Period Mappings and Period Domains

An introduction to Griffiths' theory of period maps and domains, focused on algebraic, group-theoretic and differential geometric aspects.

James Carlson (Author), Stefan Müller-Stach (Author), Chris Peters (Author)

9781108422628, Cambridge University Press

Hardback, published 24 August 2017

0 pages, 35 b/w illus. 3 tables 180 exercises
23.8 x 16 x 4.2 cm, 1.03 kg

'This monograph provides an excellent introduction to Hodge theory and its applications to complex algebraic geometry.' Gregory Pearlstein, Nieuw Archief voor Weskunde

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 Kähler. 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.

Part I. Basic Theory: 1. Introductory examples
2. Cohomology of compact Kähler 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.

Subject Areas: Algebraic geometry [PBMW], Complex analysis, complex variables [PBKD], Real analysis, real variables [PBKB]