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.

• 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

"The brilliance of the results and their broad spectrum of their applications makes this book an outstanding piece. Yet, there is more to write and to develop: the authors suggest the existence of future lines of research for a next book."--Jonathan Sanchez Hernandez, European Mathematical Society