Period mappings and period domains
著者
書誌事項
Period mappings and period domains
（Cambridge studies in advanced mathematics, 168）
Cambridge University Press, 2017
2nd ed
- : hardback
- : pbk
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注記
Includes bibliographical references (p. 540-555) and index
内容説明・目次
内容説明
This up-to-date introduction to Griffiths' theory of period maps and period domains focusses on algebraic, group-theoretic and differential geometric aspects. Starting with an explanation of Griffiths' basic theory, the authors go on to introduce spectral sequences and Koszul complexes that are used to derive results about cycles on higher-dimensional algebraic varieties such as the Noether-Lefschetz theorem and Nori's theorem. They explain differential geometric methods, leading up to proofs of Arakelov-type theorems, the theorem of the fixed part and the rigidity theorem. They also use Higgs bundles and harmonic maps to prove the striking result that not all compact quotients of period domains are Kahler. This thoroughly revised second edition includes a new third part covering important recent developments, in which the group-theoretic approach to Hodge structures is explained, leading to Mumford-Tate groups and their associated domains, the Mumford-Tate varieties and generalizations of Shimura varieties.
目次
- Part I. Basic Theory: 1. Introductory examples
- 2. Cohomology of compact Kahler manifolds
- 3. Holomorphic invariants and cohomology
- 4. Cohomology of manifolds varying in a family
- 5. Period maps looked at infinitesimally
- Part II. Algebraic Methods: 6. Spectral sequences
- 7. Koszul complexes and some applications
- 8. Torelli theorems
- 9. Normal functions and their applications
- 10. Applications to algebraic cycles: Nori's theorem
- Part III. Differential Geometric Aspects: 11. Further differential geometric tools
- 12. Structure of period domains
- 13. Curvature estimates and applications
- 14. Harmonic maps and Hodge theory
- Part IV. Additional Topics: 15. Hodge structures and algebraic groups
- 16. Mumford-Tate domains
- 17. Hodge loci and special subvarieties
- Appendix A. Projective varieties and complex manifolds
- Appendix B. Homology and cohomology
- Appendix C. Vector bundles and Chern classes
- Appendix D. Lie groups and algebraic groups
- References
- Index.
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