, F induces on gr W k an R-Hodge structure of weight k.

177

The Map ψ



J p,q such that A splitting of (W, F ) is a bigrading VC =   Wk,C = J r,q . J p,q , F p =

(1)

r≥p, q

p+q≤k

We say (W, F ) splits over R if it admits a splitting (J p,q ) satisfying J p,q = J q,p ,

(2)

(called an R-splitting). This is equivalent to the existence of a decomposition V =  k∈Z Sk , such that   F p ∩ Sk,C (∀p). (3) Sl (∀k), F p = Wk = l≤k

k

If (W, F ) splits over R, an R-splitting of (W, F ) and (Sk ) as in (3) are unique and are given by    q J p,q = F p ∩ F ∩ Wp+q,C , Sk = J p,q ∩ V , p+q=k

respectively. In [CKS, 3], with a mixed Hodge structure (W, F ), the authors canonically associate a pair (Fˆ , ε) of a decreasing filtration Fˆ = (W, F )∧ on VC , such that (W, Fˆ ) is an R-split mixed Hodge structure, and a nilpotent endomorphism ε = ε(W, F ) of VC , such that F = exp(ε)Fˆ . (In the notation of [CKS], the above Fˆ and exp(ε) are written as F˜0 and exp(iδ) exp(−ζ ), respectively.) The following hold. (4) Consider the case V = H0,R . Let (σ, Z) be a nilpotent orbit, let W = W (σ )[−w], and let F ∈ Z. Then (W, F ) is a mixed Hodge structure ([CKS]). Let Fˆ = (W, F )∧ , let H0,R = k Sk be the decomposition given by the R-splitting of (W, Fˆ ), and let ν be the action of Gm,R on H0,R given by ν(t)v = t k−w v for v ∈ Sk . Then, for any element N of the interior (0.7.7) of σ , the SL(2)-orbit (ρ, ϕ) in one variable associated to the pair (N, F ) satisfies ϕ(0) = Fˆ ,

ρ˜ = ν

(see 5.2.2).

Ad(ν(t))N = t −2 N

(∀ N ∈ σR ).

From this we have This is because N ∈ σR is written as an R-linear combination of a finite number of elements of the interior of σ . (5) If V = H0,R and W satisfies {x ∈ H0,R | x, Wk 0 = 0} = W2w−k−1 for all k, then ε(W, F ) belongs to gC . (6) Fix W and let FW be the set of all decreasing filtrations F on VC such that (W, F ) is a mixed Hodge structure, and regard FW as a real manifold in the natural way. Then the map F  → (W, F )∧ , ε(W, F ) is a real analytic function on FW . The constructions of δ and ζ are given below (we will review the definition of δ precisely, but will refer to [CKS, 3, (6.60)] for the details of the construction

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CHAPTER 6

of ζ ). Then (W, F )∧ is defined to be exp(−ε)F , where ε ∈ gC is the unique nilpotent element satisfying exp(ε) = exp(iδ) exp(−ζ ). ε can also be written using the Hausdorff formula ε = H (iδ, −ζ ), where H (x, y) = x + y + (1/2)[x, y] + · · · . p,q p,q p,q ≡ There is a unique splitting (I ) = (I (W, F )) of (W, F ) such that I q,p r,s I mod for any p, q ∈ Z, which we call Deligne splitting (see r 0 for any s ∈ S. Then (Ns )s∈S is a good basis for (A, V ) by Lemma 6.3.6. This proves Proposition 6.3.5 (i). Proposition 6.3.5 (ii) follows directly from Lemma 6.3.6. 2 

Definition 6.3.8 Let p = (A, V , Z) ∈ Dval . By an excellent basis for p, we mean a good basis (Ns )s∈S for (A, V ) such that for any F ∈ Z, ((Ns )s∈S , F ) generates a nilpotent orbit. 

Proposition 6.3.9 Let p = (A, V , Z) ∈ Dval . Then an excellent basis for p exists. Furthermore, for any σ ∈ F(A, V ) (5.3.2), there exists an excellent basis (Ns )s∈S for p such that Ns ∈ σ for any s ∈ S. The proof of 6.3.9 will be given in 6.3.14 below after preliminaries 6.3.11–6.3.13. 6.3.10 Deduction of Theorem 5.4.3 (i) from Proposition 6.3.9. Theorem 5.4.3 (i) follows

2 from Proposition 6.3.9 if we take Nj := s∈Sj as Ns (1 ≤ j ≤ n). 6.3.11 and let (as )s , (Sj )j be as in Definition 6.3.3. Fix a good basis (Ns )s∈S for (A, V ), Denote S≤j := k≤j Sk and S≥j := k≥j Sk . Let c ∈ Q>0 , and let α = (αj,s,t ) and β = (βj,s,t ) (1 ≤ j ≤ n, s, t ∈ Sj ) be families of elements of Q>0 such that αj,s,t > aast > βj,s,t for any j, s, t. We define a

rational nilpotent cone σc,α,β in gR as the set of s∈S ys Ns ∈ A with ys ∈ R≥0 satisfying the following two conditions: (1) If 1 ≤ j < n and s ∈ S≤j , t ∈ S≥j +1 , then ys ≥ cyt . (2) If 1 ≤ j ≤ n and s, t ∈ Sj , then αj,s,t yt ≥ ys ≥ βj,s,t yt . Note that σc,α,β,R = AR .

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The Map ψ

Proposition 6.3.12 Let F(A, V ) be as in 5.3.2, and fix a good basis (Ns )s∈S for (A, V ). Then, for a rational nilpotent cone σ such that σR = AR , the following (1) and (2) are equivalent. (1) σ ∈ F(A, V ). (2) σc,α,β ⊂ σ for some c, α, β as in 6.3.11. Proof. We prove that (1) implies

(2). Assume (1). Take a finite subset I of / I . Then σ = {a ∈ (σ ∩ A)∨ such that (σ ∩ A)∨ = ϕ∈I (Q≥0 )ϕ and such that 0 ∈ AR | ϕ(a) ∈ R≥0 (∀ϕ ∈ I )}. Here we denote the R-linear map AR → R induced by ϕ by the same letter ϕ. It is sufficient to prove that there exist c, α, β such that ϕ(σc,α,β ) ⊂ R≥0 for any ϕ ∈ I . Let ϕ ∈ I , and assume ϕ(Aj )  = 0, ϕ(Aj −1 ) = 0. It is sufficient to prove that if c is sufficiently large and if αj,s,t and βj,s,t are sufficiently near to aast , then, for any s∈S yλ,s Ns ∈ σc,α,β (yλ,s ∈ R≥0 ), ϕ( s∈S yλ,s Ns ) =

s∈Sj yλ,s ϕ(Ns ) + t∈S≥j +1 yλ,t ϕ(Nt ) belongs to R≥0 . If yλ,s = 0 for any s ∈ Sj , then yλ,t = 0 for any t ∈ S≥j +1 because yλ,s ≥ cyλ,t . Hence we may assume that yλ,t yλ,s > 0 for some s ∈ Sj . Then, since yλ,s is sufficiently near to 0 for s ∈ Sj the ratio of (y ) near to the ratio and t ∈ S≥j +1 and since λ,s s∈Sj is sufficiently

of (as )s∈Sj for which s∈Sj as ϕ(Ns ) > 0, we have s∈Sj yλ,s ϕ(Ns ) + t∈S≥j +1 yλ,t ϕ(Nt ) > 0. Next we prove that (2) implies (1). It is sufficient to prove that σc,α,β ∈ F(A, V ). Let ϕ ∈ A∗ and assume ϕ(σc,α,β ) ⊂ R≥0 . It is sufficient to prove ϕ ∈ V . Take y →∞ elements yλ,s (s ∈ S, λ ∈ ) of R>0 with a directed set  such that yλ,s λ,t for any s ∈ S≤j , t ∈ S≥j +1 with 1 ≤ j < n and that the ratio of (yλ,s )s∈Sj converges to the ratio of (as )s∈Sj for 1 ≤ j ≤ n. Then, for a sufficiently large λ ∈ ,

hence ϕ( s∈S yλ,s Ns ) ≥ 0. Let 1 ≤ j ≤ n and y s∈S λ,s Ns belongs to σc,α,β and

j −1 j assume ϕ ∈ VQ , ϕ ∈ / VQ . Take s∈Sj as Ns as νj . Then     yλ,s Ns = yλ,s ϕ(Ns ) + yλ,t ϕ(Nt ). 0≤ϕ s∈S

s∈Sj

t∈S≥j +1

y

Since yλ,s → ∞ for any s ∈ Sj and t ∈ S≥j +1 and since the ratio of (yλ,s )s∈Sj λ,t

converges to the ratio of (as )s∈Sj , we obtain νj (ϕ) = s∈S as ϕ(Ns ) ≥ 0. This proves ϕ ∈ V by the description of V in 6.3.1. 2 

Lemma 6.3.13 Let p = (A, V , Z) ∈ Dval and let (Ns )s∈S be a good basis for (A, V ). Let c, α, β and σc,α,β be as in 6.3.11 (defined with respect to (Ns )s∈S ). Then there exists a good basis (Ns
)s∈S for (A, V ) such that Ns
∈ σc,α,β for any s ∈ S. Proof. If Cj > 0 (1 ≤ j ≤ n) and if Cj /Cj +1 (1 ≤ j ≤ n, put Cn+1 = 1) are sufficiently large, then the following conditions (1) and (2) hold. (1) If 1 ≤ j < k ≤ n and s ∈ Sj and t ∈ Sk , then Cj as > (1 + Ck at )c.

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CHAPTER 6

(2) If 1 ≤ j ≤ n and s, t ∈ Sj and s  = t, then αj,s,t >

1 + Cj as , Cj at

Cj a s > βj,s,t . 1 + Cj at

We fix such Cj . For s ∈ S, take a rational number bs which is sufficiently near to as . Then we have the following conditions (3)–(5). (3) If 1 ≤ j < k ≤ n and s ∈ Sj and t ∈ Sk , then Cj bs ≥ (1 + Ck bt )c. (4) If 1 ≤ j ≤ n and s, t ∈ Sj and s  = t, then αj,s,t ≥

1 + Cj bs , Cj bt

(5) If 1 ≤ j ≤ n and s ∈ Sj , then as −

C j bs

Cj b s ≥ βj,s,t . 1 + Cj bt

t∈Sj

1 + Cj

at

t∈Sj

bt

> 0.

For s ∈ S, define Ns
as follows. Let j be the integer such that s ∈ Sj . Let Ns


= Ns +

j  k=1

Ck



bt Nt .

t∈Sk

From (3) and (4), we see easily that Ns
∈ σc,α,β . Furthermore, if we denote the left-hand side of (5) by as
, then   as N s ≡ as
Ns
mod Aj −1,R . s∈Sj

s∈Sj

Since as
> 0 for all s ∈ S, (Ns
)s is a good basis for (A, V ) by Definition 6.3.3 and the description of V in 6.3.1. 2

6.3.14 Proof of Proposition 6.3.9. Take τ ∈ F(A, V ) such that (τ, Z) is a nilpotent i-orbit. Replacing σ by σ ∩ τ ∈ F(A, V ), we may assume that (σ, Z) is a nilpotent i-orbit. Then any good basis (Ns )s∈S for (A, V ) such that Ns ∈ σ for all s ∈ S is an excellent basis for p. To construct such a good basis, take first any good basis (Ns
)s∈S for (A, V ) (6.3.5). Then, by 6.3.12, we may assume σ = σc,α,β for some c, α, β as in 6.3.11, where σc,α,β is defined with respect to this good basis (Ns
)s∈S . Hence we are reduced to Lemma 6.3.13. 2

195

The Map ψ

6.4 PROOFS OF THEOREM 5.4.3 (ii) AND THEOREM 5.4.4 In this section, we prove Theorem 5.4.3 (ii) and Theorem 5.4.4. Proposition 6.4.1 Let (Ns )s∈S be a finite family of mutually commuting nilpotent ˇ and assume that ((Ns )s∈S , F ) generates a nilpotent orbit. elements of gR , let F ∈ D, Let as ∈ R>0 for s ∈ S. Assume that S is the disjoint union of nonempty subsets Sj (1 ≤ j ≤ n). Let S≤j , S≥j be as in 6.3.11. For 1 ≤ j ≤ n, let Dˇ j be the subset of Dˇ consisting of all F
∈ Dˇ such that ((Ns )s∈S≤j , F
) generates a nilpotent orbit. Let  be a directed ordered set, let Fλ ∈ Dˇ (λ ∈ ), yλ,s ∈ R>0 (λ ∈ , s ∈ S), and assume that the following five conditions are satisfied. (1) Fλ converges to F . (2) yλ,s → ∞ for any s ∈ S. (3) If 1 ≤ j < n, s ∈ S≤j and t ∈ S≥j +1 , then (4) If 1 ≤ j ≤ n and s, t ∈ Sj , then

yλ,s yλ,t

yλ,s yλ,t

→ aast . exist Fλ∗ ∈

(5) For 1 ≤ j ≤ n and e ≥ 0, there , t ∈ S≥j +1 ) such that    ∗ iyλ,t Nt Fλ∗ ∈ Dˇ j exp

→ ∞.

∗ Dˇ (λ ∈ ) and yλ,t ∈ R>0 (λ ∈

(for λ: for sufficiently large),

t∈S≥j +1 e yλ,s d(Fλ , Fλ∗ ) → 0 (∀ s ∈ Sj ), e ∗ yλ,s |yλ,t − yλ,t | → 0 (∀ s ∈ Sj , ∀ t ∈ S≥j +1 ).

Here d is a metric on a neighborhood of F in Dˇ that is compatible with the analytic structure (3.1.4).

For each 1 ≤ j ≤ n, take cj ∈ Sj and denote Nj := s∈Sj aacs Ns . Let (ρ, ϕ) j

be the SL(2)-orbit in n variables associated to (N1 , . . . , Nn , F ) (6.1.3), and let r = ϕ(i). Then, for sufficiently large λ, we have   √ √ ρ(( yλ,c1 , . . . , yλ,cn )) exp iyλ,s Ns Fλ = gλ kλ · r s∈S

for some gλ ∈ GR and kλ ∈ Kr such that kλ → 1, √ √ Int(ρ(( yλ,c1 , . . . , yλ,cn )))ν (gλ ) → 1 (ν = 0, ±1). We give preliminaries 6.4.2 and 6.4.3 for the proof of Proposition 6.4.1. Lemma 6.4.2 Let the notation be as in 6.4.1. Let 1 ≤ j ≤ n. Then 3      % yλ,ck ˇ exp ρ˜k iyλ,s Ns Fλ → exp(iNj )ϕ(0j , in−j ) in D, yλ,ck+1 j ≤k≤n s∈S ≥j

where yλ,cn+1 := 1. (Recall that Nj :=

as s∈Sj ac j

Ns .)

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CHAPTER 6

Proof. We prove 6.4.2 by downward induction on j . First we show that we may assume    iyλ,s Ns Fλ ∈ Dˇ j (∀ λ). exp 

4

(1)

s∈S≥j +1



 exp s∈S≥j iy λ,s Ns . Since the Ns are nilpo tent and they mutually commute, exp( s∈S≥j iyλ,s Ns ) is a polynomial in yλ,s with coefficients in gC . From this, we see that, if e ≥ 0 is a sufficiently large integer, e then the following holds: If xλ ∈ gC converges to 0 satisfying yλ,c x → 0, then j λ ∗ ∗ Ad(uλ )(xλ ) → 0. Take such e and take Fλ (λ ∈ ) and yλ,t (λ ∈ , t ∈ S≥j +1 ) ∗ as in 6.4.1 (5). Then (1) is satisfied if we replace yλ,s by yλ,s and Fλ by Fλ∗ . Let ∗ ∗ yλ,s = yλ,s (λ ∈ , s ∈ Sj ) and define uλ in the same way as uλ replacing yλ,s e ∗ . Then Fλ = exp(xλ )Fλ∗ for some xλ ∈ gC such that yλ,c x → 0. We have by yλ,s j λ uλ Fλ = uλ exp(xλ )Fλ∗ = exp(Ad(uλ )(xλ ))(uλ (u∗λ )−1 )u∗λ Fλ∗ , and Ad(uλ )(xλ ) → 0, uλ (u∗λ )−1 → 1. Hence we can assume (1). We assume (1). By Lemma 6.1.10, we have  % 3     yλ,ck exp ρ˜k iyλ,s Ns Fλ yλ,ck+1 k≥j s∈S≥j  % 3      yλ,ck yλ,s exp i Ns ρ˜k iyλ,t Nt Fλ . = exp yλ,ck+1 yλ,cj s∈S k≥j t∈S

Let uλ =

j ≤k≤n ρ˜k

yλ,ck

yλ,ck+1

≥j +1

j

If j < n, the hypothesis of induction is  % 3     yλ,ck exp ρ˜k iyλ,t Nt Fλ yλ,ck+1 t∈S k≥j +1 ≥j +1

→ exp(iNj +1 )ϕ(0j +1 , in−j −1 ).

(2)

On the other hand, in the case j = n, consider the following convergence: Fλ → F.

(3)

Let W (j ) = W (σj ) where σj = (R≥0 )N1 + · · · + (R≥0 )Nj . By (1), in the case j < n (resp. j = n), (2) (resp. (3)) is a convergence of mixed Hodge structures for the weight filtration W (j ) [−w]. The R-split mixed Hodge structure associated to the right-hand side of (2) (resp. (3)) is ϕ(0j , in−j ), and the R-splitting 6.1.2 (3) of this R-split mixed Hodge structure is given by ρ˜j (6.1.3 (5)). By 6.1.11 (ii), it follows that  % 3     yλ,ck exp ρ˜k iyλ,t Nt Fλ → ϕ(0j , in−j ). (4) yλ,ck+1 k≥j t∈S ≥j +1

On the other hand, by 6.4.1 (4), we have  yλ,s Ns → Nj . yλ,cj s∈S

(5)

j

Applying the exponential of i-times of (5) to (4), we obtain Lemma 6.4.2.

2

197

The Map ψ

Lemma 6.4.3 be as in 6.4.1. Fix j such that 1 ≤ j ≤ n, and assume

Let the notation Fλ ∈ Dˇ j for any λ. Put iy N that exp t t∈S≥j +1 λ,t  yλ,s Ns (1 ≤ k ≤ j ), Nλ,k := y s∈Sk λ,ck    "3 #  % yλ,cl  exp  Uλ :=  ρ˜l iyλ,t Nt  Fλ . y λ,c l+1 t∈S l≥j +1 ≥j +1

For each λ, let (ρλ , ϕλ ) be the SL(2)-orbit in j variables associated with (1)

((Nλ,k )1≤k≤j , Uλ ), and let rλ := ϕλ (i). Then rλ converges to r. Note that (1) generates a nilpotent orbit by 6.1.10. Proof. By Lemma 6.4.2, Uλ → exp(iNj +1 )ϕ(0j +1 , in−j −1 ),

(2)

where we understand exp(iNn+1 )ϕ(0n+1 , i−1 ) = F . By the condition 6.4.1 (4), Nλ,k → Nk

(1 ≤ k ≤ j ).

