Classifying spaces of degenerating polarized Hodge structures / Kazuya Kato and Sampei Usui.

Format:

Book

Author/Creator:

Kato, K. (Kazuya), author.

Usui, Sampei, author.

Series:

Annals of mathematics studies ; Number 169.

Annals of Mathematics Studies ; Number 169.

Language:

English

Subjects (All):

Hodge theory.

Logarithms.

Physical Description:

1 online resource (349 pages)

Edition:

Course Book

Place of Publication:

Princeton, New Jersey ; Oxfordshire, England : Princeton University Press, 2009.

Language Note:

English

Summary:

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.

Contents:

Frontmatter

Contents

Introduction

Chapter 0. Overview

Chapter 1. Spaces of Nilpotent Orbits and Spaces of Nilpotent i-Orbits

Chapter 2. Logarithmic Hodge Structures

Chapter 3. Strong Topology and Logarithmic Manifolds

Chapter 4. Main Results

Chapter 5. Fundamental Diagram

Chapter 6. The Map ψ:D#val DSL(2)

Chapter 7. Proof of Theorem A

Chapter 8. Proof of Theorem B

Chapter 9. b-Spaces

Chapter 10. Local Structures of DSL(2) and ΓDbSL(2),≤1

Chapter 11. Moduli of PLH with Coefficients

Chapter 12. Examples and Problems

Appendix

References

List of Symbols

Index

Notes:

Description based upon print version of record.

Includes bibliographical references and index.

Description based on print version record.

ISBN:

9781400837113

1400837111

9780691138220

0691138222

OCLC:

979582209