Cyclic Coverings, Calabi-Yau Manifolds and Complex Multiplication

1. Front Matter

   Pages 1-7

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2. Introduction

   • Jan Christian Rohde

   Pages 1-9

3. An Introduction to Hodge Structures and Shimura Varieties

   • Jan Christian Rohde

   Pages 11-57

4. Cyclic Covers of the Projective Line

   • Jan Christian Rohde

   Pages 59-69

5. Some Preliminaries for Families of Cyclic Covers

   • Jan Christian Rohde

   Pages 71-78

6. The Galois Group Decomposition of the Hodge Structure

   • Jan Christian Rohde

   Pages 79-89

7. The Computation of the Hodge Group

   • Jan Christian Rohde

   Pages 91-119

8. Examples of Families with Dense Sets of Complex Multiplication Fibers

   • Jan Christian Rohde

   Pages 121-142

9. The Construction of Calabi-Yau Manifolds with Complex Multiplication

   • Jan Christian Rohde

   Pages 143-156

10. The Degree 3 Case

    • Jan Christian Rohde

    Pages 157-167

11. Other Examples and Variations

    • Jan Christian Rohde

    Pages 169-186

12. Examples of CMCY Families of 3-manifolds and their Invariants

    • Jan Christian Rohde

    Pages 187-198

13. Maximal Families of CMCY Type

    • Jan Christian Rohde

    Pages 199-208

14. Back Matter

    Pages 1-24

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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.

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

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