Modern aspects of the cohomological study of varieties

  • Meeting times: MWF 11-11:50, SEO 636
  • First meeting: Monday, August 22, 2011
  • Office hours: M, F 12-1, or by appointment
Course description: I will describe various results on the singular and coherent cohomology of smooth projective varieties, with emphasis on relatively recent techniques. I will begin by quickly reviewing more classical aspects of the topology of algebraic varieties and Hodge theory, together with their applications to important cohomological results like the Kodaira-Nakano vanishing theorem. Subsequently, I will address more recent theorems on the Betti and Hodge numbers of varieties over the complex numbers (for instance that birational Calabi-Yau manifolds have the same Hodge numbers). Part of this story is related to p-adic integration and the Weil conjectures, and will naturally take us through the positive characteristic and finite fields world as well. Further techniques that will appear will be chosen from among motivic integration, derived categories, and module theory over exterior algebras.
Lecture Notes: