Cyclic Coverings, Calabi-Yau Manifolds and Complex Multiplication
The main goal of this book is the construction of families of Calabi-Yau 3-manifolds with dense sets of complex multiplication fibers. The new families are determined by combining and generalizing two methods. Firstly, the method of E. Viehweg and K. Zuo,
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1975
Jan Christian Rohde
Cyclic Coverings, Calabi-Yau Manifolds and Complex Multiplication
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Jan Christian Rohde Institut fuer Algebraische Geometrie Leibniz Universität Hannover Welfengarten 1, GRK 1463 30167 Hannover Germany [email protected]
ISSN 0075-8434 e-ISSN 1617-9692 ISBN 978-3-642-00638-8 e-ISBN 978-3-642-00639-5 DOI 10.1007/978-3-642-00639-5 Springer Dordrecht Heidelberg London New York Library of Congress Control Number: PCN applied for Mathematics Subject Classification (2000): 14D07, 14G35, 14J32 c Springer-Verlag Berlin Heidelberg 2009 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, reuse of illustrations, recitation, broadcasting, reproduction on microfilm 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. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, 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. Cover design: SPi Publisher Services Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)
Preface
Calabi-Yau manifolds have been an object of extensive research during the last two decades. One of the reasons is the importance of Calabi-Yau 3-manifolds in modern physics - notably string theory. An interesting class of Calabi-Yau manifolds is given by those with complex multiplication (CM ). Calabi-Yau manifolds with CM are also of interest in theoretical physics, e.g. in connection with mirror symmetry and black hole attractors. It is the main aim of this book to construct families of Calabi-Yau 3-manifolds with dense sets of fibers with complex multiplication. Most examples in this book are constructed using families of curves with dense sets of fibers with CM . The contents of this book can roughly be divided into two parts. The first six chapters deal with families of curves with dense sets of CM fibers and introduce the necessary theoretical background. This includes among other things several aspects of Hodge theory and Shimura varieties. Using the first part, families of Calabi-Yau 3-manifolds with dense sets of fibers with CM are constructed in the remaining five chapters. In the appendix one finds examples of Calabi-Yau 3-manifolds with complex multiplication which are not necessarily fibers of a family with a dense set of CM fibers. The author hopes to have succeeded in writing a readable book that can also be used by non-specialists. On the other hand the expert will find new results about variations of Hodge structures and new examples of families of curves and Calab
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