Cyclic Coverings, Calabi-Yau Manifolds and Complex Multiplication (Lecture Notes in Mathematics Book 1975)

Annotation 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 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?bers with complex multiplication. Most - amples in this book are constructed using families of curves with dense sets of?bers with CM. The contents of this book can roughly be divided into two parts. The?rst six chapters deal with families of curves with dense sets of CM?bers and introduce the necessary theoretical background. This includes among other things several aspects of Hodge theory and Shimura varieties. Using the?rst part, families of Calabi-Yau 3-manifolds with dense sets of?bers withCM are constructed in the remaining?ve chapters. In the appendix one?nds examples of Calabi-Yau 3-manifolds with complex mul- plication which are not necessarily?bers of a family with a dense set ofCM?bers. The author hopes to have succeeded in writing a readable book that can also be used by non-specialists. Front Matter....Pages 1-7 Introduction....Pages 1-9 An Introduction to Hodge Structures and Shimura Varieties....Pages 11-57 Cyclic Covers of the Projective Line....Pages 59-69 Some Preliminaries for Families of Cyclic Covers....Pages 71-78 The Galois Group Decomposition of the Hodge Structure....Pages 79-89 The Computation of the Hodge Group....Pages 91-119 Examples of Families with Dense Sets of Complex Multiplication Fibers....Pages 121-142 The Construction of Calabi-Yau Manifolds with Complex Multiplication....Pages 143-156 The Degree 3 Case....Pages 157-167 Other Examples and Variations....Pages 169-186 Examples of CMCY Families of 3-manifolds and their Invariants....Pages 187-198 Maximal Families of CMCY Type....Pages 199-208 Back Matter....Pages 1-24 Jan Christian Rohde. Revision Of Work Originally Presented As The Author's Doctoral Thesis-- Universität Duisburg-essen, 2007. Includes Bibliographical References (p. 223-225) And Index.