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MumfordTate Groups and Domains: Their Geometry and Arithmetic (Am183) (Annals of Mathematics Studies)by M. Green
Synopses & ReviewsPublisher Comments:MumfordTate 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 MumfordTate 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 MumfordTate 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 finitedimensional 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 MumfordTate domains by arithmetic groups. Bringing together complex geometry, representation theory, and arithmetic, this book opens up a fresh perspective on an important subject. Synopsis:MumfordTate 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 MumfordTate 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 MumfordTate 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 finitedimensional 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 MumfordTate domains by arithmetic groups. Bringing together complex geometry, representation theory, and arithmetic, this book opens up a fresh perspective on an important subject. Synopsis:MumfordTate 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 MumfordTate 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 MumfordTate 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 finitedimensional 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 MumfordTate domains by arithmetic groups. Bringing together complex geometry, representation theory, and arithmetic, this book opens up a fresh perspective on an important subject. About the AuthorMark 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.
Table of ContentsIntroduction 1
I MumfordTate Groups 28 I.A Hodge structures 28 I.B MumfordTate groups 32 I.C Mixed Hodge structures and their MumfordTate groups 38 II Period Domains and MumfordTate Domains 45 II.A Period domains and their compact duals 45 II.B MumfordTate domains and their compact duals 55 II.C NoetherLefschetz loci in period domains 61 III The MumfordTate Group of a Variation of Hodge Structure 67 III.A The structure theorem for variations of Hodge structures 69 III.B An application of MumfordTate groups 78 III.C NoetherLefschetz 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 MumfordTate groups 132 IV.F Hodge domains 149 IV.G MumfordTate 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 MumfordTate 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 MumfordTate Domains 213 VI.A Groups stabilizing subsets of D 215 VI.B Decomposition of NoetherLefschetz 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 MumfordTate domains 239 VII Classification of MumfordTate Subdomains 240 VII.A A general algorithm 240 VII.B Classification of some CMHodge structures 243 VII.C Determination of subHodgeLiealgebras 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 Algebrogeometric examples for the NoetherLefschetzlocus 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 What Our Readers Are SayingBe the first to add a comment for a chance to win!Product Details
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