Representations of Fundamental Groups of Algebraic Varieties
Using harmonic maps, non-linear PDE and techniques from algebraic geometry this book enables the reader to study the relation between fundamental groups and algebraic geometry invariants of algebraic varieties. The reader should have a basic knowledge of
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1708
Springer Berlin Heidelberg New York Barcelona Hong Kong London Milan Paris Singapore Tokyo
Kang Zuo
Representations of Fundamental Groups of Algebraic Varieties
Springer
Author Kang Zuo Fachbereich Mathematik Universitat Kaiserslautern Postfach 3049 67653 Kaiserslautern, Germany E-mail: [email protected]
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Die Deutsche Bibliothek - CIP-Einheitsaufnahme
Zuo, Kang: Representations of fundamental groups of algebraic varietes / Kang Zuo. - Berlin; Heidelberg; New York; Barcelona; Hong Kong ; London; Milan; Paris; Singapore; Tokyo: Springer, 1999 (Lecture notes in mathematics; 1708) ISBN 3-540-66312-6
Mathematics Subject Classification (1991): 14H30, 14Jxx, 14160,32125, 58E20 ISSN 0075-8434 ISBN 3-540-66312-6 Springer-Verlag Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1999 Printed in Germany
Typesetting: Camera-ready TEX output by the author Printed on acid-free paper SPIN: 10650239 41/3143-543210
To my mother
Contents 1 Introduction 2 Preliminaries
3
1
10
2.1 Review of Algebraic groups over arbitrary fields
10
2.2 Representations of fundamental groups and Moduli spaces
12
2.3 p-adic norm on a vector space and Bruhat-Tits buildings .
20
Harmonic metrics on flat vector bundles
25
3.1
Pluriharmonic maps of finite energy
25
3.2
Pluriharmonic maps of possibly infinite energy but with controlled growth at infinity . . . . . . . . . . . . . . . . . . . . . . ..
41
4 N on-abelian Hodge theory, factorization theorems for non rigid or p-adic unbounded representations 52 4.1 4.2 4.3 4.4 4.5
Higgs bundles for archimedean representations and equivariant holomorphic L-forms for p-adic representations. . . . . ..
52
Albanese maps and a Lefschetz type theorem for holomorphic I-forms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..
63
Factorizations for nonrigid representations into almost simple complex algebraic groups. . . . . . . . . . . . . . . . . . . . . . ..
83
Factorizations for p-adic unbounded representations into almost simple p-adic algebraic groups . . . . . . . . . . . . . . . ..
97
Simpson's construction of families of nonrigid representations
101
5 Shafarevich maps for representations of fundamental groups, Kodaira dimension and Chern-hyperbolicity of Shafarevich varieties 104 104
5.1
Shafarevich maps and general discussions . . . . . . . .
5.2
Constructing automorphic forms via equivariant plurih
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