Mumford-Tate Groups and Domains

by Green, Mark; Griffiths, Phillip; Kerr, Matt

• ISBN13: 9780691154251
• ISBN10: 0691154252
• Format: Paperback
• Copyright: 2012-04-02
• Publisher: Princeton Univ Pr

Summary

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.

Table of Contents

Introduction	p. 1
Mumford-Tate Groups	p. 28
Hodge structures	p. 28
Mumford-Tate groups	p. 32
Mixed Hodge structures and their Mumford-Tate groups	p. 38
Period Domains and Mumford-Tate Domains	p. 45
Period domains and their compact duals	p. 45
Mumford-Tate domains and their compact duals	p. 55
Noether-Lefschetz loci in period domains	p. 61
The Mumford-Tate Group of a Variation of Hodge Structure	p. 67
The structure theorem for variations of Hodge structures	p. 69
An application of Mumford-Tate groups	p. 78
Noether-Lefschetz loci and variations of Hodge structure	p. 81
Hodge Representations and Hodge Domains	p. 85
Part I: Hodge representations	p. 86
The adjoint representation and characterization of which weights give faithful Hodge representations	p. 109
Examples: The classical groups	p. 117
Examples: The exceptional groups	p. 126
Characterization of Mumford-Tate groups	p. 132
Hodge domains	p. 149
Mumford-Tate domains as particular homogeneous complex manifolds	p. 168
Appendix: Notation from the structure theory of semi-simple Lie algebras	p. 179
Hodge Structures with Complex Multiplication	p. 187
Oriented number fields	p. 189
Hodge structures with special endomorphisms	p. 193
A categorical equivalence	p. 196
Polarization and Mumford-Tate groups	p. 198
An extended example	p. 202
Proofs of Propositions V.D.4 and V.D.5 in the Galois case	p. 209
Arithmetic Aspects of Mumford-Tate Domains	p. 213
Groups stabilizing subsets of D	p. 215
Decomposition of Noether-Lefschetz into Hodge orientations	p. 219
Weyl groups and permutations of Hodge orientations	p. 231
Galois groups and fields of definition	p. 234
Appendix: CM points in unitary Mumford-Tate domains	p. 239
Classification of Mumford-Tate Subdomains	p. 240
A general algorithm	p. 240
Classification of some CM-Hodge structures	p. 243
Determination of sub-Hodge-Lie-algebras	p. 246
Existence of domains of type IV(f)	p. 251
Characterization of domains of type IV(a) and IV(f)	p. 253
Completion of the classification for weight 3	p. 256
The weight 1 case	p. 260
Algebra-geometric examples for the Noether-Lefschetz-locus types	p. 265
Arithmetic of Period Maps of Geometric Origin	p. 269
Behavior of fields of definition under the period map - image and preimage	p. 270
Existence and density of CM points in motivic VHS	p. 275
Bibliography	p. 277
Index	p. 287
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