(3)

As in 6.1.3, ((Nk )1≤k≤j , exp(iNj +1 )ϕ(0j +1 , in−j −1 )) generates a nilpotent orbit and the associated SL(2)-orbit (ρ
, ϕ
) in j variables has the reference point ϕ
(i) = r. Hence, by 6.1.6, rλ converges to r as λ → ∞. 2 6.4.4 Proof of Proposition 6.4.1. Let the notation be as in 6.4.1. We prove the following ˇ assertion (Cj ) by a downward induction on j . Let 0 ≤ j ≤ n, and let Dˇ 0 := D.  ˇ (Cj ) Assume that exp t∈S≥j +1 iy λ,t Nt Fλ ∈ Dj for any λ. Then, for a sufficiently large λ, we have # "  √ √ ρ(( yλ,c1 , . . . , yλ,cn )) exp iyλ,s Ns Fλ  = exp 



h∈Nj

g− R

bλ,h

% 1≤k≤j

3

s∈S

yλ,ck yλ,ck+1

−hk

  kλ · r,

(g− R

denotes the (−1)-eigenspace of gR under the Cartan involuwhere bλ,h ∈

h tion associated with Kr ), kλ ∈ Kr , and h∈Nj bλ,h 1≤k≤j xk k (xk ∈ C) absolutely converges when |xk | (1 ≤ k ≤ j ) are sufficiently small, which satisfy the following three conditions. Here ycn+1 := 1. (1) kλ → 1. j (2) |(Ad ◦ρ˜k )1≤k≤j -weight of for the product  bλ,h | ≤ h  ν order in N . 4 y λ,c k (3) For each h ∈ Nj , Ad (bλ,h ) converges for ν = k≥j +1 ρ˜k yλ,c k+1

0, ±1. Moreover, if h = 0 then it converges to 0.

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CHAPTER 6

Note that Proposition 6.4.1 is nothing but (C0 ). 4 y 

√ √ λ,ck and We prove (Cj ). By ρ(( yλ,c1 , . . . , yλ,cn )) = 1≤k≤n ρ˜k yλ,c k+1

Lemma 6.1.10, we have

# "  √ √ ρ(( yλ,c1 , . . . , yλ,cn )) exp iyλ,s Ns Fλ s∈S

  #  yλ,s % yλ,ck  exp  =  ρ˜k i Ns  Uλ yλ,ck+1 yλ,cj +1 k≤j s∈S≤j    "3 #  yλ,c % yλ,ck k  exp  =  ρ˜k i Nλ,k  Uλ , y y λ,c λ,c k+1 j +1 k≤j k≤j 

"3

where Nλ,k and Uλ are as in Lemma 6.4.3. Let (ρλ , ϕλ ) be the SL(2)-orbit in j variables associated with ((Nλ,k )1≤k≤j , Uλ ), and put rλ := ϕλ (i). Then, by 6.2.1, # "  √ √ iyλ,s Ns Fλ ρ(( yλ,c1 , . . . , yλ,cn )) exp s∈S

#  "3 #−1 % % yλ,ck yλ,ck   ρ˜λ,k  fλ k1,λ · rλ , =  ρ˜k y y λ,c λ,c k+1 k+1 k≤j k≤j   3 % yλ,c −hk  k λ  , aλ,h ∈ g−,r fλ := exp  aλ,h R , y λ,c k+1 j k≤j 

"3

where

h∈N

k1,λ ∈ Krλ , k1,λ → 1. −,r

Here gR λ denotes the (−1)-eigenspace of gR under the Cartan involution associated to the maximal compact sugroup Krλ of GR . Claim 1 We can write rλ = gλ k2,λ · r, ρ˜λ,k = Int(gλ ) ◦ ρ˜k , gλ ∈ (G◦ )W (1) ,...,W (j ) ,R , k2,λ ∈ Kr , gλ → 1, k2,λ → 1. We prove this claim. Take a Q-parabolic subgroup P of GR such that (G◦ )W (1) ,...,W (j ) ,R ⊂ P , GW (1) ,...,W (j ) ,R,u ⊂ Pu ,  j Image ρ((Gm,R × 1n−j )) → P /Pu ⊂ (center of P /Pu ). For the existence of P , see [KU2, Proof of 4.12]. Since rλ → r by Lemma 6.4.3, we can write rλ = gλ k2,λ · r, gλ ∈ P , k2,λ ∈ Kr , gλ → 1, k2,λ → 1. We show that gλ ∈ (G◦ )W (1) ,...,W (j ) ,R . In fact, since both (ρ˜λ,k )1≤k≤j and (ρ˜k )1≤k≤j split (W (k) )1≤k≤j , there exists uλ ∈ GW (1) ,...,W (j ) ,R,u such that ρ˜λ,k = Int(uλ ) ◦ ρ˜k

(1 ≤ k ≤ j ).

199

The Map ψ

On the other hand, since the homomorphisms Gm,R → P /Pu induced by ρ˜λ,k and ρ˜k coincide and since ρ˜λ,k and ρ˜k are the Borel-Serre liftings at rλ and r, respectively, of this common induced homomorphism, we have ρ˜λ,k = Int(gλ ) ◦ ρ˜k

(1 ≤ k ≤ j ).

Hence Int(u−1 λ gλ ) ◦ ρ˜k = ρ˜k

(1 ≤ k ≤ j ).

Therefore ◦ u−1 λ gλ ∈ (G )W (1) ,...,W (j ) ,R ,

gλ ∈ (G◦ )W (1) ,...,W (j ) ,R .

This proves Claim 1. We go back to the proof of (Cj ). By Claim 1, we have # "  √ √ iyλ,s Ns Fλ ρ(( yλ,c1 , . . . , yλ,cn )) exp 

%

= Int 

"3 ρ˜k

k≤j

s∈S

yλ,ck yλ,ck+1

#  (gλ ) Int(gλ )−1 (fλ ) Int(gλ )−1 (k1,λ )k2,λ · r.

(4)

−,r

Here Int(gλ )−1 (k1,λ ) ∈ Kr and, concerning aλ,h ∈ gR λ in the definition of fλ , we −,r have Ad(gλ )−1 (aλ,h ) ∈ g− R = gR . Furthermore, if we write    gλ,−h  , gλ,−h ∈ Lie(GW (1) ,...,W (j ) ,R ), (5) gλ = exp  h∈Nj

((Ad ◦ρ˜k )1≤k≤j -weight of gλ,−h ) = −h,

gλ,−h → 0,

we have

  "3 # 3 % yλ,c  % yλ,ck k  (gλ ) = exp  gλ,−h Int  ρ˜k y y λ,c λ,c k+1 k+1 j k≤j k≤j

−hk

 .

(6)

h∈N

By Proposition 6.2.2, |(Ad ◦ρ˜k )1≤k≤j -weight of Ad(gλ )−1 (aλ,h )| < h.

(7)

From (4)–(7), we obtain

# "  √ √ iyλ,s Ns Fλ ρ(( yλ,c1 , . . . , yλ,cn )) exp  = exp 



h∈Nj

bλ,h

% k≤j

3

s∈S

yλ,ck yλ,ck+1

−hk

  kλ · r,

where

kλ := Int(gλ )−1 (k1,λ )k2,λ ∈ Kr , kλ → 1, bλ,h ∈ g− R , bλ,h converges for each h, bλ,h (±h) → 0, |(Ad ◦ρ˜k )1≤k≤j -weight of bλ,h | ≤ h.

(8)

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CHAPTER 6

Here bλ,h (±h) denotes the parts of bλ,h of weight ±h with respect to (Ad ◦ρ˜k )1≤k≤j . In the case j = n, (8) already completes the proof of (Cj ). If 0 ≤ j < n, it remains to prove (3) of (Cj ). This is shown by downward induction on j , as we have mentioned. First we show the following.  ˇ Claim 2 To prove (Cj ) (3), we may assume exp t∈S≥j +1 iy λ,t Nt Fλ ∈ Dj +1 for any λ. We prove Claim 2. By the condition (5) of 6.4.1 for j + 1, there exist e ≥ 0, ∗ ˇ yλ,t Fλ∗ ∈ D, ∈ R (t ∈ S≥j +2 ) satisfying the following conditions (9)–(13). ∗  ∗ ∗ ˇ (9) exp t∈S≥j +1 iy λ,t Nt Fλ ∈ Dj +1 . Here yλ,t := yλ,t (t ∈ Sj +1 ). 2e (10) yλ,c d(Fλ , Fλ∗ ) → 0. j +1 ∗ (11) yλ,t − yλ,t → 0 for any t ∈ S≥j +1 . 4 y  

 λ,ck (12) Let αλ = and define αλ∗ exp ρ ˜ k k≥j +1 t∈S≥j +1 iy λ,t Nt yλ,c k+1

∗ (t ∈ S≥j +1 ). (Note Uλ = αλ Fλ .) Then, analogously by replacing yλ,t by yλ,t e and yλ,c log(αλ (αλ∗ )−1 ) → 0 in gC . j +1

αλ (αλ∗ )−1 → 1 in GC , Furthermore,

−e yλ,c Ad(αλ ) → 0 j +1

in the space of linearendomorphisms of gC .  4 y

λ,ck and define βλ∗ analogously by replacing yλ,ck (13) Let βλ = k≥j +1 ρ˜k yλ,c ∗ by yλ,c (k ≥ j + 1). Then, k

k+1

Ad(βλ )ν − Ad(βλ∗ )ν → 0

−e yλ,c Ad(βλ )ν → 0 j +1

and

for ν = 0, ±1

in the space of linear endomorphisms of gC . ∗ ∗ Define yλ,s := yλ,s for s ∈ S≤j +1 . Define bλ,h (h ∈ Nj ) just as bλ,h by replacing ∗ ∗ yλ,s , Fλ by yλ,s , Fλ , respectively. To prove Claim 2, it is sufficient to prove that ∗ Ad(βλ )ν (bλ,h ) − Ad(βλ∗ )ν (bλ,h )→0

for ν = 0, ±1.

The left-hand side of this is equal to −e e ∗ ∗ yλ,c (bλ,h − bλ,h )) + (Ad(βλ )ν − Ad(βλ∗ )ν )(bλ,h ). Ad(βλ )ν (yλ,c j +1 j +1

Hence, by (13), it is sufficient to prove that e ∗ yλ,c (bλ,h − bλ,h ) → 0. j +1

Since bλ,h is a real analytic function in ((Nλ,k )1≤k≤j , Uλ ) (6.1.6), it is sufficient to prove that e yλ,c d(Uλ , Uλ∗ ) → 0 j +1

(14)

where Uλ∗ = αλ∗ Fλ∗ . By (10), we can write Fλ = exp(xλ )Fλ∗

with

xλ ∈ gC ,

2e yλ,c x → 0. j +1 λ

201

The Map ψ

We have Uλ = αλ Fλ = exp(Ad(αλ )(xλ ))αλ (αλ∗ )−1 Uλ∗ . Using (12) and −e 2e e Ad(αλ ))(yλ,c x ) → 0, Ad(αλ )(xλ ) = (yλ,c yλ,c j +1 λ j +1 j +1

we obtain (14). Thus Claim  2 is proved. ˇ Now we assume exp t∈S≥j +1 iy λ,t Nt Fλ ∈ Dj +1 for any λ. Then, by 6.2.1, we have elements bλ,h
for h
∈ Nj +1 and we have, for h ∈ Nj , 3 −k ∞  yλ,cj +1 bλ,(h,k) . (15) bλ,h = yλ,cj +2 k=0 Using (8), applied to bλ,(h,k) replacing j by j + 1, we have (16) |(Ad ◦ρ˜j +1 )-weight of bλ,(h,k) | ≤ k. If h = 0, the parts of bλ,(h,k) of (Ad ◦ρ˜j +1 )-weight ±k converge to 0. By the hypothesis of induction, 4 y ν 

λ,ck (bλ,(h,k) ) converges for ν = 0, ±1. (17) Ad k≥j +2 ρ˜k yλ,c k+1

By (15)–(17), we have (3) of (Cj ).

2

6.4.5 Proof of Theorem 5.4.3 (ii). Let F be the filter on the set AR for which the sets σ + y (σ ∈ F(A, V ), y ∈ σ ) (5.3.2) is a base. For F ∈ Z, let F
F be the filter on the set D whose basis is given by the sets {exp(iy)F | y ∈ U } where U ranges over sufficiently small elements in F. Then F
F is independent of the choice of F ∈ Z. So we denote F
F by F
. By Proposition 6.4.1, for any (Nj )1≤j ≤n and F ∈ Z as in the hypothesis of 5.4.3 (ii), the class [ρ, ϕ] ∈ DSL(2) of the SL(2)-orbit (ρ, ϕ) associated with (N1 , . . . , Nn , F ) (6.1.3) is the limit of F
in DSL(2) (5.2.16) and hence it is independent of the choice of (N1 , . . . , Nn , F ). 2 We use the notion of “regular space,” which we review here. Definition 6.4.6 [Bn, Ch. 1, §8, no. 4, Definition 2] A topological space is called regular if it is Hausdorff and satisfies the following axiom: Given any closed subset F of X and any point x ∈ / F , there is a neighborhood of x and a neighborhood of F that are disjoint. Lemma 6.4.7 [Bn, Ch. 1, §8, no. 5, Theorem 1] Let X be a topological space, A a dense subset of X, f : A → Y a map from A into a regular space Y . A necessary and sufficient condition for f to extend to a continuous map f : X → Y is that, for each x ∈ X, f (y) tends to a limit in Y when y tends to x while remaining in A. The continuous extension f of f to X is then unique.

202

CHAPTER 6

6.4.8 Our method for the proof of Theorem 5.4.4 is as follows. It is sufficient to prove ψ  that, for σ ∈ , the composite map ψ˜ : Eσ,val → Dσ,val − → DSL(2) is continuous.  ˜ Let p ∈ Eσ,val , and let W be the family of weight filtrations associated with ψ(p) ∈ DSL(2) . We show in 6.4.10 that there is an open neighborhood U of p such that ˜ ) ⊂ DSL(2) (W ). Now since DSL(2) (W ) is a regular space (5.2.15 (iii)), to prove ψ(U the continuity of ψ, it is sufficient to prove the following (1). 

(1) Let (xλ )λ be a directed family in Eσ,val such that the image of xλ in |toric|σ,val  is contained in |torus|σ for any λ, and assume xλ converges to p in Eσ,val . Then ψ(xλ ) converges to ψ(p) in DSL(2) . We will prove this by using Proposition 6.4.1. 

Lemma 6.4.9 Let p = (A, V , Z) ∈ Dval , and let W be the family of weight filtrations associated to ψ(p). Let (Ns )s∈S be an excellent basis for p, and let Sj (1 ≤ j ≤ n) be as in 6.3.3. Let τ ∈ F(A, V ) and assume that τ satisfies the following condition (1).

(1) Let ys ∈ R and assume s∈S ys Ns ∈ τ . Then ys ≥ 0 for all s. Furthermore, if s ∈ Sj and ys  = 0, then yt  = 0 for any k ≤ j and any t ∈ Sk . Then for any p
= (A
, V
, Z
) ∈ Dval such that τ ∩ A
∈ F(A
, V
), we have ψ(p
) ∈ DSL(2) (W ). 

Proof. Let W
be the family of weight filtrations associated to ψ(p
). By Proposition 6.3.9, there exists an excellent basis (Ns
)s∈S
for p
such that Ns
∈ τ for any s ∈ S
. Let Sj
⊂ S
(1 ≤ j ≤ n
) be as in 6.3.3. Recall that W and W
are described as follows (6.1.3 (6)). Let J (resp. J
) be the subset of {1, . . . , n} (resp. {1, . . . , n
}) consisting of elements k such that       W (R≥0 )Ns  = W (R≥0 )Ns , s∈S≤k−1

W

 

(R≥0 )Ns


s∈S≤k

 = W


s∈S≤k−1

 

(R≥0 )Ns


 .


s∈S≤k

Write J = {b(1), . . . , b(m)} with b(1) < · · · < b(m), J
= {b
(1), . . . , b
(m
)} with b
(1) < · · · < b
(m
). Then W = (W (j ) )1≤j ≤m where W (j ) = W

 

 (R≥0 )Ns ,

s∈S≤b(j )




W = (W


(j )

)1≤j ≤m
where W


(j )

=W

  s∈S

≤b (j )

(R≥0 )Ns


 .

203

The Map ψ

Let c : {1, . . . , n} → {1, . . . , m} and d : {1, . . . , m
} → {1, . . . , n} be the  maps defined as follows. Let c(k) = max{j | 1 ≤ j ≤ m, b(j ) ≤ k}. Then W s∈S≤k

(R≥0 )Ns = W (c(k)) for any k. Let 1 ≤ j ≤ m
, write s∈S
Ns
= s∈S ys Ns , ≤b (j ) and let d(j ) = max{k | 1 ≤ k ≤ n, ys  = 0 for some s ∈ Sk }. Then, by the assumption (1) on τ , ys  = 0 for any s ∈ S≤d(j ) . Hence by Cattani and Kaplan (5.2.4), 
(c(d(j ))) . Since c(d(j )) ≤ c(d(k)) if 1 ≤ j ≤ k ≤ W
(j ) = W s∈S≤b
(j ) Ns = W

m , this shows ψ(p ) ∈ DSL(2) (W ). 2 

Lemma 6.4.10 Let p ∈ Eσ,val , and let W be the family of weight filtrations asso˜ ciated to ψ(p) ∈ DSL(2) . Then there exists an open neighborhood U of p such that ˜ ) ⊂ DSL(2) (W ). ψ(U 

Proof. Let p = (A, V , Z) be the image of p in Dσ,val . Take an excellent basis (Ns )s∈S for p such that Ns ∈ σ for any s ∈ S (Proposition 6.3.9). Then, τ = σc,α,β for any c, α, β as in 6.3.11 (σc,α,β is defined here with respect to (Ns )s∈S ) satisfies the condition (1) in 6.4.9. Take τ satisfying (1) in 6.4.9, and let U be the subset of   Eσ,val consisting of all elements whose images (A
, V
, Z
) in Dσ,val satisfy τ ∩ A
∈

F(A , V ). It is sufficient to prove the following (1)–(3). (1) p ∈ U . ˜ ) ⊂ DSL(2) (W ). (2) ψ(U (3) U is open. (1) is clear. (2) follows from Lemma 6.4.9. We prove (3). The inclusion
(τ ) ⊂
(σ ) induces
(σ )∨ ⊂
(τ )∨ . Recall that |toric|σ,val is identified with the set of all pairs (V

, h) where V

a valuative submonoid of
(σ )∨ gp =
(τ )∨ gp contaning mult such that h−1 (R>0 ) = (V

)× (5.1.11).
(σ )∨ and h is a homomorphism V

→ R≥0 It is easy to see that the set U coincides with the inverse image of the open set of |toric|σ,val consisting of all (V

, h) ∈ |toric|σ,val such that V

⊃
(τ )∨ , and hence is open. 2 Lemma 6.4.11 Let x ∈ |toric|σ,val and let (A, V , d) be the triple corresponding to x as in 5.3.6. Take a good basis (Nj )j ∈S for (A, V ), and let (as )s and (Sj )j be as in 6.3.3. Let B be any R-vector subspace of σR such that σR = AR ⊕ B, and let d˜ ∈ B be the unique representative of d ∈ σR /AR in σR contained in B. Then e(iy) (y ∈ σR ) converges to x if and only if the following (1)–(4) are satisfied. Write y = s∈S ys Ns + b ∈ σR (ys ∈ R, b ∈ B). (1) ys → ∞ for any s ∈ S. (2) If 1 ≤ j < k ≤ n and s ∈ Sj , t ∈ Sk , then yyst → ∞. (3) If 1 ≤ j ≤ n and s, t ∈ Sj , yyst converges to aast . ˜ (4) b converges to d. Proof. Write x = (V
, h) where V
is a valuative submonoid of σ ∨gp containing σ ∨ mult and h is a homomorphism V
→ R≥0 such that h−1 (R>0 ) = (V
)× (5.1.11). Assume e(iy) converges to x. We prove that (1)–(4) are satisfied.

