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

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.

Mark Green is professor of mathematics at the University of California, Los Angeles and is Director Emeritus of the Institute for Pure and Applied Mathematics. Phillip A. Griffiths is Professor Emeritus of Mathematics and former director at the Institute for Advanced Study in Princeton. Matt Kerr is assistant professor of mathematics at Washington University in St. Louis.