Mumford-Tate Groups and Domains ：Their Geometry and Arithmetic (AM-183) ( Annals of Mathematics Studies )

Publication subTitle ：Their Geometry and Arithmetic (AM-183)

Publication series ：Annals of Mathematics Studies

Author： Green Mark;Griffiths Phillip A.;Kerr Matt;  

Publisher： Princeton University Press‎

Publication year： 2012

E-ISBN: 9781400842735

P-ISBN(Paperback): 9780691154244

Subject： O187 algebraic geometry

Keyword： 数学

Language： ENG

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Description

Mumford-Tate groups are the fundamental symmetry groups of Hodge theory, a subject which rests at the center of contemporary complex algebraic geometry. This book is the first comprehensive exploration of Mumford-Tate groups and domains. Containing basic theory and a wealth of new views and results, it will become an essential resource for graduate students and researchers.

Although Mumford-Tate groups can be defined for general structures, their theory and use to date has mainly been in the classical case of abelian varieties. While the book does examine this area, it focuses on the nonclassical case. The general theory turns out to be very rich, such as in the unexpected connections of finite dimensional and infinite dimensional representation theory of real, semisimple Lie groups. The authors give the complete classification of Hodge representations, a topic that should become a standard in the finite-dimensional representation theory of noncompact, real, semisimple Lie groups. They also indicate that in the future, a connection seems ready to be made between Lie groups that admit discrete series representations and the study of automorphic cohomology on quotients of Mumford-Tate domains by arithmetic groups. Bringing together complex geometry, representation theory, and arithmetic, this book opens up a fresh perspective on an important subject.

Chapter

Cover

Title

Copyright

Contents

Introduction

I: Mumford-Tate Groups

I.A: Hodge structures

I.B: Mumford-Tate groups

I.C: Mixed Hodge structures and their Mumford-Tate groups

II: Period Domains and Mumford-Tate Domains

II.A: Period domains and their compact duals

II.B: Mumford-Tate domains and their compact duals

II.C: Noether-Lefschetz loci in period domains

III: The Mumford-Tate Group of a Variation of Hodge Structure

III.A: The structure theorem for variations of Hodge structures

III.B: An application of Mumford-Tate groups

III.C: Noether-Lefschetz loci and variations of Hodge structure

IV: Hodge Representations and Hodge Domains

IV.A: Part I: Hodge representations

IV.B: The adjoint representation and characterization of which weights give faithful Hodge representations

IV.C: Examples: The classical groups

IV.D: Examples: The exceptional groups

IV.E: Characterization of Mumford-Tate groups

IV.F: Hodge domains

IV.G: Mumford-Tate domains as particular homogeneous complex manifolds

Appendix: Notation from the structure theory of semi-simple Lie algebras

V: Hodge Structures with Complex Multiplication

V.A: Oriented number fields

V.B: Hodge structures with special endomorphisms

V.C: A categorical equivalence

V.D: Polarization and Mumford-Tate groups

V.E: An extended example

V.F: Proofs of Propositions V.D.4 and V.D.5 in the Galois case

VI: Arithmetic Aspects of Mumford-Tate Domains

VI.A: Groups stabilizing subsets of D

VI.B: Decomposition of Noether-Lefschetz into Hodge orientations

VI.C: Weyl groups and permutations of Hodge orientations

VI.D: Galois groups and fields of definition

Appendix: CM points in unitary Mumford-Tate domains

VII: Classification of Mumford-Tate Subdomains

VII.A: A general algorithm

VII.B: Classification of some CM-Hodge structures

VII.C: Determination of sub-Hodge-Lie-algebras

VII.D: Existence of domains of type IV(f)

VII.E: Characterization of domains of type IV(a) and IV(f)

VII.F: Completion of the classification for weight 3

VII.G: The weight 1 case

VII.H: Algebro-geometric examples for the Noether-Lefschetz-locus types

VIII: Arithmetic of Period Maps of Geometric Origin

VIII.A: Behavior of fields of definition under the period map — image and preimage

VIII.B: Existence and density of CM points in motivic VHS

Bibliography

Index

A

B

C

D

E

F

G

H

I

K

L

M

N

O

P

Q

R

S

T

U

V

W

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