204

CHAPTER 6

Let v be an element of V which sends Ns to 1 and Nt (t  = s) to 0. Then exp

v


→Q there exists an element v
∈ V
such that the composition A −→
(σ )gp ⊗ Q − coincides with a positive multiple of v. Then v
∈ / (V
)× . Hence h(v
) = 0. Since e(iy) converges to x, v(y) converges to ∞ (see 5.1.11 (3)). This shows that (1) is satisfied. We prove that (2) is satisfied. Let c > 0, and let v ∈ V be the element that sends Ns to 1, Nt to −c, and Nu to 0 (u ∈ S − {s, t}). As above, we have v(y) = ys − cyt → ∞. Since c is arbitrary, this shows that (2) is satisfied. We prove that (3) is satisfied. Take ε > 0. Let v be the element of V which sends Ns to −at , Nt to as + ε, and Nu to 0 (u ∈ S − {s, t}). As above, we have v(y) = −at ys + (as + ε)yt → ∞.

(5)

Next let v be the element of V which sends Ns to at , Nt to −as + ε, and Nu to 0 (u ∈ S − {s, t}). As above, we have v(y) = at ys + (−as + ε)yt → ∞.

(6)

Since ε is arbitrary, these results (5) and (6) prove that (3) is satisfied. ˜ We prove that (4) is satisfied. For any v
∈ (V
)× , v
(e(ib)) converges to v
(e(i d)) ˜ in R>0 . This shows that b converges to d. On the other hand, it is easy to see that e(iy) converges to x if (1)–(4) are satisfied. 2 6.4.12 Proof of Theorem 5.4.4. We prove Theorem 5.4.4. It is sufficient to prove that   ψ˜ : Eσ,val → DSL(2) is continuous. Let p ∈ Eσ,val and let W be the family of weight ˜ filtrations associated with ψ(p). By the fact that DSL(2) (W ) is a regular space, and by Lemmas 6.4.7 and 6.4.10, it is sufficient to prove (1) in 6.4.8. Take an R-subspace B of σR such that σR = AR ⊕ B. Write xλ = (e(yλ + bλ ), Fλ
) ˇ Let Fλ = exp(bλ )Fλ
. Fix an excellent basis (Ns )s∈S with yλ ∈ AR , bλ ∈ B, Fλ
∈ D. for

(A, V ), and let Sj (1 ≤ j ≤ n) and as (s ∈ S) be as in 6.3.3. Write yλ = s∈S yλ,s Ns . Then from Lemma 6.4.11, we see that the assumption of Proposition 6.4.1 is satisfied. (The condition (5) in 6.4.1 is satisfied by Proposition 3.1.6.) ˜ λ ) = exp(yλ )Fλ converges to ψ(p). ˜ Hence, by Propositions 6.4.1 and 5.2.16, ψ(x 2

Chapter Seven Proof of Theorem A

In this chapter we prove Theorem A. In Section 7.1, we prove Theorem A (i). In  Section 7.2, we study the actions of σC (resp. iσR ) on Eσ , Eσ,val (resp. Eσ , Eσ,val ), gp and then, using these results, we prove Theorem A for
(σ ) \Dσ in Section 7.3. In Section 7.4, we complete the proof of Theorem A for
\D . In Chapter 7, let  be a fan in gQ and let
be a subgroup of GZ which is strongly compatible with .

7.1 PROOF OF THEOREM A (i) In this section, we prove Theorem A (i). We first prove ˜ ) be the closed analytic subset of Dˇ Proposition 7.1.1 Let σ ∈ , let A(σ defined by ˜ ) = {F ∈ Dˇ | N (F p ) ⊂ F p−1 (∀N ∈ σ, ∀p ∈ Z)}, A(σ and let A(σ ) be the subset of Dˇ consisting of all elements F such that (σ, exp(σC )F ) ˜ ). is a nilpotent orbit. Then A(σ ) is an open set of A(σ ˜ ) is sufficiently close Proof. Let F(0) ∈ A(σ ). It is enough to prove that, if F ∈ A(σ to F(0) , then F ∈ A(σ ). ˜ ). In the rest of the proof of Proposition 7.1.1, F always denotes an element of A(σ We prove the following (Ak ) and (Bk ) by downward induction on k. Let W = W (σ ). (Ak ) If F converges to F(0) , then F (Wk ) converges to F(0) (Wk ). W (Bk ) If F converges to F(0) , then F (gr W k ) converges to F(0) (gr k ). If k is sufficiently large, (Ak ) and (Bk ) obviously hold. Furthermore, we can deduce (Ak−1 ) from (Ak ) and (Bk ), since F p (Wk−1 ) = Ker(F p (Wk ) → F p (gr W k )) p p p )). Hence it is sufficient to prove (B and F(0) (Wk−1 ) = Ker(F(0) (Wk ) → F(0) (gr W k ), k W assuming (Al ) for all l ≥ k. Note that, since F(0) (gr k ) is an R-Hodge structure of weight w + k, we have p

q

W W gr W k,C = F(0) (gr k ) ⊕ F (0) (gr k )
p

p

p

if p + q = w + k + 1. p

(1)

W Take a basis e(0) of F(0) (gr W k ) and a basis e(0) of F(0) (Wk ) whose image in gr k con
p p tains e(0) . If F converges to F(0) , F p (Wk ) converges to F(0) (Wk ) by our hypothesis

206

CHAPTER 7 p

(Ak ), and hence some basis ep of F p (Wk ) converges to e(0) . Applying this also replacing p by q = w + k + 1 − p, we have q

p W W gr W k,C = F (gr k ) + F (gr k )

(2)

if F is sufficiently close to F(0) . We show that q

p W W gr W k,C = F (gr k ) ⊕ F (gr k )

(3)

if F is sufficiently close to F(0) . Assume first that k ≥ 0. If (3) is not true, we should have by (2) that q

W F p (gr W k ) ∩ F (gr k )  = 0.

(4) ∼

Take an element N in the interior (0.7.7) of σ . Then N k induces gr W → gr W k − −k . Since q q−k k p p−k k ˜ ), we have N F ⊂ F and N F ⊂ F , and hence (4) shows that F ∈ A(σ F p−k (gr W −k ) ∩ F

q−k

(gr W −k )  = 0.

(5)

On the other hand,  , 0 induces a nondegenerate pairing W gr W k,C × gr −k,C → C, p

W W q p−k and gr W (gr W −k ) ∩ F k,C = F (gr k ) + F (gr k ) annihilates F q−k

(6) q−k

(gr W −k ) in this pair-

(gr W ing since p + q − k > w. Hence F p−k (gr W −k ) ∩ F −k ) = 0, contradicting (5). This proves (3) in the case k ≥ 0. Assume next that k ≤ 0. If (3) is not satisfied, then we should have (4). Since −k ≥ k, (A−k ) is assumed and hence p−k q−k (gr W (gr W gr W −k ) + F −k ) if F is sufficiently close to F(0) . But this shows −k,C = F q W W p that gr −k,C annihilates F (gr k ) ∩ F (gr W k ) in the pairing (6), for p + q − k > w, p contradicting (4). Thus (3) is proved. Now (3) and the fact ep converges to e(0) prove (Bk ). By (Bk ), we have (7) (W [−w], F ) is a mixed Hodge structure if F is sufficiently close to F(0) . Now we prove that F ∈ A(σ ) if F is sufficiently close to F(0) . By [CKS 4.66], it is enough to prove that, if F is sufficiently close to F(0) , then (8) (W [−w], F,  , 0 , N) is a polarized mixed Hodge structure for any element N of the interior (0.7.7)of σ . For k ≥ 0, let Pk,N be the primitive part of gr W k with respect to N , let bk,N be the k nondegenerate bilinear form on gr W k defined by (x, y)  → x, N (y)0 , and let hk,N,F be the nondegenerate Hermitian form on Pk,N,C defined by (x, y)  → bk,N (i p−q x, y) for x in the (p, q)-component of the Hodge structure F (Pk,N,C ). Then (8) is satisfied if and only if the following conditions (9) and (10) are satisfied for any k ≥ 0. (9) bk,N (F p (Pk,N,C ), F q (Pk,N,C )) = 0 if p + q > k + w. (10) hk,N,F is positive definite.

207

PROOF OF THEOREM A

˜ ), N k (F p ) ⊂ F p−k . Since (p − k) + q > w, We prove (9). Since F ∈ A(σ p−k q F , F 0 = 0. This proves (9). We show (11) If (10) is satisfied by one element N of the interior (0.7.7) of σ , it is satisfied for any N in the interior of σ . Let N, N
be elements of the interior of σ and assume that (10) is true. For a ∈ R, 0 ≤ a ≤ 1, consider the elements Na := aN + (1 − a)N
of the interior of σ . Take bases of Pk,Na ,C continuously in a. Then hk,Na ,F are regarded as nondegenerate Hermitian matrices. If s(a) denotes the minimum of all eigenvalues of the matrix hk,Na ,F , s(a) is a continuous function from the interval [0, 1] to R which does not have value 0. Since hk,N,F is positive definite, we have s(1) > 0. Hence s(a) > 0 for any a ∈ [0, 1] by the continuity. Hence s(0) > 0 and this shows that hk,N
,F is also positive definite. For a fixed element N in the interior of σ , since (9) is satisfied, the condition (10) is an open condition on F (Pk,N,C ), satisfied by F(0) (Pk,N,C ). Hence, by (Bk ), it is satisfied if F is sufficiently close to F(0) . 2 7.1.2 Proof of Theorem A (i). We have E˜ σ,val = toricσ,val ×toricσ E˜ σ as a topological space. Since E˜ σ,val → E˜ σ is a proper surjective map, the topology of E˜ σ is the quotient topology of the one of E˜ σ,val . Hence it is sufficient to prove that Eσ,val is open in E˜ σ,val . Assume (xλ )λ is a directed family of elements of E˜ σ,val which converges to x ∈ Eσ,val . We prove (1) xλ ∈ Eσ,val if λ is sufficiently large. ˇ Let x = Write xλ = (qλ , Fλ
), x = (q, F
) (qλ , q ∈ toricσ,val , Fλ
, F
∈ D).  (A, V , Z) ∈ Dσ,val be the image of x and take an excellent basis (Ns )s∈S for x such that Ns ∈ σ (q) for any s (6.3.9). Let Sj (1 ≤ j ≤ n) be as in 6.3.3. For 1 ≤ j ≤ n, let σj = s∈S≤j (R≥0 )Ns . Take an R-subspace B of σR such that σR = AR ⊕ B. We have a unique injective open continuous map S )val × B → |toric|σ,val (R≥0

−2πys which sends ((e )s∈S , b) (ys ∈ R, b ∈ B) to e(( s∈S iys Ns ) + ib) (cf. 3.3.5). Let U be the image of this map. Define the maps ts : U → R≥0 (s ∈ S) and b : U → B by S S (t, b) = ((ts )s∈S , b) : U  (R≥0 )val × B → R≥0 × B.

Let | | : toricσ,val → |toric|σ,val be the canonical projection induced by C → R, z  → |z|. Then, |q| ∈ U and t (|q|) := (ts (|q|))a∈S = 0. Since |qλ | → |q|, we may assume |qλ | ∈ U . By Lemma 6.4.11, for each sufficiently large λ, there

208

CHAPTER 7

exists j with 0 ≤ j ≤ n such that ts (|qλ |) = 0 if s ∈ S≤j and ts (|qλ |)  = 0 if s ∈ S≥j +1 . Dividing (xλ )λ into finite subfamilies, we may assume this j is common for any λ. For s ∈ S≥j +1 , define yλ,s ∈ R by ts (|qλ |) = e−2πyλ,s . Define Fλ := exp(b(|qλ |))Fλ
and F := exp(b(|q|))F
. Note that, since xλ ∈ E˜ σ,val , we have

˜ j ) (notation in 7.1.1). Fλ , exp( s∈S≥j +1 iyλ,s Ns )Fλ ∈ A(σ We prove (1) by downward induction on this j . ˜ n ) converges to F and (σn , F ) generates a nilpotent orbit If j = n, Fλ ∈ A(σ and hence, by Proposition 7.1.1, if λ is sufficiently large then (σn , Fλ ) generates a nilpotent orbit, that is, xλ ∈ Eσ,val . Assume j < n. Let d be a metric on a neighborhood of F in Dˇ which is compatible with the analytic structure (3.1.4). Take any l such that j < l ≤ n and take any e ≥ 0. Then, since E˜ σ is an analytically constructible subset of Eˇ σ (3.5.9, 3.5.10), we can deduce the following fact from Propositions 3.1.5 and 3.1.6: When λ is sufficiently large, there exist xλ∗ = (qλ∗ , Fλ∗
) ∈ E˜ σ,val (λ ∈ ) satisfying the following conditions (2)–(4). Let Fλ∗ = exp(b(|qλ∗ |))Fλ∗
. (2) For s ∈ S, ts (|qλ∗ |) = 0 if s ∈ S≤l , and ts (|qλ∗ |)  = 0 if s ∈ S≥l+1 . ∗ ∗ (3) Let s ∈ S≥l+1 and define yλ,s ∈ R by ts (|qλ |) = e−2πyλ,s . Then e yλ,s d(Fλ , Fλ∗ ) → 0 e yλ,s |yλ,t

∗ − yλ,t |

(∀ s ∈ Sl ),

→0

(∀ s ∈ Sl , ∀ t ∈ S≥l+1 ).

(4) xλ∗ converges to x in E˜ σ,val . By the hypothesis of downward induction on j , xλ∗ belongs to Eσ,val if λ is sufficiently large. Fix N in the interior (0.7.7) of σj . Let k ≥ 0 and let Pk,N be the primitive part of W (σ )

gr k j with respect to N . It is enough to prove that, when λ is sufficiently large, Fλ (Pk,N,C ) is a polarized Hodge structure with respect to the intersection form bk,N : (x, y)  → x, N k (y)0 (as in the proof of 7.1.1). (Though we have to prove this for any N in the interior of σj , it is enough to consider one fixed N by 7.1.1 (11).) By replacing our H0,R with  , 0 by Pk,N with bk,N , we are reduced to the case j = 0. Then the assumptions of Proposition 6.4.1 are satisfied, and hence we have exp( s∈S iyλ,s Ns )Fλ ∈ D if λ is sufficiently large. This shows that (qλ , Fλ ) ∈ Eσ,val if λ is sufficiently large (3.3.7). Thus we have proved Theorem A (i). 2 The following result will be used in §7.2. Corollary 7.1.3 The map 

Eσ,val → Eσ,val ,

(q, F )  → (|q|, F ),

is continuous. Proof. Since E˜ σ is an analytically constructible subset of Eˇ σ and Eσ is an open set of E˜ σ in the strong topology, we can use Proposition 3.1.5 to check the continuity of Eσ → Eσ , (q, F )  → (|q|, F ). The assertion follows from this. 2

209

PROOF OF THEOREM A

7.2 ACTION OF σC ON Eσ 7.2.1 Let σ ∈ . Consider the action of σC on Eσ for which a ∈ σC sends (q, F ) to ˇ (q, F ) ∈ Eσ . This action is con(e(a)q, exp(−a)F ) where q ∈ toricσ , F ∈ D, tinuous. In fact, by a formal argument, we see that this action σC × Eσ → Eσ is continuous in the strong topology of σC × Eσ in σC × Eˇ σ . By 3.1.8 (2), this proves the continuity of the action σC × Eσ → Eσ in the product topology on σC × Eσ . From this continuity, we see that the action of σC on Eσ,val , for which a ∈ σC ˇ (q, F ) ∈ Eσ,val , sends (q, F ) to (e(a)q, exp(−a)F ) where q ∈ toricσ,val , F ∈ D, and e in 3.3.5, is also continuous.  Consider also the induced action of iσR on Eσ (resp. Eσ,val ). The quotient spaces for these actions are identified as σC \Eσ =
(σ )gp \Dσ , iσR \Eσ = Dσ ,

σC \Eσ,val =
(σ )gp \Dσ,val , 



iσR \Eσ,val = Dσ,val .

In this section, we prove the following theorem. Theorem 7.2.2 (i) The action of σC on Eσ (resp. Eσ,val ) is proper.  (ii) The action of iσR on Eσ (resp. Eσ,val ) is proper. (In fact (ii) is a corollary of (i), because Eσ is a closed topological subspace of Eσ and iσR is a closed topological subgroup of σC .) Before proving this theorem, we recall the notion of “proper action" and some related results for our later use. The proof of theorem 7.2.2 will be given in 7.2.13. Definition 7.2.3 [Bn, Ch. 3, §4, no. 1, Definition 1] Let H be a Hausdorff topological group acting continuously on a topological space X. H is said to act properly on X if the map H × X → X × X, (h, x)  → (x, hx), is proper. Recall that, in the convention in this book (0.7.5), the meaning of properness of a continuous map is slightly different from that in [Bn]. However, for a continuous action of a Hausdorff topological group H on a topological space X, the map H × X → X × X, ; (h, x)  → (x, hx) is always separated, and hence this map is proper in our sense if and only if it is proper in the sense of [Bn]. Lemma 7.2.4 (cf. [Bn, Ch. 3, §4, no. 2, Proposition 3]) If a Hausdorff topological group H acts properly on a topological space X, then the quotient space H \X is Hausdorff. Lemma 7.2.5 (cf. [Bn, Ch. 3, §4, no. 4, Corollary]) If a discrete group H acts properly and freely on a Hausdorff space X, then the projection X → H \X is a local homeomorphism.

