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Full Description
The period matrix of a curve effectively describes how the complex structure varies; this is Torelli's theorem dating from the beginning of the nineteenth century. In the 1950s during the first revolution of algebraic geometry, attention shifted to higher dimensions and one of the guiding conjectures, the Hodge conjecture, got formulated. In the late 1960s and 1970s Griffiths, in an attempt to solve this conjecture, generalized the classical period matrices introducing period domains and period maps for higher-dimensional manifolds. He then found some unexpected new phenomena for cycles on higher-dimensional algebraic varieties, which were later made much more precise by Clemens, Voisin, Green and others. This 2003 book presents this development starting at the beginning: the elliptic curve. This and subsequent examples (curves of higher genus, double planes) are used to motivate the concepts that play a role in the rest of the book.
Contents
Part I. Basic Theory of the Period MapCohomology 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. The Period Map: Algebraic Methods: 6. Spectral sequences; 7. Koszul complexes and some applications; 8. Further applications: Torelli theorems for hypersurfaces; 9. Normal functions and their applications; 10. Applications to algebraic cycles: Nori's theorem; Part III: Differential Geometric Methods: 11. Further differential geometric tools; 12. Structure of period domains; 13. Curvature estimates and applications; 14. Harmonic maps and Hodge theory; Appendix A. Projective varieties and complex manifolds; Appendix B. Homology and cohomology; Appendix C. Vector bundles and Chern classes.
