Christian Rohde (auth.)3642006388, 9783642006388
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.
Table of contents :
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
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