210

CHAPTER 7

Here “free” means that any element of H − {1} has no fixed point. Lemma 7.2.6 (cf. [Bn, Ch. 3, §2, no. 2, Proposition 5]) Let H be a Hausdorff topological group acting continuously on topological spaces X and X
. Let ψ : X → X
be an equivariant continuous map. (i) If ψ is proper and surjective and if H acts properly on X, then H acts properly on X
. (ii) If H acts properly on X
and if X is Hausdorff, then H acts properly on X. Lemma 7.2.7 Assume that a Hausdorff topological group H acts on a topological space X continuously and freely. Let X
be a dense subset of X. Then, the following two conditions (1) and (2) are equivalent. (1) The action of H on X is proper. (2) Let x, y ∈ X,  be a directed set, (xλ )λ∈ be a family of elements of X
and (hλ )λ∈ be a family of elements of H , such that (xλ )λ (resp. (hλ xλ )λ ) converges to x (resp. y). Then (hλ )λ converges to an element h of H and y = hx. If X is Hausdorff, these equivalent conditions are also equivalent to the following condition (3). (3) Let  be a directed set, (xλ )λ∈ be a family of elements of X
and (hλ )λ∈ be a family of elements of H , such that (xλ )λ and (hλ xλ )λ converge in X. Then (hλ )λ converges in H . Proof. Since the action of H on X is free, the action of H on X is proper if and only if the injection H × X → X × X, (h, x)  → (x, hx), is closed. From this, it is easily seen that (1) implies (2). Assume (2). We prove (1). It is sufficient to prove that, for a closed subset W of H and a closed subset Z of X, the image I of W × Z in X × X is closed. Let (x, y) be an element of the closure of I in X × X. For a neighborhood U of x and a neighborhood V of y, there exist elements hU,V ∈ W and zU,V ∈ Z such that zU,V ∈ U and hU,V zU,V ∈ V . For an element xU,V of X
which is sufficiently near to zU,V , we have xU,V ∈ U and hU,V xU,V ∈ V . Let  be the set of all pairs (U, V ). Since (xλ )λ converges to x and (hλ xλ )λ converges to y, there exists h ∈ H such that y = hx and such that (hλ )λ converges to h. The last property of (hλ )λ shows h ∈ W . Since (zλ )λ converges to x, x belongs to Z. Hence y = hx shows (x, y) ∈ I . Assume finally X is Hausdorff. It is clear that (2) implies (3). Conversely, assume (3) and let x, y ∈ X, , (xλ )λ∈ , (hλ )λ∈ be as in the hypothesis in (2). Then (hλ )λ converges to some element h of H . Since (hλ xλ )λ converges both to y and to hx and since X is Hausdorff, we have y = hx. 2 By Theorem 7.2.2 and Lemma 7.2.4, we have  Corollary 7.2.8 The spaces
(σ )gp \Dσ ,
(σ )gp \Dσ,val , Dσ , Dσ,val are Hausdorff.

This Corollary will be generalized in Sections 7.3 and 7.4 by replacing σ and
(σ )gp by  and
.

211

PROOF OF THEOREM A

We first prove Proposition 7.2.9 (i) The action of σC on Eσ (resp. Eσ,val ) is free.  (ii) The action of iσR on Eσ (resp. Eσ,val ) is free. Here and in the following, free action always means set-theoretic free action as in 7.2.5. Proof. We show here that the action of σC on Eσ is free. The rest is proved in the similar way. If a · (q, F ) = (q, F ) (a ∈ σC , (q, F ) ∈ Eσ ) then, by e(a)q = q, we have a = b + c (b ∈ σ (q)C , c ∈ log
(σ )gp ). On the other hand, taking ε( ) = ε(W (σ (q)), ) of exp(−a)F = F , we have ε(F ) = ε(exp(−a)F ) = −i Im(b) + ε(F ) by 6.1.7 (2). It follows Im(b) = 0 and hence exp(Re(b) + c)F = F . Take an element y of the interior (0.7.7) of σ (q) such that F
:= exp(iy)F ∈ D. Then exp(Re(b) + c)F
= F
, that is, exp(Re(b) + c) belongs to the compact group KF

. Since Re(b) + c is nilpotent, we have Re(b) + c = 0 and hence a = 0. 2 The following proposition is a key for the proof of Theorem 7.2.2. Proposition 7.2.10 Let σ and σ
be rational nilpotent cones. Let α ∈ Eσ,val   ˇ (resp. and α
∈ Eσ
,val . Assume that (e(iyλ ), Fλ ) ∈ Eσ,val,triv (yλ ∈ σR , Fλ ∈ D)  ˇ converges to α (resp. α
) in the strong (e(iy
λ ), Fλ
) ∈ Eσ
,val,triv (yλ
∈ σR
, Fλ
∈ D)) topology, and that 

exp(iyλ )Fλ = exp(iy
λ )Fλ
in D. Then we have  (i) The images of α and α
in Dval coincide.
(ii) yλ − yλ converges in gR . 





ψ

→ DSL(2) is conProof. Since the composite map Eσ,val (resp. Eσ
,val ) → Dval − tinuous by Definition 5.3.8 and Theorem 5.4.4, the image of α (resp. α
) under this composite map is the limit of exp(iyλ )Fλ (resp. exp(iy
λ )Fλ
) in DSL(2) . Since exp(iyλ )Fλ = exp(iy
λ )Fλ
and since DSL(2) is Hausdorff ([KU2, 3.14 (ii)]), these images coincide, say p ∈ DSL(2) . Let m be the rank of p, let (ρ, ϕ) be an SL(2)orbit in m variables representing p and let (W (k) )1≤k≤m be the associated family of weight filtrations. Let (A, V , Z) (resp. (A
, V
, Z
)) be the image of α (resp.  α
) in Dval . Take an excellent basis (Ns )s∈S (resp. (Ns
)s∈S
) for (A, V , Z) (resp. (A
, V
, Z
)) (6.3.8) and let (as )s∈S and (Sj )1≤j ≤n (resp. (as
)s∈S
and (Sj
)1≤j ≤n
) be as in 6.3.3. For each l with 1 ≤ l ≤ m, let f (l) (resp. f
(l)) be the unique integer such that 1 ≤ f (l) ≤ n (resp. 1 ≤ f
(l) ≤ n
), that W (l) = W ( s∈S≤f (l) (R≥0 )Ns )

(resp. = W ( s∈S

(R≥0 )Ns
)) and that W (l)  = W ( s∈S≤f (l)−1 (R≥0 )Ns ) (resp. ≤f (l)

 = W ( s∈S

(R≥0 )Ns
)). Let g(l) (resp. g
(l)) be any element of Sf (l) ≤f (l)−1

(resp. Sf

(l) ).

212

CHAPTER 7

Take an R-subspace B of σR (resp. B
of σR
) such that σR = AR ⊕ B (resp. = A
R ⊕ B
). Write  
yλ,s Ns + bλ , yλ
= yλ,s Ns
+ bλ
yλ =

σR


s∈S


s∈S

with yλ,s ,


yλ,s

∈ R, bλ ∈ B,

bλ



> 0. Then, by ∈ B . We may assume yλ,s , yλ,s



(s ∈ S
), Lemma 6.4.11, yλ,s (s ∈ S), yλ,s

(s ∈ S≤j , t ∈ S≥j +1 ) tend to ∞, as
, at


yλ,s yλ,t

yλ,s yλ,t

(s ∈ S≤j , t ∈ S≥j +1 ) and

(s, t ∈ Sj ) tends to


yλ,s

as ,
at yλ,t


yλ,s
yλ,t

(s, t ∈ Sj
) tends to

and bλ and bλ
converge.

Claim A

y

λ,g (l) yλ,g(l)

converges to an element of R>0 for 1 ≤ l ≤ m.

We prove this claim. The assumption of Proposition 6.4.1 is satisfied if we take (Ns )s∈S , as , Sj , yλ,s as above the same ones in 6.4.1, the above exp(bλ )Fλ as Fλ in 6.4.1, the limit of the above exp(bλ )Fλ as F in 6.4.1. By 6.4.1, we have  2 2 yλ,g(1) yλ,g(m) exp(iyλ )Fλ → r. ,..., ρ˜ yλ,g(2) yλ,g(m+1) Similarly we have 5 5  6
6
6 yλ,g
(1) 6 yλ,g
(m)  exp(iy
λ )Fλ
→ ρ(t) ρ˜ 7
,...,7
˜ ·r yλ,g
(2) yλ,g
(m+1) m
for some t = (t1 , . . . , tm ) ∈ R>0 . Here yλ,g(m+1) = yλ,g
(m+1) = 1. As in [KU2, 4.12], take a continuous map m m ). such that β(ρ(t) ˜ · x) = tβ(x) (x ∈ D, t ∈ R>0 β : D → R>0

Applying β to the above convergences, taking their ratios and using exp(iyλ )Fλ = exp(iy
λ )Fλ
, we have
yλ,g
(l)

→ tl2

yλ,g(l)

(1 ≤ l ≤ m).

Claim A is proved. Next, we prove the following Claims Bl and Cl (1 ≤ l ≤ m + 1) by induction on l.

−1
Claim Bl yλ,g(l) ( s∈S≤f (l)−1 yλ,s Ns − s∈S

yλ,s Ns
) converges. Claim Cl

s∈S≤f (l)−1

QNs =

s∈S

≤f (l)−1

≤f (l)−1

QNs
.



Here yλ,g(m+1) := 1, S≤f (m+1)−1 := S, S≤f
(m+1)−1 := S . Note that Proposition 7.2.10 follows from Claims Bm+1 and Cm+1 . In fact, A = A
follows from Cm+1 , V = V
follows from Bm+1 , and Z = Z
follows from the

213

PROOF OF THEOREM A

facts that the limit of exp(bλ )Fλ (resp. exp(bλ
)Fλ
) is an element of Z (resp. Z
) and that these limits coincide by Bm+1 , Cm+1 and the assumption exp(iyλ )Fλ = exp(iy
λ )Fλ
. We prove these claims. First, Claims B1 and C1 are trivial. Assume l > 1. By the hypothesis Claim Cl−1 of induction, Ns (s ∈ S≤f (l−1)−1 ) and Ns
(s ∈ S
) are commutative. Hence, by the formula exp(x1 + x2 ) = exp(x1 ) exp(x2 ) if x1 x2 = x2 x1 and by the assumption exp(iyλ )Fλ = exp(iy
λ )Fλ
, we have       iyλ,s Ns Fλ = exp iy
λ − iyλ,s Ns Fλ
. exp iyλ − s∈S≤f (l−1)−1

Applying

m k=l

ρ˜k

4

yλ,g(k) yλ,g(k+1)

s∈S≤f (l−1)−1



to this and using 6.1.10, we obtain

 yλ,s Ns yλ,g(l) f (l−1)≤j 1, and points (0, 0)1,z with 0 < z ≤ ∞.

Proposition 12.4.6 We have commutative diagrams βval

2 (R≥0 )val (P ) −−−−→ DSL(2),val     δ  3 R≥0

βP

−−−−→

DBS ,

βval

2 )val (Q) −−−−→ DSL(2),val (R≥0     δ
  3 R≥0

βQ

−−−−→

DBS .

Here the map δ sends (r1 , r2 ) with r2  = 0 to   r1 2 2 , r2 , r2 , r2 (0, 0)s and (0, 0)s,z with s < 1 to (0, 0, 0), and (0, 0)1,z to (z, 0, 0). The map δ
sends (r1 , r2 ) with r1  = 0 to   r2 2 2 ,r ,r , r1 1 2 (0, 0)s and (0, 0)s,z with s > 1 to (0, 0, 0), and (0, 0)1,z to (z−1 , 0, 0). 2

Proof. Easy.

12.4.7 

Contrary to the case of §12.1, there is no continuous map Dσ,val → DBS and there is no continuous map DSL(2) → DBS which extend the identity map of D. In fact, by 12.4.2, 12.4.3, when t ∈ R>0 converges to 0, β(t 2 , t) ∈ D converges to 3 ) · r) in DBS and β(t, t 2 ) ∈ D converges to βQ (0, 0, 0) = βP (0, 0, 0) = (P , d(R>0 3 3 ) · r) in DBS . On the other hand, both β(t 2 , t) = (Q, d(R>0 ) · r)  = (P , d(R>0 −6 −2 exp(it N1 + it N2 )F(0) and β(t, t 2 ) = exp(it −6 N1 + it −4 N2 )F(0) converge to  αval ((0, 0)0 ) in Dσ,val and also both converge to β(0, 0) in DSL(2) .

299

EXAMPLES AND PROBLEMS

12.5 RELATIONSHIP WITH [U2] We describe the relationship of this book with the work [U2], which is the prototype of the present work. 12.5.1 Let 
be the subset of  consisting of {0} and all elements of the form (R≥0 )N where N is any element in gQ satisfying the following condition (1). 9 1 if w is odd, 2 N = 0, dim(Image(N )) = (1) 2 if w is even. In [U2], the space D
, extended period maps into
\D
(
= GZ is considered there), and their differentials are constructed. The notation and the topology of
\D
in [U2] are different from those of this book. We describe the relationship with the formulation in this book, together with additional comments. Note that N, as in (1) with w = 2, arises, for example, as the logarithm of the local monodromy of degenerations of surfaces with a simple elliptic singularity. The motivation of [U2] was the hope to start to extend Griffiths’ theory of period maps and their differentials in such situations. 12.5.2 Let I be the set of rational increasing filtrations W on H0,R satisfying the following (1) and (2). W1 = H0,R , W−2 = 0, W0 , W−1 0 = 0. 9 1 if w is odd, W dim(gr 1 ) = 2 if w is even.

(1) (2)

We will consider the open set ∪W ∈I DSL(2) (W ) of DSL(2) . Proposition 12.5.3 (i) The continuous map ψ

D
= D
,val − → ∪W ∈I DSL(2) (W ) is bijective. (ii) The canonical map $ $ DSL(2) (W ) = DSL(2),val (W ) → DBS W ∈I

W ∈I

is injective, and the topology of induced from the topology of DBS .



W ∈I

DSL(2) (W ) coincides with the topology

Proof. (i) is proved in [U2, (3.12) (ii)]. We reprove (i) here using terminologies in the this book. Let W ∈ I , and define a rational R-subspace LW of gR by LW = {X ∈ gR | X(W0 ) = 0}.

(1)

300

CHAPTER 12

Then if w is odd, LW is identified with the space of all R-linear maps from W the one-dimensional R-space gr W 1 to the one-dimensional R-space gr −1 . In the case w is even, the map from LW to the set of all antisymmetric bilinear forms W gr W 1 × gr 1 → R which sends X ∈ LW to the bilinear form (x, y)  → X(x), y0 , is bijective. Hence, in any case, we have dimR LW = 1. We define a map of the converse direction $  DSL(2) (W ) → D
f :

(2)

(3)

W ∈I

as follows. Let W ∈ I and p ∈ DSL(2) (W) − D, let (ρ, ϕ) be an SL(2)-orbit in one variable representing p, and let N = ρ∗ 00 10 . Let σ = (R≥0 )N . Then, since N ∈ LW , we have by (2) that σ is rational. Define f (p) := (σ, ϕ(iR)) ∈ D
. Then f (p) is independent of the choice of the representative (ρ, ϕ) of p. We show that ψ ◦ f and f ◦ ψ are the identity maps. This is clear for ψ ◦ f by 6.1.1 (8). We consider  f ◦ ψ. Let ((R≥0 )N, Z) ∈ D
− D, and let W = W (σ ) ∈ I for σ := (R≥0 )N . Let F ∈ Z, let (ρ, ϕ) be the SL(2)-orbit in one variable associated to (N, F ) and let ε = ε(W [−w], F ) ∈ gC . Since ε(Wk,C ) ⊂ Wk−2,C for any k ∈ Z, we have ε ∈ LW,C . By (2), we have ε = (a + ib)N for some a, b ∈ R, and F = exp((a + ib)N )ϕ(0). Hence (σ, Z) = (σ, exp(aN ) exp(iRN )ϕ(0)) and hence [ρ, ϕ] = ψ(σ, Z) = exp(aN )ψ(σ, exp(iRN )ϕ(0)) = exp(aN )[ρ, ϕ]. This shows that exp(aN )ϕ(0) = ϕ(0). But aN ∈ L−1,−1 (W [−w], ϕ(0)) and hence a = 0 by Lemma 6.1.8 (iii). This shows that (σ, Z) = (σ, exp(iRN )ϕ(0)) = (σ, ϕ(iR)) = f ([ρ, ϕ]) = f (ψ(σ, Z)). The proof of (ii) is easy. For W ∈ I , DSL(2),val (W ) is the intersection of DSL(2),val with the open set DBS,val (P ) with P = (G◦ )W,R . By the definition of the topology of DSL(2),val , this shows that W ∈I DSL(2),val (W ) is a topological subspace of DBS,val . Furthermore, for W ∈ I and P = (G◦ )W,R , the map DBS,val (P ) → DBS (P ) is a homeomorphism since dim SP = 1. This proves (ii). 2 12.5.4

  The continuous bijection ψ : D
→ W ∈I DSL(2) (W ) (12.5.3 (i)) is not necessarily a homeomorphism in general. In [U2], the topology of
\D
for
= GZ is not  the one defined in this book, but is defined as the quotient of the topology of W ∈I DSL(2) (W ). (In [U2], the topology of ∪W ∈I DSL(2) (W ) is defined in terms of Siegel sets, but this definition is equivalent to the one as a subset of DBS in the present case.) 12.5.5 The relationship of notation. D
in this book is the union of D and B(W (N ) [−w], p, N ) of [U2], where N ranges over all elements of gQ satisfying 12.5.1 (1), and p ranges over all possible choices of Hodge numbers on gr W (N) .

301

EXAMPLES AND PROBLEMS

For N as above and for
= GZ ,
(σ )gp \Dσ for σ = (R≥0 )N of this book is ˜ DW,N with W = W (N)[−w] in [U2].
\D
of this book is denoted as D/
in [U2]. Proposition 12.5.6 Let p ∈ D
− D, let (ρ, ϕ) be an SL(2)-orbit in one variable whose class coincides with ψ(p) ∈ DSL(2) , let r = ϕ(i) and let W be the weight filtration associated to (ρ, ϕ). Let
be a subgroup of GZ of finite index. Then the following conditions (1)–(4) are equivalent. (1) (2) (3) (4)

The image of p in
\D
has a compact neighborhood. ψ(p) ∈ DSL(2) has a compact neighborhood in DSL(2) . GW,R · r is a neighborhood of r in D. dim Kr = dim Kr
.

