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Cyclic Coverings, Calabi-Yau Manifolds and Complex Multiplication

  • Authors
  • Christian┬áRohde

Part of the Lecture Notes in Mathematics book series (LNM, volume 1975)

Table of contents

  1. Front Matter
    Pages 1-7
  2. Jan Christian Rohde
    Pages 1-9
  3. Jan Christian Rohde
    Pages 59-69
  4. Jan Christian Rohde
    Pages 71-78
  5. Jan Christian Rohde
    Pages 91-119
  6. Jan Christian Rohde
    Pages 157-167
  7. Jan Christian Rohde
    Pages 169-186
  8. Jan Christian Rohde
    Pages 199-208
  9. Back Matter
    Pages 1-24

About this book

Introduction

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, who have constructed a deformation of the Fermat quintic with a dense set of CM fibers by a tower of cyclic coverings. Using this method, new families of K3 surfaces with dense sets of CM fibers and involutions are obtained.

Secondly, the construction method of the Borcea-Voisin mirror family, which in the case of the author's examples yields families of Calabi-Yau 3-manifolds with dense sets of CM fibers, is also utilized. Moreover fibers with complex multiplication of these new families are also determined.

This book was written for young mathematicians, physicists and also for experts who are interested in complex multiplication and varieties with complex multiplication. The reader is introduced to generic Mumford-Tate groups and Shimura data, which are among the main tools used here. The generic Mumford-Tate groups of families of cyclic covers of the projective line are computed for a broad range of examples.

Keywords

Calabi-Yau manifolds Complex multiplication Shimura varieties Variations of Hodge structures manifold

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-642-00639-5
  • Copyright Information Springer-Verlag Berlin Heidelberg 2009
  • Publisher Name Springer, Berlin, Heidelberg
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-3-642-00638-8
  • Online ISBN 978-3-642-00639-5
  • Series Print ISSN 0075-8434
  • Series Online ISSN 1617-9692
  • Buy this book on publisher's site