Compactifying Moduli Spaces

This book focusses on a large class of objects in moduli theory and provides different perspectives from which compactifications of moduli spaces may be investigated.Three contributions give an insight on particular aspects of moduli problems. In the firs

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Paul Hacking Radu Laza Dragos Oprea

Compactifying Moduli Spaces

Advanced Courses in Mathematics CRM Barcelona Centre de Recerca Matemàtica Managing Editor: Enric Ventura

More information about this series at http://www.springer.com/series/5038

Paul Hacking • Radu Laza • Dragos Oprea

Compactifying Moduli Spaces Editors for this volume: Gilberto Bini, Università degli Studi di Milano Martí Lahoz, Université Paris Diderot – Paris 7 Emanuele Macrì, Northeastern University Paolo Stellari, Università degli Studi di Milano

Paul Hacking Department of Mathematics University of Massachusetts Amherst, MA, USA

Radu Laza School of Mathematics Institute for Advanced Study Princeton, NJ, USA

Dragos Oprea Department of Mathematics University of California La Jolla, CA, USA

ISSN 2297-0312 (electronic) ISSN 2297-0304 Advanced Courses in Mathematics - CRM Barcelona ISBN 978-3-0348-0920-7 ISBN 978-3-0348-0921-4 (eBook) DOI 10.1007/978-3-0348-0921-4 Library of Congress Control Number: 2015960183 Mathematics Subject Classification (2010): 14D06 Springer Basel Heidelberg New York Dordrecht London © Springer Basel 2016 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. 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. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. Printed on acid-free paper Springer Basel AG is part of Springer Science+Business Media (www.birkhauser-science.com)

Contents

Foreword . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

vii

1 Perspectives on the Construction and Compactification of Moduli Spaces Radu Laza Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 The GIT approach to constructing moduli spaces . 1.1.1 Basic GIT and moduli . . . . . . . . . . . . 1.1.2 Applications of GIT to moduli . . . . . . . 1.2 Moduli and periods . . . . . . . . . . . . . . . . . . 1.2.1 Period maps . . . . . . . . . . . . . . . . . . 1.2.2 Applications of locally symmetric varieties . 1.2.3 Comparison to GIT compactifications . . . 1.3 The KSBA approach to moduli spaces . . . . . . . 1.3.1 The KSBA approach . . . . . . . . . . . . . 1.