Proof. Write p = ((R≥0 )N, exp(iRN )F ) where N is an element of gQ that satisfies 12.5.1 (1). Let (ρ, ϕ) be the SL(2)-orbit in one variable associated with (N, F ), let r = ϕ(i), and let W = W (N). By Theorem 10.1.6, (2) is equivalent to (3). Assume (3). We prove (1). By the proof of Proposition 12.5.3 (i), we have r ∈ Z. By Theorem 7.4.12, it is sufficient to prove that, if F
∈ D is sufficiently near to r, then (N, F
) satisfies the small Griffiths transversality. By (3), F
= g · r for some g ∈ GW,R . Since Ad(g)(N ) ∈ LW = RN with LW in 12.5.3 (1), we have Ad(g)(N ) = aN for some a ∈ R × . Hence N F
p = NgF p = a −1 gN F p ⊂ a −1 gF p−1 = F
p−1 . Hence we have (1). Assume (1). We prove (3). If F
∈ D is near to r, then, by Theorem 7.4.12, (N, F
) generates a nilpotent orbit. If F
converges to r, the class of the SL(2)-orbit (ρ
, ϕ
) associated to (N, F
) converges to ψ(p) in DSL(2) . By Theorem 10.2.2, ϕ
(i) = g · r for some g ∈ GW,R . Furthermore, since ε(W [−w], F
) ∈ iRN by the proof
of 12.5.3 (i) and since it converges√to ε(W [−w], r)√= iN , F
= exp(iaN (0) √ )ϕ −1 −1 −1

˜ a) g · r with ρ( ˜ a) g ∈ ˜ a) ϕ (i) = ρ( for some a ∈ R>0 . Hence F = ρ( GW,R . Finally by [U2, (3.12) (iii)], (3) and (4) are equivalent. 2

12.6 COMPLETE FANS As is seen from the examples in Sections 12.2 and 12.3, it is often impossible to construct a compact
\D (it is often even impossible to construct a locally compact
\D which has a point outside
\D). We think that the right generalization of the compactness of toroidal compactifications to the general case is the “completeness” in the following sense. Definition 12.6.1 A fan  in gQ is complete if D,val = Dval . 



As is easily seen,  is complete if and only if D,val = Dval .

302

CHAPTER 12

12.6.2 In the examples in Sections 12.2 and 12.3, the fan  is complete; whereas, in Section 12.1 (the case D = hg ),  is complete if g = 1, but  is not complete if g ≥ 2. Conjecture 12.6.3 There exists a complete fan which is strongly compatible with GZ (and hence with any subgroup of GZ of finite index).∗ Proposition 12.6.4 Let
be a neat subgroup of GZ , let  be a fan which is strongly compatible with
, and assume that
\D is compact. Then  is complete. 







Proof. Let p ∈ Dval . We prove p ∈ D,val . Since
\D →
\D and
\D,val → 
\D are proper,
\D,val is compact.   Take a rational nilpotent cone τ such that p ∈ Dτ,val . Since D is dense in Dτ,val ,  there exists a directed family (xλ )λ of points of D which converges to p in Dτ,val .  Since
\D,val is compact, replacing (xλ )λ by a cofinal subfamily if necessary, we may assume that the image of (xλ )λ in
\D converges to an element (α mod
) in  
\D,val for some α ∈ D,val . Since
\DSL(2) is Hausdorff, the images of p and α in
\DSL(2) coincide. Replacing α by a suitable translation of α by
, we may assume that the images of p and α in DSL(2) coincide.   Since D,val →
\D,val is a local homeomorphism, there exist γλ ∈
such that  (γλ xλ )λ converges to α in D,val . Then, in DSL(2) , both (xλ )λ and (γλ xλ )λ converge to ψ(p). Since DSL(2) →
\DSL(2) is a local homeomorphism, we see that γλ = 1 if λ is sufficiently large.   Thus (xλ )λ converges to p in Dτ,val and also converges to α in D,val . Take σ ∈    such that α ∈ Dσ,val . Then (xλ )λ converges to α in Dσ,val by 7.3.2 (ii). Hence p = α   in Dval by 7.3.10 (1). Thus p ∈ D,val , as desired. 2 12.6.5 In the classical situation (0.4.14), [AMRT] constructed a fan  in gQ which is strongly compatible with GZ (and hence strongly compatible with any subgroup
of GZ of finite index) and for which GZ \D (and hence
\D for such
) is compact. For a subgroup
of GZ of finite index,
\D is the toroidal compactification of
\D by Mumford and others associated to . By 12.6.4, this  is complete. Theorem 12.6.6 Let X be a connected, logarithmically smooth, fs logarithmic analytic space and let U = Xtriv be the open set of X consisting of all points at which the logarithmic structure is trivial. Let H be a variation of polarized Hodge structure on U with unipotent local mondromy along X − U . Let u ∈ U and let (H0 ,  , 0 ) = (HZ,u ,  , u ). Let
be a neat subgroup of GZ of finite index, let  be ∗ See

the end of section 12.7.

EXAMPLES AND PROBLEMS

303

a fan in gQ , and assume that
contains the global monodromy Image(π1 (U, u) → GZ ), that  is strongly compatible with
, and that  is complete. Then X() in 4.3.5 is a logarithmic modification of X and the period map U →
\D associated to H extends to a morphism X() →
\D of logarithmic manifolds. By the nilpotent orbit theorem of Schmid interpreted as Theorem 2.5.14, H extends to a PLH on X. Hence Theorem 12.6.6 is reduced to Proposition 12.6.7 Let
be a neat subgroup of GZ and let  be a fan in gQ which is strongly compatible with
. Assume  is complete. Let X be an object of B(log) and let H be a polarized logarithmic Hodge structure of weight w and of Hodge type (hp,q ) endowed with a
-level structure. Then: (i) The object X() of B(log) over X (4.3.5) is a logarithmic modification of X (3.6.12). (ii) The inverse image of H on X() is of type (w, (hp,q ), H0 ,  , 0 ,
, ) and hence we have the associated period morphism X() →
\D . Proof. To prove (i), consider the maps Xval →
\Dval =
\D,val →
\D , where the first arrow is as in 8.4.1. Let (x, V , h) ∈ Xval . Let y be a point of Xlog lying over x, and take a representative ∼ → (H0 ,  , 0 ) of the germ µy of the
-level structure µ of H . µ˜ y : (HZ,y ,  , y ) − The image of (x, V , h) ∈ Xval in
\Dval has the form ((A, V
, Z) mod
) where (A, V
, Z) ∈ Dval is defined as in 8.4.1 with respect to the above choice of µ˜ y . The image of (A, V
, Z) under Dval = D,val → D is the pair (σ, Z
), where σ is the smallest element of  such that {h ∈ HomQ (A, Q) | h(σ ∩ A) ⊂ Q≥0 } ⊂ V
(5.3.2 (3)), and Z
= exp(σC )Z. Via µ˜ y , the local monodromy action of π1 (x log ) on HZ,y induces a homomorphism π1 (x log ) → GZ and its logarithm sx : π1 (x log ) → gQ . Let σ1 = π1+ (x log ) ∩ sx−1 (σ ) gp × and, as in 3.6.14, let P (σ1 ) = {a ∈ (MX /OX )x | hγ (a) ≥ 0 (∀ γ ∈ σ1 )}, where hγ gp × is the homomorphism (MX /OX )x → Z induced by γ . (Here we use the duality gp × )x .) Then it is easy to prove (x, σ1 ) ∈ Q
1 (X) (3.6.14) of π1 (x log ) and (MX /OX gp × and P (σ1 ) ⊂ V . As in 4.3.5, µ˜ y defines a lifting to (MX /OX )x ⊗ gQ of the gp × germ of N ∈
(X,
\((MX /OX ) ⊗ gQ )) at x, and this lifting coincides with sx gp × )x ⊗ gQ = Hom(π1 (x log ), gQ ). Hence the inclusion via the identification (MX /OX P (σ1 ) ⊂ V shows that the condition (6) of 3.6.28 is satisfied. By Theorem 3.6.28, X() is a logarithmic modification of X and (i) is proved. Furthermore, the image of (x, V , h) under Xval = X()val → X() coincides with (x, σ1 , h1 ) ∈ X() ⊂ Q
(X), where h1 is the restriction of h. To prove (ii), let x1 = (x, σ1 , h1 ) be any element of X() ⊂ Q
(X). Take a point (x, V , h) of Xval lying over x1 . Fix µ˜ y as above, and define (A, V
, Z) ∈ Dval and (σ, Z
) ∈ D from (x, V , h) as above with respect to µ˜ y . Also let (σ1
, Z1
) ∈ Dˇ orb be the representative of the image of x1 under the period map X() →
\Dˇ orb associated to the pullback of H to X(), defined with respect to µ˜ y . Then σ1
is

304

CHAPTER 12

generated by sx (σ1 ) as a cone. It is sufficient to prove the following (1) and (2). (1) σ1
⊂ σ . Furthermore, σ is the smallest element of  which contains σ1
. (2) exp(σC )Z1
is a σ -nilpotent orbit. We prove (1). Since σ1 = π1+ (x log ) ∩ sx−1 (σ ), we have σ1
⊂ σ . If τ ∈  contains σ1
, then σ1
⊂ σ ∩ τ and hence (σ ∩ τ ∩ A)∨ ⊂ (σ1
∩ A)∨ ⊂ V
. Since σ is the smallest element of  satisfying (σ ∩ A)∨ ⊂ V
, we have σ ∩ τ = σ , that is, σ ⊂ τ . We prove (2). We have Z
= exp(σC )Z,


)Z. Z1
= exp(σ1,C

Hence exp(σC )Z1
coincides with the σ -nilpotent orbit Z
.

2

Proposition 12.6.8 Assume we are in the classical situation 0.4.14. Let  be a fan in gQ , let
be a subgroup of GZ of finite index, and assume that
is strongly compatible with . Then,
\D is compact if and only if  is complete. Proof. The “only if” part is shown in 12.6.4. We prove the “if” part. Replacing
by a neat subgroup of
of finite index, we may assume that
is neat. By the existence of a toroidal compactification proved in [AMRT], there is a fan 
which is strongly compatible with
such that
\D
is compact. Write
\D
by X. By 12.6.6, the universal PLH on X defines the period map X() →
\D . The image of this map is dense since it contains
\D. Furthermore, X() is compact since X() is a logarithmic modification of X and X is compact. Hence X() →
\D is surjective, and hence
\D is compact. 2 12.7 PROBLEMS 12.7.1 Construct in general a complete fan (12.6.1) that is strongly compatible with GZ (Conjecture 12.6.3).∗ 12.7.2 Relate our theory to the theory of Hodge modules of Morihiko Saito. 12.7.3 Study p-adic analogues of our theory. 12.7.4 Give structures of ringed spaces on DBS and on DSL(2) and extend the standard Hodge filtration over these spaces. What kind of functors would DBS and DSL(2) with ∗ See

the end of section 12.7.

305

EXAMPLES AND PROBLEMS 

these structures represent? What kind of morphism of functors would Dval → DSL(2) represent? 12.7.5 Let f : Y → X be a logarithmically smooth morphism of logarithmically smooth, fs logarithmic analytic spaces (2.1.11), whose underlying morphism of analytic spaces is projective, and assume that f −1 (Xtriv ) = Ytriv and that the cokernel of gp gp × (MX /OX )f (y) → (MY /OY× )y is torsion free for any y ∈ Y . Assume that there exists the corresponding period map ϕ : X →
\D . Then, ϕ gives a global invariant of degenerations of the fibers of f . We have a decomposition
\D = Z , according to the
-equivalence class of the cone σ of a (σ, Z) ∈ D . j j For each j , investigate the common property among the fibers f −1 (x) for x ∈ ϕ −1 (Zj ). For each z ∈ Zj , describe geometric relation among the fibers f −1 (x) for x ∈ ϕ −1 (z). 12.7.6 As we have seen in this book, our moduli space
\D is not necessarily an analytic space and not necessarily locally compact. However, can we expect that, if X is an algebraic variety over C, the closure ϕ(X) in
\D of the image of a period map ϕ : X →
\D is always an algebraic variety (and hence is locally compact)? Is ϕ(X) compact when  is complete? Assume, furthermore, that X is a moduli space of algebraic varieties, and ϕ is the corresponding period map. Assume  is complete and ϕ is an immersion. Then, can we expect that ϕ(X) is a natural compactification of the the moduli space X? 12.7.7 Generalize the present results to the case when D is the classifying space of mixed Hodge structures with polarized graded quotients. Added in the Proof Recently, Kenta Watanabe proved that Conjecture 12.6.3 was false. The authors expect that the following modified version of that conjecture is correct. The idea of the following new definition is due to Chikara Nakayama. Let N be the set of all rational nilpotent cones σ in gR such that (σ, Z) is a ˇ nilpotent orbit for some subset Z of D. For a fan  in gQ , we say  is complete in the new sense (resp. in the weak sense) if $ $ $ $ σ (resp. σ ⊂ σ ). σ = σ ∈N

σ ∈

σ ∈N

σ ∈

Then,  is complete in the weak sense if  is either complete (in the sense of 12.6.1) or complete in the new sense.

306

CHAPTER 12

Conjecture (modified version) There is a fan  in gQ which is complete in the new sense and strongly compatible with GZ . Problem 12.7.1 should be rewritten as follows.  Problem (1) For each given 0 = w, (hp,q )p,q , H0 ,  , 0 , determine whether there exists or not a complete fan in gQ which is strongly compatible with GZ . (2) Construct a fan in gQ which is complete in the new sense and strongly compatible with GZ .

Appendix

A1 POSITIVE DIRECTION OF LOCAL MONODROMY A1.1 The points at infinity of the classifying space of polarized logarithmic Hodge structures appear in various directions of degeneration of polarized Hodge structures. To consider the direction of the degeneration, it is important to define the monodromy cone in the group of local monodromy consisting of local monodromy of positive direction. For this, we have to fix which direction in the local monodromy group is positive. In particular, we have to fix one of the two isomorphisms π1 (∗ )  Z and call the element of π1 (∗ ) corresponds to 1 the positive generator of π1 (∗ ), and define the monodromy cone in π1 (∗ ) as the submonoid generated by this element, that is, as the image of N ⊂ Z under this isomorphism. This is in fact a rather delicate point. The authors think that the following is the best definition which is compatible with the works of many people (Schmid [Sc], etc.). In our definition: (i) When we regard the upper half plane h as the universal covering of ∗ via the projection h → ∗ , τ  → e2π iτ , the positive generator of π1 (∗ ) is the automorphism τ  → τ + 1 of h. (ii) The positive generator of π1 (∗ ) is the class of a route in ∗ in the counterclockwise directed circle [0, 1] → , t  → re2π it for any fixed r (0 < r < 1). (iii) Let q be the coordinate function of . Then the logarithms of q form a local system on ∗ and hence the group π1 (∗ ) acts on stalks of this sheaf. If γ is the positive generator of π1 (∗ ), γ acts on the germs l of logarithms of q by l  → l − 2πi, not l  → l + 2π i. The reader may feel that (iii) is strange. In what follows, we explain that (i)–(iii) are compatible. A1.2 Let X be a topological space and fix a ∈ X. The fundamental group π1 (X, a) of X with base point a is the group of all equivalence classes of continuous maps γ : [0, 1] → X such that γ (0) = γ (1) = a. (We omit the definition of the equivalence (see [Sp]).) For such γ , δ : [0, 1] → X, the product class(γ )class(δ) is the class of the route γ δ : [0, 1] → X which is “go through γ first, and then go through δ”. A1.3 The universal covering X˜ of X for base point a is defined to be the set of all equivalence classes of continuous maps f : [0, 1] → X such that f (0) = a. (We

308

APPENDIX

omit the definition of the equivalence (see [Sp]).) We have the canonical projection p : X˜ → X, f  → f (1). The fundamental group π1 (X, a) is identified with the group of all automorphisms of X˜ over X. In this identification, the class of γ : [0, 1] → X (γ (0) = γ (1) = a) in π1 (X, a) sends the class of f : [0, 1] → X (f (0) = a) to the class of the route γf : [0, 1] → X which is “go through γ first and then go through f ”. A1.4 Example 1. Regard the upper half plane h as the universal covering of ∗ via the projection τ  → e2π iτ . Fix any element a ∈ ∗ . Then the class in π1 (∗ , a) of [0, 1] → , t  → ae2π it , is identified (by A1.3) with the automorphism h → h, τ  → τ + 1. Example 2. The class of [0, 1] → R/Z, t  → (t mod Z), in π1 (R/Z, 0) is identified with the automorphism x  → x + 1 of the universal covering R of R/Z. A1.5 Let X and a be as in A1.2, and let L be a locally constant sheaf on X. Then π1 (X, a) acts on the stalk La of L at a. For the class of γ : [0, 1] → X (γ (0) = γ (1) = a) in π1 (X, a), the action of γ is the composition, from the left to the right, of all isomorphisms or their inverses in ∼

∼

∼

∼

−
([0, 1], γ −1 (L)) − → γ −1 (L)0 ← − La . → γ −1 (L)1 ← La − This action coincides with the composition, from the left to the right, of all isomorphisms or their inverses in ∼

∼

∼

∼

˜ p −1 (L)) − La − → p−1 (L)γ ← −
(X, → p−1 (L)a˜ ← − La . ˜ and a˜ denotes the point of X˜ which is the Here we regard γ as an element of X, class of the constant map [0, 1] → X with value a. Thus we have a group homomorphism: ˜ → Aut(La ), π1 (X, a) = Aut X (X)

γ  → (rewinding along γ ).

˜ on La as the composition, Be careful that if we define the action of γ ∈ Aut X (X) from the left to the right, of all isomorphisms or their inverses in ∼

∼

∼

∼

˜ p −1 (L)) − → p−1 (L)a˜ ← −
(X, → p−1 (L)γ ← − La , La − ˜ could not be preserved (the order of the product the group structure of Aut X (X) should become the converse). A1.6 Let x be an fs logarithmic point. Let a = (x, h0 ) ∈ x log .

