Cyclic coverings, Calabi-Yau manifolds and complex multiplication /
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| Author / Creator: | Rohde, Jan Christian. |
|---|---|
| Imprint: | Dordrecht ; London : Springer, 2009. |
| Description: | ix, 228 p. : ill. ; 24 cm. |
| Language: | English |
| Series: | Lecture notes in mathematics, 0075-8434 ; 1975 Lecture notes in mathematics (Springer-Verlag) ; 1975. |
| Subject: | |
| Format: | Print Book |
| URL for this record: | http://pi.lib.uchicago.edu/1001/cat/bib/7727037 |
Table of Contents:
- Introduction
- 1. An introduction to Hodge structures and Shimura varieties
- 1.1. The basic definitions
- 1.2. Jacobians, Polarizations and Riemann's Theorem
- 1.3. The definition of the Shimura datum
- 1.4. Hermitian symmetric domains
- 1.5. The construction of Shimura varieties
- 1.6. The definition of complex multiplication
- 1.7. Criteria and conjectures for complex multiplication
- 2. Cyclic covers of the projective line
- 2.1. Description of a cyclic cover of the projective line
- 2.2. The local system corresponding to a cyclic cover
- 2.3. The cohomology of a cover
- 2.4. Cyclic covers with complex multiplication
- 3. Some preliminaries for families of cyclic covers
- 3.1. The generic Hodge group
- 3.2. Families of covers of the projective line
- 3.3. The homology and the monodromy representation
- 4. The Galois group decomposition of the Hodge structure
- 4.1. The Galois group representation on the first cohomology
- 4.2. Quotients of covers and Hodge group decomposition
- 4.3. Upper bounds for the Mumford-Tate groups of the direct summands
- 4.4. A criterion for complex multiplication
- 5. The computation of the Hodge group
- 5.1. The monodromy group of an eigenspace
- 5.2. The Hodge group of a general direct summand
- 5.3. A criterion for the reaching of the upper bound
- 5.4. The exceptional cases
- 5.5. The Hodge group of a universal family of hyperelliptic curves
- 5.6. The complete generic Hodge group
- 6. Examples of families with dense sets of complex multiplication fibers
- 6.1. The necessary condition SINT
- 6.2. The application of SINT for the more complicated cases
- 6.3. The complete lists of examples
- 6.4. The derived variations of Hodge structures
- 7. The construction of Calabi-Yau manifolds with complex multiplication
- 7.1. The basic construction and complex multiplication
- 7.2. The Borcea-Voisin tower
- 7.3. The Viehweg-Zuo tower
- 7.4. A new example
- 8. The degree 3 case
- 8.1. Prelude
- 8.2. A modified version of the method of Viehweg and Zuo
- 8.3. The resulting family and its involutions
- 9. Other examples and variations
- 9.1. The degree 3 case
- 9.2. Calabi-Yau 3-manifolds obtained by quotients of degree 3
- 9.3. The degree 4 case
- 9.4. Involutions on the quotients of the degree 4 example
- 9.5. The extended automorphism group of the degree 4 example
- 9.6. The automorphism group of the degree 5 example by Viehweg and Zuo
- 10. Examples of CMCY families of 3-manifolds and their invariants
- 10.1. The length of the Yukawa coupling
- 10.2. Examples obtained by degree 2 quotients
- 10.3. Examples obtained by degree 3 quotients
- 10.4. Outlook onto quotients by cyclic groups of high order
- 11. Maximal families of CMCY type
- 11.1. Facts about involutions and quotients of K3-surface
- 11.2. The associated Shimura datum
- 11.3. The examples
- A. Examples of Calabi-Yau 3-manifolds with complex multiplication
- A.1. Construction by degree 2 coverings of a ruled surface
- A.2. Construction by degree 2 coverings of P2
- A.3. Construction by a degree 3 quotient
- References
- Index