Mumford-Tate Groups and Domains

Mark Green; Phillip A. Griffiths; Matt Kerr

Mark Green, Phillip A. Griffiths and Matt Kerr

Mumford-Tate Groups and Domains

Their Geometry and Arithmetic (AM-183)

In: Annals of Mathematics Studies, 183

Princeton University Press | 2012

https://doi.org/10.1515/9781400842735

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Contents

About this book

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.

Author information

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.

Reviews

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

Frontmatter Free access PDF PDF	i
Contents Free access PDF PDF	v
Introduction Access restricted Content is available PDF PDF	1
Chapter I. Mumford-Tate Groups Access restricted Content is available PDF PDF	28
Chapter II. Period Domains and Mumford-Tate Domains Access restricted Content is available PDF PDF	45
Chapter III. The Mumford-Tate Group of a Variation of Hodge Structure Access restricted Content is available PDF PDF	67
Chapter IV. Hodge Representations and Hodge Domains Access restricted Content is available PDF PDF	85
Chapter V. Hodge Structures With Complex Multiplication Access restricted Content is available PDF PDF	187
Chapter VI. Arithmetic Aspects of Mumford-Tate Domains Access restricted Content is available PDF PDF	213
Chapter VII. Classification of Mumford-Tate Subdomains Access restricted Content is available PDF PDF	240
Chapter VIII. Arithmetic of Period Maps of Geometric Origin Access restricted Content is available PDF PDF	269
Index Access restricted Content is available PDF PDF	287

(Deutsch)

Mumford-Tate Groups and Domains

Their Geometry and Arithmetic (AM-183)

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