309

APPENDIX

We have an isomorphism π1 (x log , a)  Hom(Mxgp /Ox× , Z)

(1)

defined as follows. We have an exact sequence 0 → Hom(Mxgp /Ox× , Z) → Hom(Mxgp /Ox× , R) → Hom(Mxgp /Ox× , S1 ) → 0 where the third arrow is induced by R → S1 , t  → e2π it , and we have a homeomor∼ gp → Hom(Mx /Ox× , S1 ), (x, h)  → hh−1 phism x log − 0 . Via this exact sequence and gp the last isomorphism, we identify Hom(Mx /Ox× , R) with the universal covering gp of x log , and Hom(Mx /Ox× , Z) with the fundamental group of x log acting on the gp universal covering Hom(Mx /Ox× , R) by translations. gp In the isomorphism (1), for f ∈ Hom(Mx /Ox× , Z), the corresponding element log of π1 (x , a) coincides with the class of [0, 1] → x log , t  → (x, ht ) where ht (q) = e2π itf (q) h0 (q) for q ∈ MX,x . Example. Consider the logarithmic point x = 0 ∈  whose logarithmic structure is the inverse image of M . Let a = (x, h0 ) ∈ x log with h0 : Mx → S1 , q  → 1, where q is the coordinate function of  regarded as a section of Mx . Then gp x log  Hom(Mx /Ox× , S1 )  S1 , (x, h)  → hh−1 0  → h(q). Let f0 be an isomor∼ gp phism Mx /Ox× − → Z, (class of q)  → 1. Then, under the above isomorphism (1), the corresponding element of π1 (x log , a) is the class of γ : [0, 1] → x log = S1 , t  → e2π it , since f0 (class of q) = 1 and h0 (q) = 1. From (1), it follows that π1 (x log , a)  Z, (class of γ )  → 1. This isomorphism sends 1 ∈ Z to the automorphism ξ  → ξ + 1 of R which is regarded as the universal covering of x log = S1 via R → S1 , ξ  → e2π iξ . A1.7 Let x be an fs logarithmic point. We define the monodromy cone π1+ (x log ) ⊂ π1 (x log ) as the image of Hom(Mx /Ox× , N) under the isomorphism (1) in A1.6. For example, in the case x = 0 ∈ , let a ∈ x log be as in Example in A1.6, b ∈ ∗ be any point, and δ be a path in log joining from a to b. Then π1 (log , a)  π1 (log , b), γ  → δ −1 γ δ, and via the isomorphisms π1 (x log , a)  π1 (log , a)  π1 (log , b)  π1 (∗ , b), the generator of π1+ (x log , a)  N corresponds to the class in π1 (∗ , b) of [0, 1]  → ∗ , t  → be2π it . A1.8 Example. Let L be the locally constant sheaf on X = ∗ defined to be the inverse × image of q Z under OX → OX , f  → exp(2π if ), where q is the coordinate function of . Then L = Zτ + Z where τ is a local section of OX such that q = exp(2π iτ ). Note that L is the local system of the first homology groups H1 of the standard family of elliptic curves parametrized by ∗ (see 0.1.4, 0.2.1, 0.2.10). Let x = 0 ∈ , a ∈ x log and b ∈ ∗ be as in A1.6 and A1.7, and let γ ∈ π1 (∗ , b) be the element corresponding to the generator of π1+ (x log , a). Then the action of γ on the stalk Lb sends 1 ∈ Lb to 1 and τ ∈ Lb to τ − 1.

310

APPENDIX

Let L∗ = HomZ (L, Z) which is the local system of the first cohomology groups H of the standard family of elliptic curves. Let (e1 , e2 ) be the local Z-basis of L∗ which is dual to (τ, 1). Then the action of γ on the stalk L∗b sends e1 ∈ L∗b to e1 and e2 ∈ L∗b to e1 + e2 . 1

A1.9 log

Let x be an fs logarithmic point. Then Ox is a locally constant sheaf on x log and log hence π1 (x log ) acts on the stalk Ox,y at y ∈ x log . For example, in the case x = 0 ∈ , the action of the generator of π1+ (x log ) log on Ox,y (y ∈ x log ) sends (2π i)−1 log(q) (q is the coordinate function of ) to (2πi)−1 log(q) − 1 (A1.8). Note that the corresponding element of π1 (∗ ) is the automorphism of h which sends τ ∈ h to τ + 1 ∈ h (A1.4, Example 1). This is a little confusing since the coordinate function τ of h is (2π i)−1 log(q). There is duality between the point and the coordinate function. A1.10 Let x be an fs logarithmic point. For an abelian group A, normalize the isomorphism H 1 (x log , A)  Hom(π1 (x log ), A) as follows. Let F be an A-torsor on x log . Since F is a locally constant sheaf, π1 (x log ) acts on stalks of F. We define the homomorphism π1 (x log ) → A corresponding to the class of F in H 1 (x log , A) by γ  → a (γ ∈ π1 (x log )) where a is the element of A such that f = a + γ (f ) (not γ (f ) = a + f ) for any element f of any stalk of F. Here a+ denotes the action of a on F. Consider the exact sequence 2π i

exp

0 → Z −→ Lx −→ τ −1 (Mxgp ) → 0 gp

and let δ : Mx → H 1 (x log , Z) be the connecting homomorphism which sends gp gp q ∈ Mx to the Z-torsor of (2π i)−1 log(f ). Then the composition Mx → ∼ → Hom(π1 (x log ), Z), where the second arrow is as in our normalH 1 (x log , Z) − gp ization, sends q ∈ Mx to the homomorphism π1 (x log ) → Z induced by q via the isomomorphism (1) in Section A1.6. This normalization is useful in Chapter 8.

A2 PROPER BASE CHANGE THEOREM FOR TOPOLOGICAL SPACES The following proper base change theorem A2.1 is well-known (Deligne [SGA4 12 , page 39], Kashiwara and Schapira [KS, Remark 2.5.3]). In [KS, Prop. 2.6.7] and in [V] by Verdier (see also Godement [Go, II Theorem 4.11.1]), this theorem is proved for locally compact spaces. But in this book, we use this theorem for more general topological spaces. The proof of this theorem in the general case is outlined by Kajiwara and Nakayama [KjNc, §2]. For the convenience of the reader, we give here the details of the proof in the general case.

311

APPENDIX

Theorem A2.1 Let X and Y be topological spaces and let f : X → Y be a proper (cf. Section 0.7) continuous map. Let Y
be a topological space, let g : Y
→ Y be a continuous map, let f
: X
= X ×Y Y
→ Y
be the map induced by f , and let g
: X
→ X be the map induced by g. (i) Let F be a sheaf of abelian groups on X. Then ∼

→ R m f∗
(g
)−1 F for any m ≥ 0. g −1 R m f∗ F − (ii) Let F be a sheaf of groups on X. Then ∼

g −1 R 1 f∗ F − → R 1 f∗
(g
)−1 F. (iii) Let F be a sheaf of sets on X. Then ∼

g −1 f∗ F − → f∗
(g
)−1 F. Here in (ii), R 1 f∗ F is defined to be the sheaf on Y associated to the presheaf U  → H 1 (f −1 (U ), F ) where H 1 is the set of isomorphism classes of F-torsors. The authors learned the proof of this theorem, given below, from Chikara Nakayama. Corollary A2.2 Let X and Y be topological spaces, let f : X → Y be a proper continuous map, and let y ∈ Y . Let i : f −1 (y) → X be the inclusion map. (i) Let F be a sheaf of abelian groups on X. Then ∼

→ H m (f −1 (y), i −1 F ) for any m ≥ 0. (R m f∗ F )y − (ii) Let F be a sheaf of groups on X. Then ∼

(R 1 f∗ F )y − → H 1 (f −1 (y), i −1 F ). (iii) Let F be a sheaf of sets on X. Then ∼

(f∗ F )y − →
(f −1 (y), i −1 F ). This corollary is the case Y
= {y} of the theorem. Conversely, the theorem follows from this corollary easily. This corollary is deduced from the following proposition. Proposition A2.3 Let X be a topological space and let K be a compact subspace of X satisfying the following condition (H). (H) If x, y ∈ K and x  = y, there are open subsets U , V of X such that x ∈ U , y ∈ V , and U ∩ V = ∅. Let i : K → X be the iclusion map. In the following, limU denotes the inductive − → limit where U ranges over all open neighborhoods of K in X. (i) Let F be a sheaf of abelian groups on X. Then ∼

→ H m (K, i −1 F ) for any m ≥ 0. lim H m (U, F ) − − → U

312

APPENDIX

(ii) Let F be a sheaf of groups on X. Then ∼

→ H 1 (K, i −1 F ). lim H 1 (U, F ) − − → U

(iii) Let F be a sheaf of sets on X. Then ∼

→
(K, i −1 F ). lim
(U, F ) − − → U

A2.4 We can deduce Corollary A2.2 from Proposition A2.3 as follows. Apply A2.3 by taking K to be f −1 (y) inA2.2. In this situation, f −1 (V ) for open neighborhoods V of y form a basis of neighborhoods of f −1 (y). (In fact, if U is an open neighborhood of f −1 (y), f (X − U ) is closed in Y since f is a closed map, and V := Y − f (X − U ) is an open neighborhood of y and satisfies f −1 (V ) ⊂ U .) Hence (R m f∗ F )y = limU H m (U, F ), where U ranges over all open neighborhoods of f −1 (y) (we take − → m = 1 in (ii), and m = 0 in (iii)). In the rest of A2, we prove Proposition A2.3. A2.5 We first give the proof of (iii) of Proposition A2.3. Proof of (iii) The injectivity of limU
(U, F ) →
(K, i −1 F ) is easily seen by − → looking at stalks at points of K. We prove the surjectivity. Let s ∈
(K, i −1 F ). Claim 1 There are a finite family (Uk )1≤k≤n of open sets of X such that K ⊂ ∪nk=1 Uk and elements sk ∈
(Uk , F ) such that the germ of sk at any point of K ∩ Uk coincides with the germ of s. Proof. For each x ∈ K, there is an open neighborhood U (x) of x in X and s(x) ∈
(U (x), F ) such that at any point of K ∩ U (x), the germ of s(x) and that of s coincide. Since K is compact, a finite subfamily of (U (x))x∈K covers K. 2 Claim 2 Let U and V be open sets of X such that K ⊂ U ∪ V . Then there are open sets U
, V
of X such that K − V ⊂ U
⊂ U and K − U ⊂ V
⊂ V and that U
∩ V
= ∅. Proof. For each x ∈ K − V and y ∈ K − U , since x  = y, the condition (H) shows that there are open sets U (x, y) and V (x, y) of X such that x ∈ U (x, y) and y ∈ V (x, y) and that U (x, y) ∩ V (x, y) = ∅. For each x ∈ K − V , since K − U ⊂ ∪y∈K−U V (x, y) and K − U is compact, there is a finite family y1 , . . . , ym(x) of  elements of K − U such that K − U is contained in V (x) := m(x) k=1 V (x, yk ). Let -m(x) U (x) = k=1 U (x, yk ). Then U (x) ∩ V (x) = ∅. Since K − V is compact and K − V ⊂ ∪x∈K−V U (x), there xr of elements of K − V such  is a finite family x1 , . . . , that K − V ⊂ U
:= U ∩ rk=1 U (xk ). Let V
= V ∩ rk=1 V (xk ). Then K − V ⊂ U
⊂ U , K − U ⊂ V
⊂ V , and U
∩ V
= ∅. 2

313

APPENDIX

Now we can prove (iii) of proposition A2.3. Let (Uk )1≤k≤n and sk be as in Claim 1. We prove by induction on n. Assume n ≥ 2. Let K
= K − Un . Since K
is compact n−1
and K ⊂ k=1 Uk , by the induction hypothesis, there are an open neighborhood W of K − Un in X and t ∈
(W, F ) such that at any point of K
, the germ of t and that of s coincide. By shrinking W , we may assume that the germ of s and that of t coincide at any point of K ∩ W . Hence we are reduced to the case n = 2. Now assume n = 2 (so K ⊂ U1 ∪ U2 ). By Claim 2, there are open sets U1
and
U2 of X such that K − U2 ⊂ U1
⊂ U1 , K − U1 ⊂ U2
⊂ U2 , and U1
∩ U2
= ∅. Let U3
be the open subset of U1 ∩ U2 consisting of all points at which the germ of s1 and that of s2 coincide. Let U = U1
∪ U2
∪ U3
. Then since K − U1
− U2
⊂ K ∩ U1 ∩ U2 ⊂ U3
, we have K ⊂ U . Since U1
∩ U2
= ∅, there is an element s
∈
(U, F ) whose restriction to Uk
coincides with that of sk for k = 1, 2, and whose restriction to U3
coincides with that of s1 (= that of s2 ). Thus, the germ of s
at any point of K coincides with that of s. 2 A2.6 We review flabby sheaves and soft sheaves. A sheaf F on a topological space X is said to be flabby if, for any open set U of X, the map
(X, F ) →
(U, F ) is surjective. A sheaf F on a compact topological space X is said to be soft if for any closed set C of X, the map
(X, F ) →
(C, i −1 F ) is surjective, where i : C → X is the inclusion map. By Proposition A2.3 (iii), which we just proved, we have Lemma A2.7 Let X be a topological space and let K be a compact subspace of X satisfying the condition (H) in A2.3. Let i : K → X be the inclusion map. Then if 2 F is a flabby sheaf on X, i −1 F is a soft sheaf on K. A2.8 We review the cohomology of sheaves. Let X be a topological space. If F is a sheaf of abelian groups on X, there is an exact sequence 0 → F → Q0 → Q1 → Q2 → · · ·

(1)

of sheaves of abelian groups with Qk (k ≥ 0) flabby [Go, II 4.3]. For such sequence, the m-th cohomology group H m (X, F ) is identified with the m-th cohomology group H m (
(X, Q• )) of the complex
(X, Q• ) of abelian groups. In the case X is compact, if we have an exact sequence (1) with Qk (k ≥ 0) soft, then the m-th cohomology group H m (X, F ) is identified with H m (
(X, Q• )) (see [KS, Exercise II.5]). If F is a sheaf of groups on X, there is an injective homomorphism F → Q of sheaves of groups with Q flabby. For such injective homomorphism, H 1 (X, F ) is identified with the quotient set
(X, F\Q)/
(X, Q) of
(X, F\Q) under the natural right action of
(X, Q).

314

APPENDIX

In the case X is compact, if we have an injective homomorphism F → Q with Q soft, then H 1 (X, F ) is identified with
(X, F\Q)/
(X, Q). (These assertions for sheaves of groups can be proved in the same way as in the above case of sheaves of abelian groups.) A2.9 We prove (i) and (ii) of Proposition A2.3. Proof of (i). Take an exact sequence A2.8 (1) with Qk (k ≥ 0) flabby. Then we have a commutative diagram ∼

limU H m (U, F ) −−−−→ limU H m (
(U, Q• )) − → − →       ∼

H m (K, i −1 F ) −−−−→ H m (
(K, i −1 Q• )). Here the lower horizontal isomorphism is due to the fact that i −1 Qk are soft by ∼ → H m (limU
(U, Q• )), the right A2.7 and to A2.8. Since limU H m (
(U, Q• )) − − → − → vertical arrow is an isomorphism by (iii) of A2.3. Hence the left vertical arrow is an isomorphism. 2 Proof of (ii). Take an injective homomorphism F → Q with Q flabby. Then we have a commutative diagram ∼

limU H 1 (U, F ) −−−−→ − →   

limU
(U, F\Q)/
(U, Q) − →   

∼

H 1 (K, i −1 F ) −−−−→
(K, i −1 F\i −1 Q)/
(K, i −1 Q). Here the lower horizontal isomorphism is due to the fact that i −1 Q is soft ∼ → (limU
(U, F\Q))/ by A2.7 and to A2.8. Since limU (
(U, F\Q)/
(U, Q)) − − → − → (limU
(U, Q)), the right vertical arrow is a bijection by (iii) of A2.3. Hence the − → left vertical arrow is a bijection. 2

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List of Symbols CHAPTER 0

•X : de Rham complex of an analytic space X 0.1.7 ( , )F : HC × HC → C, Hermitian form associated with F 0.1.8 || := {r ∈ R | 0 ≤ r < 1} 0.2.9 × } ⊂ X 0.4.30 Xtriv := {x ∈ X | MX,x = OX,x N := Z≥0 0.7.1 LR forZ-module L and R = Q, R, C 0.7.3 0 = w, (hp,q )p,q∈Z , H0 ,  , 0 0.7.3 (hp,q )p,q∈Z 0.7.3 H0 : Z-module 0.7.3  , 0 : Q-rational non-degenerate (−1)w -symmetric C-bilinear form on H0,C GR := Aut(H0,R ,  , 0 ) for R = Z, Q, R, C 0.7.3 gR := Lie GR for R = Q, R, C 0.7.3 G◦ : connected component of G in Zariski topology containing 1 0.7.3 A : category of analytic spaces 0.7.4 A(log) : category of fs logarithmic analytic spaces 0.7.4 f −1 (F) : sheaf-theoretic inverse image 0.7.6 f ∗ (F) : module-theoretic inverse image 0.7.6

CHAPTER 1 (HZ , F ) : Hodge structure 1.1.1, 0.1.5 F = (F p )p∈Z : Hodge filtration 1.1.1, 0.1.5  ,  : bilinear form on HQ 1.1.2, 0.1.8 CF ∈ Aut(HC ) : Weil operator associated with F 1.1.2 D : classifying space of polarized Hodge structures of type 0 1.2.1 Dˇ : compact dual of D 1.2.2 σ : a nilpotent cone in gR 1.3.5 ˇ 1.3.6 Dˇ orb := {(σ, exp(σC )F ) | nilpotent cone σ in gR , F ∈ D}  ˇ 1.3.6 Dˇ orb := {(σ, exp(iσR )F ) | nilpotent cone σ in gR , F ∈ D}  : a fan in gQ 1.3.8 D : space of nilpotent orbits in directions in  1.3.8  D : space of nilpotent i-orbits in directions in  1.3.8 Dσ := D{face of σ } 1.3.9  Dσ := D{face of σ } 1.3.9

0.7.3

322

LIST OF SYMBOLS



D → D , canonical projection 1.3.9 D = ∪σ ∈ Dσ 1.3.9  D = ∪σ ∈ Dσ 1.3.9
(σ ) :=
∩ exp(σ ) 1.3.10  : fan in gQ of rank ≤ 1 1.3.11

CHAPTER 2 α : M → OX , logarithmic structure 2.1.1, 0.2.4 f ∗ (M, α) : inverse image of logarithmic structure 2.1.3 S × ⊂ S : group of invertible elements in a monoid S 2.1.4 S gp ⊃ S : group associated to an integral monoid S 2.1.4 (Sλ , θλ )λ : chart of a logarithmic structure 2.1.5 C[S] : monoid ring of S over C 2.1.6 q ωX/Y : sheaf of logarithmic differential q-forms on X over Y 2.1.7 • , d): logarithmic de Rham complex of X over Y 2.1.7 (ωX/Y X ×Z Y : fiber product in the category A(log) 2.1.10 A1 : category 2.2.1 A1 (log) : category 2.2.1 Zan : analytic space associated to a scheme Z locally of finite type over C X log for X ∈ A1 (log) 2.2.3 τ : Xlog → X, canonical projection 2.2.3 gp L : sheaf of logarithms of MX 2.2.4 log log 2.2.4 OX : sheaf of rings on X log (Xlog , OX ) : ringed space associated with X ∈ A1 (log) 2.2.4 log τ : (Xlog , OX ) → (X, OX ), canonical projection 2.2.4 log log f log : (Xlog , OX ) → (Y log , OY ), associated morphism 2.2.4 q,log ωX/Y : sheaf of logarithmic differential q-forms on X log over Y log 2.2.6 •,log

