Period Maps and Torelli Problems
In this section we take a more geometric point of view on the orthogonal decomposition $$\mathbf{\tilde{H}}_{-l_{\Gamma }} =\displaystyle\bigoplus _{ p=0}^{l_{\Gamma }-1}{\mathbf{H}}^{p}$$ resulting from (2.51).
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Igor Reider
Nonabelian Jacobian of Projective Surfaces Geometry and Representation Theory
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Igor Reider Universit´e d’Angers Angers, France
ISBN 978-3-642-35661-2 ISBN 978-3-642-35662-9 (eBook) DOI 10.1007/978-3-642-35662-9 Springer Heidelberg New York Dordrecht London Lecture Notes in Mathematics ISSN print edition: 0075-8434 ISSN electronic edition: 1617-9692 Library of Congress Control Number: 2013932343 Mathematics Subject Classification (2010): 14J60, 14C05, 16G30 c Springer-Verlag Berlin Heidelberg 2013 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’s location, in its current version, and permission for use must always be obtained from Springer. Permissions for use may be obtained through RightsLink at the Copyright Clearance Center. Violations are liable to prosecution under the respective Copyright Law. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)
Preface
The monograph studies representation theoretic aspects of a nonabelian version of the Jacobian for a smooth complex projective surface X introduced in [R1]. The sheaf of reductive Lie algebras GQ associated to the nonabelian Jacobian is determined and its Lie algebraic properties are explicitly related to the geometry of configurations of points on X . In particular, it is shown that the subsheaf of centres of GQ determines a distinguished decomposition of configurations into the disjoint union of subconfigurations. Furthermore, it is shown how to use sl2 subalgebras associated to certain nilpotent elements of GQ to write equat
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