Classifying Spaces of Degenerating Polarized Hodge Structures. (AM-169) ：Classifying Spaces of Degenerating Polarized Hodge Structures. (AM-169) ( Annals of Mathematics Studies )

Publication subTitle ：Classifying Spaces of Degenerating Polarized Hodge Structures. (AM-169)

Publication series ：Annals of Mathematics Studies

Author： Kato Kazuya;Usui Sampei;;  

Publisher： Princeton University Press‎

Publication year： 2008

E-ISBN: 9781400837113

P-ISBN(Paperback): 9780691138213

Subject： O187 algebraic geometry

Keyword： 数学

Language： ENG

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Description

In 1970, Phillip Griffiths envisioned that points at infinity could be added to the classifying space D of polarized Hodge structures. In this book, Kazuya Kato and Sampei Usui realize this dream by creating a logarithmic Hodge theory. They use the logarithmic structures begun by Fontaine-Illusie to revive nilpotent orbits as a logarithmic Hodge structure.

The book focuses on two principal topics. First, Kato and Usui construct the fine moduli space of polarized logarithmic Hodge structures with additional structures. Even for a Hermitian symmetric domain D, the present theory is a refinement of the toroidal compactifications by Mumford et al. For general D, fine moduli spaces may have slits caused by Griffiths transversality at the boundary and be no longer locally compact. Second, Kato and Usui construct eight enlargements of D and describe their relations by a fundamental diagram, where four of these enlargements live in the Hodge theoretic area and the other four live in the algebra-group theoretic area. These two areas are connected by a continuous map given by the SL(2)-orbit theorem of Cattani-Kaplan-Schmid. This diagram is used for the construction in the first topic.

Chapter

Cover

Title

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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 : D^♯val → DSL(2)

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) and Г\D^♭SL(2),≤1

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

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