(ωX/Y , d) : logarithmic de Rham complex of Xlog over Y log 2.2.6 π1+ (x log )  Hom(Mx /Ox× , N) : positive monoid in π1 (x log ) 2.2.9 γ (log(f )) = log(f ) − 2π [γ , f ] for γ ∈ π1 (x log ) and f ∈ Mx 2.2.9, A1 ∼ log log → OX ⊗A L0 2.3.2 ν : OX ⊗A L −

n ∼ log log → OX ⊗A L0 2.3.2 ξ = exp( j =1 (2πi)−1 log(qj ) ⊗ Nj ) : OX ⊗A L0 − ∼

ν:L− → ξ −1 (1 ⊗ L0 ) 2.3.2 νy (1 ⊗ v) = ξy−1 (1 ⊗ v) for v ∈ L0 2.3.2 gp × N : L → (MX /OX ) ⊗ L, logarithm of local monodromy actions (HZ , F ) : prelogarithmic Hodge structure 2.4.1 (HZ ,  , , F ) : prepolarized logarithmic Hodge structure 2.4.3 H : category 2.4.4 H
: category 2.4.4 P : category 2.4.4 P
: category 2.4.4

2.3.4

2.2.1

323

LIST OF SYMBOLS log

sp(y) := HomC-alg (Ox,y , C), space of specializations over y ∈ x log 2.4.6 gp s˜ : Mx → C× , a  → exp(s(log(a))) 2.4.6 F (s) : specializaion of F at s ∈ sp(y) 2.4.6 (HZ ,  , , F ) : polarized logarithmic Hodge structure 2.4.8, 2.4.7 ϕˇ : X →
\Dˇ orb , period map 2.5.3  := (w, (hp,q )p,q∈Z , H0 ,  , 0 ,
, ) 2.5.8 ϕ : X →
\D , period map (of H with respect to ) 2.5.10  ϕ log : Xlog →
\D 2.5.10 add R≥0 : monoid 2.6.2 (HZ , W, F ) : logarithmic mixed Hodge structure 2.6.4, 2.6.2

CHAPTER 3 Sweak/X 3.1.1 Sstr/X 3.1.1 U (ε) : a neighborhood in the strong topology 3.1.3 U (s, ε) : a neighborhood in the strong topology 3.1.4, 3.1.5 D : categorical closure of a category D 3.2.2 A : category 3.2.3 A(log) : category 3.2.3
A(log) : category 3.2.3
A(log)fs : category 3.2.3 A2 : category 3.2.4 B : category 3.2.4 B∗ : category 3.2.4 A1 (log) : category 3.2.4 A2 (log) : category 3.2.4 A(log) : category 3.2.4 B(log) : category 3.2.4 B ∗ (log) : category 3.2.4 S ∨ := Hom(S, N), dual monoid 3.3.1 ∼ → {face of S ∨ }, S
 → {h ∈ S ∨ | h(S
) = {0}} 3.3.1 δ : {face of S} − ∼ → {face of
(σ )}, τ  →
(τ ), bijection 3.3.1 {face of σ } − mult C : monoid 3.3.2 mult R≥0 : monoid 3.3.2 toricσ := Spec(C[
(σ )∨ ])an , toric variety 3.3.2 torusσ := C× ⊗
(σ )gp , torus 3.3.2 Eˇ σ := toricσ ×Dˇ 3.3.2, 3.4.1 gp π1 (toriclog 3.3.2 σ )  π1 (torusσ ) 
(σ ) , canonical isomorphisms + log log π1 (q ) = π1 (q ) ∩
(σ ) for q ∈ toricσ 3.3.2 σ (q) : face of σ corresponding to a point q ∈ toricσ 3.3.2 Hσ = (Hσ,Z ,  , σ , Fσ ) : canonical pre-PLH on Eˇ σ 3.3.3 ˇ log 3.3.3 (Hσ,Z ,  , σ ) : canonical local system on toriclog σ , and also on Eσ

324

LIST OF SYMBOLS log

log

ν : Otoricσ ⊗Z Hσ,Z  Otoricσ ⊗Z H0 log OEˇ σ

log OEˇ ⊗Z σ gp

3.3.3

H0 , identification via ν 3.3.3 ⊗Z Hσ,Z = µσ : canonical
(σ ) -level structure of Hσ 3.3.3 ξ(1 ⊗ v) = 1 ⊗ µσ (v) (v ∈ Hσ,Z ), expression of µσ 3.3.3 E˜ σ ⊂ Eˇ σ 3.3.4 Eσ ⊂ E˜ σ ⊂ Eˇ σ 3.3.4, 3.4.1 e : σC → torusσ , surjective homomorphism 3.3.5 mult ) ⊂ toricσ 3.3.9 |toric|σ := Hom(
(σ )∨ , R≥0 |torus|σ := R>0 ⊗
(σ )gp ⊂ torusσ 3.3.9 Eˇ σ := |toric|σ ×Dˇ 3.3.9 Eσ ⊂ Eˇ σ 3.3.9, 3.4.3 ϕ : Eσ →
(σ )gp \Dσ , period map 3.3.10 ϕ  : Eσ → Dσ , period map 3.3.10 Eσ : topological space 3.4.1 OEσ : sheaf of rings of Eσ 3.4.1 MEσ : logarithmic structure of Eσ 3.4.1
(σ )gp \Dσ : topological space 3.4.2
\D : topological space 3.4.2 O
\D : sheaf of rings of
\D 3.4.2 M
\D : logarithmic structure of
\D 3.4.2 X ×Z Y : fiber product in the category B(log) 3.5.1 3.5.1 X ×cl Z Y : fiber product in the category of topological spaces (cl : classical) 1 ωX : sheaf of logarithmic differential 1-forms on X for X ∈ B(log) 3.5.2 • logarithmic de Rham complex on X for X ∈ B(log) 3.5.2 ωX •,log ωX : logarithmic de Rham complex on X log for X ∈ B(log) 3.5.2 θX : sheaf of logarithmic vector fields for logarithmically smooth X ∈ B(log) 3.5.4 TX : logarithmic tangent bundle for logarithmically smooth X ∈ B(log) 3.5.6 BI (X) : logarithmic blow-up of X with respect to I for X ∈ B(log) 3.6.6 BI∗ (X) : variant of BI (X) for X ∈ B(log) 3.6.6 (NQ , s) 3.6.7 gp × s ∈
(X, MX /OX ) ⊗ NQ for X ∈ B(log) 3.6.7 X(σ ) for X ∈ B(log) and a finitely generated rational cone σ in NR 3.6.8 X() for X ∈ B(log) and a rational fan  in NR 3.6.10 Q1 (X) : a set of (x, P ) for X ∈ B(log) 3.6.14 Q(X) : a set of (x, P , h) for X ∈ B(log) 3.6.14 Q
1 (X) : a set of (x, σ ) for X ∈ B(log) 3.6.14 Q
(X) : a set of (x, σ, h) for X ∈ B(log) 3.6.14 qY : Y → Q(X) for X ∈ B(log) and Y ∈ B(log) over X 3.6.14 qY
: Y → Q
(X) for X ∈ B(log) and Y ∈ B(log) over X 3.6.14 qY,1 : Y → Q1 (X) for X ∈ B(log) and Y ∈ B(log) over X 3.6.14
: Y → Q
1 (X) for X ∈ B(log) and Y ∈ B(log) over X 3.6.14 qY,1 Xval for X ∈ B(log) 3.6.18 log Xval for X ∈ B(log) 3.6.26

LIST OF SYMBOLS

CHAPTER 4 PLH : A2 (log) → (Sets), functor 4.2.1 PLH |A(log) : A(log) → (Sets), functor 4.2.2 gp × N : HQ → (MX /OX ) ⊗ HQ for a pre-PLH H on X ∈ B(log) 4.3.5 gp × ) ⊗ gQ )) for H on X ∈ B(log) with a
-level N ∈
(X,
\((MX /OX structure 4.3.5 1 ∇ : M → ωX ⊗OX M for a PLH H on a logarithmically smooth X ∈ B(log) 4.4.2 θX → p HomOX (Mp , M/Mp ) 4.4.2 θX  End  ,  (M)/F 0 End  ,  (M) 4.4.3 TXh ⊂ TX : horizontal logarithmic tangent bundle 4.4.4 θXh ⊂ θX : horizontal submodule 4.4.4 4.4.4 θXh  gr −1 F End  ,  (M)

CHAPTER 5 X : set of all maximal compact subgroups of GR 5.1.1 KF : maximal compact subgroup of GR associated to F 5.1.2 KF
: isotropy subgroup of GR at F ∈ D 5.1.2 Pu : unipotent radical of a parabolic subgroup P 5.1.3 C : center of P /Pu 5.1.3 θK : Cartan involuton of GR at K 5.1.3 aK : Borel-Serre lifting of a ∈ C ⊂ P /Pu at K 5.1.3 a ◦ F := aKF F : Borel-Serre action of a ∈ C on D 5.1.3 a ◦ K := Int(aK ) : Borel-Serre action of a ∈ C on X 5.1.3 SP : maximal Q-split torus of the center C of P /Pu 5.1.4 AP : connected component of R-valued points of SP containing 1 5.1.4 DBS ⊃ D : Borel-Serre space 5.1.5 XBS ⊃ X : Borel-Serre space 5.1.5 V × : set of invertible elements of V 5.1.5 DBS,val : valuative Borel-Serre space 5.1.6 T>0 for  a torus T 5.1.6 Wα := χ ∈V H0,R (αχ −1 ) ⊂ H0,R for α ∈ X(T ) and V ⊂ X(T ) : valuative 5.1.6 PT ,V : parabolic subgroup associated with (T , V ) 5.1.6 DBS (P ) ⊂ DBS 5.1.8 XBS (P ) ⊂ XBS 5.1.8 DBS,val (P ) ⊂ DBS,val 5.1.8 
P ⊂ X(SP ) 5.1.9 P ⊂ 
P ⊂ X(SP ) 5.1.9 X(SP )+ ⊂ X(SP ) 5.1.10 AP ⊃ AP 5.1.10 r AP  Map(P , R≥0 )  R≥0 5.1.10

325

326

LIST OF SYMBOLS

D ×AP AP 5.1.10 DBS (P )  D ×AP AP 5.1.10 Spec(C[S])an,val = Hom(S, Cmult )val , valuative toric variety 5.1.11 mult )val 5.1.11 Hom(S, R≥0 mult (AP )val := Hom(X(SP )+ , R≥0 )val 5.1.12 AP DBS,val (P )  D × (AP )val 5.1.12 XBS (P )  X ×AP AP 5.1.13 r GR  Pu × R>0 × K, Iwasawa decomposition 5.1.15 R SP : maximal R-split torus of the center C of P /Pu 5.1.15 R 5.1.15 AR P : connected component of R-valued points of SP r X  Pu × R>0 5.1.15 r D  Pu × R>0 × (KF /KF
) 5.1.15 r XBS (Q)  Pu × (R>0 ×AQ AQ ) 5.1.15 r DBS (Q)  Pu × (R>0 ×AQ AQ ) × (KF /KF
) 5.1.15 r DBS,val (Q)  Pu × (R>0 ×AQ AQ,val ) × (KF /KF
) 5.1.15 r XBS (P )  Pu × R≥0 5.1.15 r × (KF /KF
) 5.1.15 DBS (P )  Pu × R≥0 r DBS,val (P )  Pu × (R≥0 )val × (KF /KF
) 5.1.15 (ρ, ϕ) : SL(2)-orbit 5.2.1 i ∈ hn 5.2.1   0 1 Nj := ρ∗j ∈ gR 5.2.2 0 0 ρ∗j : sl(2, C) → gC 5.2.2 n  : Gm,R → SL(2, R)n 5.2.2 n ρ˜ : Gm,R → GR 5.2.2 ρ˜j : Gm,R → GR 5.2.2 W (N) : weight filtration associated with a nilpotent endomorphism N 5.2.4 W (σ ) : weight filtration associated with a nilpotent orbit (σ, Z) 5.2.4 (W (σj ))1≤j ≤n : family of weight filtrations associated to an SL(2)-orbit 5.2.5 DSL(2) : space of SL(2)-orbits 5.2.6 DSL(2),n : space of SL(2)-orbits of rank n 5.2.6 [ρ, ϕ] ∈ DSL(2) 5.2.6 DSL(2),val,n : space of valuative SL(2)-orbits of rank n 5.2.7 DSL(2),val : space of valuative SL(2)-orbits 5.2.7 n n )+ ⊂ X(Gm,R ) 5.2.7 X(Gm,R r = ϕ(i) ∈ D : reference point 5.2.8 W ( j ) := W (σj ) 5.2.9 GW,R ⊂ GR 5.2.12 DSL(2) (W ) ⊂ DSL(2) 5.2.12 DSL(2),val (W ) ⊂ DSL(2),val 5.2.12 B(U, U
, U

) : a basis of the filter in D of a point of DSL(2) 5.2.16 V : a set of pairs of a subspace of gQ and a valuative submonoid in its dual 5.3.1 F(A, V ) : set of rational nilpotent cones in gR associated to (A, V ) ∈ V 5.3.2 Dˇ val ⊃ Dval 5.3.3   Dˇ val ⊃ Dval 5.3.3

327

LIST OF SYMBOLS

Dval : space of valuative nilpotent orbits 5.3.3  Dval : space of valuative nilpotent i-orbits 5.3.3  Dval → Dval 5.3.3 D,val ⊂ Dval : space of valuative nilpotent orbits in directions in  5.3.5   D,val ⊂ Dval : space of valuative nilpotent i-orbits in directions in  5.3.5 Dσ,val = D{face of σ },val 5.3.5   Dσ,val = D{face of σ },val 5.3.5 

D,val → D,val 5.3.5 D,val → D 5.3.5   D,val → D 5.3.5 toricσ,val := Spec(C[
(σ )∨ ])an,val , valuative toric variety |toric|σ,val ⊂ toricσ,val 5.3.6 Eσ,val := toricσ,val ×toricσ Eσ 5.3.7  Eσ,val := |toric|σ,val ×|toric|σ Eσ ⊂ Eσ,val 5.3.7 Eσ,val → Eσ 5.3.7  Eσ,val → Eσ 5.3.7  ψ : Dval → DSL(2) , CKS map 5.4.3

5.3.6

CHAPTER 6 wN : Gm,R → P /Pu , weight map 6.1.1 W [l] : shift of filtration 6.1.2 ε = ε(W, F ) ∈ gC 6.1.2 Fˆ = (W, F )∧ : associated R-split Hodge filtration 6.1.2 FW : set of decreasing filtrations F on HC such that (W, F ) is a MHS 6.1.2 I p,q : (p, q)-component of Deligne splitting 6.1.2 L−1,−1 = L−1,−1 (W, F ) ⊂ End C (VC ) 6.1.2 6.1.2 δ ∈ L−1,−1 R I˜p,q := I p,q (W, exp(−iδ)F ), R-splitting of R-split MHS (W, exp(−iδ)F ) 6.1.2 δp,q : (p, q)-component of δ with respect to (I˜p,q ) 6.1.2 ζ ∈ L−1,−1 6.1.2 R ζp,q : (p, q)-component of ζ with respect to (I˜p,q ) 6.1.2 Fˆ(j ) := (W (σj )[−w], exp(iNj +1 )Fˆ(j +1) )∧ 6.1.3 0j ∈ Cj 6.1.3 ik ∈ hk 6.1.3 Nˆ j ∈ gR 6.1.3 ah ∈ gR (h ∈ Nn ) 6.1.5 bh ∈ gR (h ∈ Nn ) 6.1.5 ±,r g± 6.2.1 R = gR ⊂ gR − ch ∈ gR (h ∈ Nn ) 6.2.1 n 6.2.1 kh ∈ g+ R (h ∈ N ) 1 0 V = V
V
· · ·
V n = {0}, for a valuative submonoid V of A∗ 6.3.1 j −1 j νj : VQ /VQ → R 6.3.1

328

LIST OF SYMBOLS

{0} = A0  A1  · · ·  An = A, annihilators of the V j 6.3.1 S = 1≤j ≤n Sj : a subdivision of index set S 6.3.3 ( s∈Sj as Ns )1≤j ≤n : A∗ → R n 6.3.3 S≤j := k≤j Sk ⊂ S 6.3.11 S≥j := k≥j Sk ⊂ S 6.3.11 σc,α,β : a nilpotent cone in gR associated with c, α, β 6.3.11 Dˇ j : set of all F
∈ Dˇ such that ((Ns )s∈S≤j , F
) generates a nilpotent orbit ψ˜ :

 Eσ,val

ψ

→ Dσ,val − → DSL(2)

6.4.1

6.4.8

CHAPTER 7 E˜ σ,val := toricσ,val ×toricσ E˜ σ 7.1.2 | | : toricσ,val → |toric|σ,val 7.1.2 m m , continuous map s. t. β(ρ(t) ˜ · x) = tβ(x) (x ∈ D, t ∈ R>0 ) β : D → R>0 U (τ ) ⊂ toricσ , for a face τ of σ 7.3.4 |U |(τ ) ⊂ |toric|σ , for a face τ of σ 7.3.4

7.2.10

CHAPTER 8 LS
: sheaf of classes of logarithmic local systems of type (H0 ,  , 0 ,
) 8.1.3 LS
 R 1 τ∗ (
) 8.1.3 gp × R 1 τ∗ (Z)  MX /OX 8.1.4 gp × ) 8.1.5 LS

