Math 582GA/583GA
Hodge Theory

Charles Doran

Winter/Spring 2005, Monday/Wednesday/Friday 10:30-11:20
This will be a two-quarter introduction to Hodge theory, with particular emphasis on its applications to algebraic geometry.

Outline:

  1. Motivating examples and background results
  2. Cohomology of compact Kaehler manifolds
  3. The Hodge decomposition
  4. Families and deformations: Kodaira-Spencer-Kuranishi theory
  5. Cohomology of manifolds varying in a family
  6. Variations of Hodge structure: Infinitesimal and global
  7. Period maps: Infinitesimal, algebraic, and differential geometric methods
  8. Hypersurfaces and the Torelli theorems
  9. Gauss-Manin connection and Picard-Fuchs equations
  10. Calabi-Yau hypersurfaces and introduction to mirror symmetry
Texts: C. Voisin, Hodge Theory and Complex Algebraic Geometry, I and II; J. Bertin, J.-P. Demailly, L. Illusie, and C. Peters, Introduction to Hodge Theory;J. Carlson, S. Muller-Stach, C. Peters, Period Mappings and Period Domains; C. H. Clemens and D. R. Morrison, eds. Variations on Hodge Structure; Y. Shimizu and K. Ueno, Advances in Moduli Theory.