{"product_id":"mumfordtate-groups-and-domains-9780691154244","title":"MumfordTate Groups and Domains","description":"\u003cb\u003eBook Synopsis\u003c\/b\u003e\u003cbr\u003eMumford-Tate groups are the fundamental symmetry groups of Hodge theory, a subject which rests at the center of contemporary complex algebraic geometry. This book provides a comprehensive exploration of Mumford-Tate groups and domains.\u003cbr\u003e\u003cbr\u003e\u003cb\u003eTrade Review\u003c\/b\u003e\u003cbr\u003e\"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\u003cbr\u003e\u003cbr\u003e\u003cb\u003eTable of Contents\u003c\/b\u003e\u003cbr\u003eIntroduction 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","brand":"Princeton University Press","offers":[{"title":"Default Title","offer_id":49403790295383,"sku":"9780691154244","price":170.0,"currency_code":"GBP","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0817\/1739\/5799\/files\/9780691154244.jpg?v=1730484551","url":"https:\/\/bookcurl.com\/products\/mumfordtate-groups-and-domains-9780691154244","provider":"Book Curl","version":"1.0","type":"link"}