⊗ (MX /OX  := (H0 ,  , 0 ,
, ) 8.1.6 σ := (H0 ,  , 0 ,
(σ )gp , {face of σ }) 8.1.6 ∼ → (H0 ,  , 0 ), a lifting of µy 8.1.7 µ˜ y : (HZ,y ,  , y ) − (Hσ ,  , σ , µσ ) : canonical logarithmic local system of type σ on toricσ 8.1.7 LS : sheaf of classes of logarithmic local systems of type  8.1.8 C := PLH : A2 (log) → (Sets), functor 8.2.1 Cσ := PLHσ : A2 (log) → (Sets), functor 8.2.1 σ := (w, (hp,q )p,q∈Z , H0 ,  , 0 ,
(σ )gp , {face of σ }) 8.2.1 Bσ : A2 (log) → (Sets), functor 8.2.1 (Hθ,Z ,  , θ , µθ ) : logarithmic local system induced by θ : X → toricσ 8.2.1 Bˇ σ : A2 (log) → (Sets), functor 8.2.3 log F log := OX ⊗τ −1 (OX ) τ −1 (F ) 8.2.3 C := LS : A2 (log) → (Sets), functor 8.2.4 C σ := LSσ : A2 (log) → (Sets), functor 8.2.4 ∼ → C 8.2.6 ( σ ∈ Cσ )/ ∼ − ∼ ( σ ∈ C σ )/ ∼ − → C 8.2.6 Xval →
\Dval , period map 8.4.1 log  Xval →
\Dval , period map 8.4.1 log Xval →
\DSL(2) , period map 8.4.1

329

LIST OF SYMBOLS

CHAPTER 9 

XBS : -Borel-Serre space 9.1.1  DBS : -Borel-Serre space 9.1.1  DBS,val : -valuative Borel-Serre space 9.1.1  DSL(2) : -SL(2)-orbit space 9.1.1   DSL(2),≤1 ⊂ DSL(2) : -SL(2)-orbit space of rank ≤ 1 ∗ D : space of Cattani-Kaplan 9.1.1   XBS (P ) ⊂ XBS 9.1.3   DBS (P ) ⊂ DBS 9.1.3   DBS,val (P ) ⊂ DBS,val 9.1.3   DSL(2),≤1 (W ) ⊂ DSL(2),≤1 9.1.3   DSL(2),1 = DSL(2),≤1 − D 9.1.5  D → DSL(2),≤1 , period map 9.4.1

9.1.1

CHAPTER 10 W (J ) for J ⊂ {1, . . . , n} 10.1.1 gR (χ ) ⊂ gR 10.1.1 n XJ ⊂ X = X(Gm,R ) 10.1.1 X+ ⊂ X 10.1.1 n × gR × Kr · r 10.1.1 Y0 := R>0 n Y3 := (R≥0 )val × gR × Kr · r 10.1.1 n Y1 ⊂ R≥0 × gR × Kr · r 10.1.1 Y2 ⊂ Y3 10.1.1 η0 : Y0 → D, continuous map 10.1.1 L ⊂ gR : an R-subspace 10.1.2 Yj,L ⊂ Yj (j = 0, 1, 2, 3) 10.1.2 η1 : Y1 → DSL(2) 10.1.3, 10.3.3 η2 : Y2 → DSL(2),val 10.1.3, 10.3.2 η3 : Y3 → DBS,val 10.1.3, 10.3.1 Uj : an open set of Yj,L (j = 1, 2, 3) 10.1.3 pJ = [ρJ , ϕJ ] ∈ DSL(2) (W (J ) ) for J ⊂ {1, . . . , n} 10.2.1 U (p) := ∪J ⊂{1,...,n} GW (J ) ,R · pJ , special open neighborhood of p ∈ DSL(2) µ : P × (AP )val → P 10.2.8  : a set of Q-parabolic subgroups of GR 10.2.10 n n )val (P ) ⊂ (R≥0 )val : open subset 10.2.10 (R≥0 n (R≥0 )val (P ) → (AP )val 10.2.10 n n )val (P ) → D ×AP (AP )val = DBS,val (P ) 10.2.10 D ×R>0 (R≥0 ◦ PV ⊂ (G )R : parabolic subgroup associated with a valuative submonoid V 10.2.11 n n κP : D ×R>0 (R≥0 )val (P ) → D 10.2.12

10.2.1

330 κ : DSL(2) (W ) → D 10.2.13 Dˇ W ⊂ Dˇ 10.2.15 U (p)val ⊂ DSL(2),val : inverse image of U (p) ⊂ DSL(2) 10.3.2 C(X) : cone over a topological space X 10.4.2  U (p) ⊂ DSL(2),≤1 : image of U (p) ⊂ DSL(2),≤1 10.4.3 CHAPTER 11 A ⊂ End Q (H0,Q ) : a semisimple Q-algebra 11.1.1 Dˇ A ⊂ Dˇ 11.1.2 D A := D ∩ Dˇ A in Dˇ 11.1.2 11.1.2 GA R ⊂ GR (R = Z, Q, R, C) gA 11.1.2 R ⊂ gR (R = Q, R, C)
\D A 11.1.4 DA ⊂ D 11.1.5 DσA ⊂ Dσ 11.1.5 ,A  D ⊂ D 11.1.5 A Eˇ σ ⊂ Eˇ σ 11.1.6 EσA := Eσ ∩ Eˇ σA 11.1.6 Eσ,A := Eσ ∩ Eˇ σA in Eˇ σ 11.1.6
\DA 11.1.6 11.3.1 PLHA  : A2 (log) → (Sets), functor A PLHA 11.3.1   Mor( ,
\D ) CHAPTER 12 Sp(g) : symplectic group 12.1.1 hg : Siegel upper half space 12.1.3 |∗ | := || − {0} 12.4.1

LIST OF SYMBOLS

Index

A-fan in gQ 11.1.5 A-nilpotent cone 11.1.5 A-PLH 11.2.4, 0.4.33 A-PLH of type  11.2.2  action of iσR on Eσ 7.2.1  action of iσR on Eσ,val 7.2.1 action of σC on Eσ 7.2.1 action of σC on Eσ,val 7.2.1 analytic space 0.7.4 analytically constructible 3.1.4 argument function h 2.2.3 associated logarithmic structure 2.1.1, 0.2.11 associated SL(2)-orbit in one variable 6.1.1 associated SL(2)-orbit in several variables 6.1.3 big Griffiths transversality 2.4.9, 4.4.6, 0.4.22 Borel-Serre action (on D and on X ) 5.1.3 Borel-Serre lifting 5.1.3 Borel-Serre space DBS ⊃ D 5.1.5 Borel-Serre space XBS ⊃ X 5.1.5 χ -Hodge numbers 10.2.3 canonical logarithmic local system of type σ on toricσ  canonical map D → DSL(2),≤1 9.4.1 canonical pre-PLH Hσ on Eˇ σ 3.3.3 canonical projection τ : Xlog → X 2.2.3, 2.2.4 Cartan involution 5.1.3, 0.5.7 categorical closure D 3.2.2 categorically generalized analytic space 3.2.4 category A 0.7.4 category Ared 3.2.1 category A 3.2.3 category A1 2.2.1 category A2 3.2.4 category A(log) 0.7.4 category Ared (log) 3.2.1 category A(log) 3.2.4 category A(log) 3.2.3
3.2.3 category A(log)
category A(log)fs 3.2.3 category A1 (log) 2.2.1, 3.2.4 category A2 (log) 3.2.4 category B 3.2.4 category B∗ 3.2.4 category B(log) 3.2.4 category B∗ (log) 3.2.4

3.3.3, 8.1.7

332

INDEX

category H 2.4.4 category H
2.4.4 category P 2.4.4 category P
2.4.4 category of analytic spaces A 0.7.4 category of fs logarithmic analytic spaces A(log) chart of a logarithmic structure 2.1.5 CKS map 5.4.3 classical situation 8.2.7, 0.4.14 classifying space D of polarized Hodge structures compact 0.7.5 compact dual Dˇ 1.2.2 compatible family of filtrations 5.2.12 complete fan 12.6.1 condition (A0 ) 8.3.5 condition (A1 ) 2.2.1 condition (A
1 ) 2.2.1 condition (A2 ) 3.2.4 condition (A3 ) 8.3.5 condition (A4 ) 8.3.6 condition (B) 3.2.4 condition (B∗ ) 3.2.4 cone 1.3.1 Deligne splitting 6.1.2 distributive family of filtrations

0.7.4

1.2.1

5.2.12

equivalence of SL(2)-orbits 5.2.6 evaluation homomorphism OX,x → C for X ∈ A1 excellent basis for (A, V , Z) 6.3.8 exponential sequence on X for X ∈ A1 2.2.2 extended period map of Cattani-Kaplan 9.4.3

2.2.2

-Borel-Serre space 9.1.1 -SL(2)-orbit space 9.1.1 -valuative Borel-Serre space 9.1.1 face of a monoid 3.3.1 face of a finitely generated cone 1.3.2 face σ (q) of σ corresponding to q ∈ toricσ 3.3.2 face of a nilpotent cone 0.4.3 family of weight filtrations associated to an SL(2)-orbit fan 1.3.3 fan in gQ 1.3.8, 0.4.4 fiber product in A(log) 2.1.10 fiber product in B(log) 3.5.1 finitely generated cone 1.3.1 flabby sheaf A2.6 free action 7.2.5 fs logarithmic analytic space 2.1.5, 0.2.11, 0.7.4 fs logarithmic point 2.1.9, 0.2.16 fs logarithmic ringed space 2.1.5 fs logarithmic structure 2.1.5, 0.2.11 fs monoid 2.1.4, 0.2.11 functor Bσ : A2 (log) → (Sets) 8.2.1 functor Bˇ σ : A2 (log) → (Sets) 8.2.3 functor C = PLH : A2 (log) → (Sets) 8.2.1, 4.2.1 functor Cσ = PLHσ : A2 (log) → (Sets) 8.2.1

5.2.5, 0.5.12

333

INDEX functor C = LS : A2 (log) → (Sets) 8.2.4, 8.1.8 functor C σ = LSσ : A2 (log) → (Sets) 8.2.4, 8.1.9 functor LS : A2 (log) → (Sets) 8.1.8 functor LSσ : A2 (log) → (Sets) 8.1.9 functor PLH : A2 (log) → (Sets) 4.2.1 functor PLH |A(log) : A(log) → (Sets) 4.2.2 11.3.1 functor PLHA  : A2 (log) → (Sets) fundamental diagram 5.0.1, 0.5.25 fundamental group π1 (x log ) 2.2.9
-level structure 2.5.2, 0.3.6 generate a nilpotent orbit 5.4.1 geometrically generalized analytic space good basis for (A, V ) 6.3.3 Griffiths domain 0.3.1 Griffiths transversality 2.4.9, 4.4.6 Griffiths transversality over X 2.4.5 Griffiths transversality on x 0.4.22 Hodge structure 1.1.1, 0.1.5 Hodge numbers 0.1.5 Hodge metric 0.1.8 homomorphism of monoids 0.7.7 horizontal tangent bundle 0.3.8, horizontal logarithmic tangent bundle 

3.2.4

4.4.4



iσR -torsor Eσ → Dσ 7.3.2 infinitesimal period map dϕ 4.4.7, 0.4.31 injective map DSL(2),val → DBS,val 5.2.11 integral monoid 0.2.11 interior 0.7.7 inverse image of logarithmic structure 2.1.3, 0.2.12 Iwasawa decomposition 5.1.15 LH 2.6.5 LMH 2.6.2, 2.6.4 local monodromy 2.3.1 locally compact 0.7.5 logarithmic blow-up 3.6.6 • , d) on X logarithmic de Rham complex (ωX

2.1.7 •,log logarithmic de Rham complex (ωX , d) on Xlog 2.2.6 logarithmic differential form 2.1.7 logarithmic Hodge structure 2.6.5 logarithmic Kodaira-Spencer map 4.4.7 logarithmic local system of type  8.1.7 logarithmic local system on X of type (H0 ,  , 0 ,
) for X ∈ A1 (log) 8.1.1 logarithmic manifold 3.5.7, 0.4.17 logarithmic modification 3.6.12 logarithmic mixed Hodge structure 2.6.2, 2.6.4 logarithmic ringed space 2.1.1 logarithmic structure 2.1.1, 0.2.4 logarithmic structure associated with a normal crossing divisor 2.1.2, 0.2.5 logarithmic structure of Eσ 3.4.1 logarithmic structure of
\D 3.4.2 logarithmic structure of toric variety 2.1.6 logarithmic tangent bundle of a logarithmic smooth object of B(log) 3.5.6

334

INDEX

logarithmic variation of polarized Hodge structure logarithmically smooth 2.1.11, 3.5.4 LVPH 2.4.9, 0.2.19 LVPH arising from geometry 0.2.21

2.4.9, 0.2.19

monoid 0.7.7 monoid Cmult 3.3.2 add monoid R≥0 2.6.2 mult monoid R≥0 3.3.2 morphism f log 2.2.4 n-level structure 2.5.2 neat 4.1.1, 0.4.1 nilpotent cone 1.3.5, 0.4.2 nilpotent i-orbit 1.3.7, 0.5.3 nilpotent orbit 1.3.7, 0.4.7 nilpotent orbit theorem 2.5.13, 2.5.14, 0.2.22 open subobject

3.6.1

parabolic subgroup 0.7.3 parabolic subgroup PT ,V associated with (T , V ) period domain 0.3.1 period map ϕˇ : X →
\Dˇ orb 2.5.3 period map ϕ : X →
\D 2.5.10, 0.4.28  period map ϕ log : Xlog →
\D 2.5.10 period map Xval →
\Dval 8.4.1 log  period map Xval →
\Dval 8.4.1 log

5.1.6

period map Xval →
\DSL(2) 8.4.1 period map ϕ of H with respect to  2.5.10, period map ϕ log of H with respect to  2.5.10, PH on X 0.3.5 PH on X of type 0 0.3.5 PH on X of type 1 0.3.6 PLH 2.4.7, 2.4.8, 0.4.22 PLH of type  2.5.8, 2.5.16 polarized Hodge structure 1.1.2, 0.1.8 polarized Hodge structure on X 0.3.5 polarized Hodge structure on X of type 0 0.3.5 polarized Hodge structure on X of type 1 0.3.6 polarized logarithmic Hodge structure 2.4.7, 2.4.8, 0.4.22 polarized logarithmic Hodge structure with coefficients in A 11.2.4, 0.4.33 polarized logarithmic Hodge structure of type  2.5.8, 2.5.16 polarized logarithmic Hodge structure of type  with coefficients in A 11.2.2 positivity 2.4.7, 0.4.22 positivity on x 0.4.22 pre-LH 2.4.1 pre-LMH 2.6.1 prelogarithmic Hodge structure 2.4.1 prelogarithmic mixed Hodge structure 2.6.1 prelogarithmic structure 2.1.1 pre-PLH 2.4.3, 0.4.21 prepolarized logarithmic Hodge structure 2.4.3, 0.4.21 proper 0.7.5 proper action 7.2.3

335

INDEX proper base change theorem push-out 2.1.1

A2

R-split mixed Hodge structure 6.1.2 R-splitting of (W, F ) 6.1.2 rank of SL(2)-orbit 5.2.3 rational cone 1.3.4 rational fan 1.3.4 rational family of weight filtrations associated with an SL(2)-orbit rational nilpotent cone 0.4.2 reductive Borel-Serre space 9.1.1 regular topological space 6.4.6 Riemann-Hodge bilinear relations 1.1.2, 0.1.8 ring 0.7.2 log ringed space (X log , OX ) 2.2.4

5.2.5

σ -nilpotent i-orbit 1.3.7 σ -nilpotent orbit 1.3.7 σC -torsor Eσ →
(σ )gp \Dσ 4.1.1 saturated monoid 0.2.11 separated 0.7.5 set of local monodromy cones of H in gR 2.5.11 sharp 1.3.1, 0.2.11, 0.7.7 sheaf-functor 8.2.5 sheaf of logarithmic vector fields of a logarithmic smooth object of B(log) 3.5.4 gp sheaf of logarithms L of MX 2.2.4 sheaf of rings of Eσ 3.4.1 sheaf of rings of
\D 3.4.2 log 2.2.4 sheaf of rings OX q sheaf ωX of logarithmic differential q-forms on X 2.1.7, 3.5.2 q,log sheaf ωX of logarithmic differential q-forms on X log 2.2.6, 3.5.2 Shimura variety 11.1.4 SL(2)-orbit 5.2.1 SL(2)-orbit determined by (W, r) 5.2.10 SL(2)-orbit of rank r 5.2.3, 0.5.11 SL(2)-orbit in n variables 5.2.1, 0.5.14 small Griffiths transversality 2.4.9, 0.4.22 soft sheaf A2.6 space D of nilpotent orbits in the direction of  1.3.8  space D of nilpotent i-orbits in the direction of  1.3.8 space DSL(2) of SL(2)-orbits 5.2.6 space Dval of valuative nilpotent orbits 5.3.3 space D,val of valuative nilpotent orbits in the directions over  5.3.5  space Dval of valuative nilpotent i-orbits 5.3.3  space D,val of valuative nilpotent i-orbits in the directions over  5.3.5 space DSL(2),val of valuative SL(2)-orbits 5.2.7 A of A-nilpotent orbits in the direction of  space D 11.1.5 ,A space D of A-nilpotent i-orbits in the direction of  11.1.5 specialization of F at s ∈ sp(y) (y ∈ x log ) 0.2.17, 2.4.6 specializations sp(y) over y ∈ x log 2.4.6 splitting of (W, F ) 6.1.2 strict morphism 2.1.10 strong topology 3.1.1, 0.4.15 strongly compatible 1.3.10, 0.4.10 submonoid 0.7.7

336

INDEX

subring 0.7.2 surjective homomorphism e : σC → torusσ

3.3.5

Theorem A 4.1.1 Theorem B 4.2.1 topology of
\D 3.4.2 topology of
(σ )gp \Dσ 3.4.2 topology of
\D,val 5.3.8 topology of
(σ )gp \Dσ,val 5.3.8  topology of D 3.4.3  topology of Dσ 3.4.3  topology of D,val 5.3.8 

topology of Dσ,val 5.3.8 topology of DBS 5.1.13 topology of DBS,val 5.1.13 topology of DSL(2) 5.2.13 topology of DSL(2),val 5.2.13  topology of
\DBS 9.1.2  topology of
\DBS,val 9.1.2 

topology of
\DSL(2),≤1 9.1.2 topology of Eσ,val 5.3.7  topology of Eσ,val 5.3.7 topology of XBS 5.1.13  topology of
\XBS 9.1.2 toric variety 3.3.2, 0.2.12 torus 3.3.2 trivial logarithmic structure 2.1.1 unipotent local monodromy 2.3.1 upper half plane 1.2.3, 0.3.2 upper half space 1.2.3, 0.3.2 valuative Borel-Serre space DBS,val 5.1.6 valuative submonoid 3.6.16 valuative toric variety 5.1.11, 5.3.6 variation of Hodge structure 0.1.10 variation of polarized Hodge structure 0.1.10 VH 0.1.10 VPH 0.1.10 (W, F ) splits over R 6.1.2 weak topology 3.1.1 weight filtration associated with a nilpotent endomorphism weight filtration associated with a nilpotent orbit 5.2.4 weight map 5.2.9, 6.1.1 (2) Weil operator 1.1.2 zeros in the new sense

0.4.17

5.2.4