SPRING 2003

OFFICE HOURS: TU-TH: 11:15-12:45 MAT P-143.

This is a continuation of MAT 614 Fall 2002. However, I will be glad to assist those interested in catching-up.

The main goal is to overview some of the topological and geometric methods needed to study algebraic varieties and maps.

We will prove a version of the so-called Decomposition Theorem. This is the most general result known about the homology of maps between algebraic or Kaehler varieties. It ties the topology of a map to a fundamental invariant of the target space: intersection cohomology.

Intersection cohomology is Goresky-MacPherson's generalization of singular cohomology to singular spaces. Its basic property is that it satisfies Poincar\'e Duality. Cohomology does not.

There will be strong emphasis on concrete examples from topology and complex geometry to see how the methods work in concrete situations.

The following material will be covered (possibly, with minor omissions due to lack of time).

Derived categories, the six operations, Poincare'-Verdier duality. (Survey style.)

Line bundles/divisors/hypersurfaces, Bertini's Theorem, first Chern class, the Weak Lefschetz Theorem, the Kodaira Vanishing Theorem, pencils, monodromy, local systems.

Intersection cohomology, the Decomposition Theorem of Beilinson-Bernstein-Deligne-Gabber. (Survey style.)

Borel-Moore homology, refined cup-products, the attaching triangle and interpretation.

Semismall maps, the Lefschetz-Hodge-Kodaira package and the decomposition theorem.

I will encourage students to give-in class presentations (perhaps, the best way to learn the material) and will assist them in preparing them. See last semester's presentations.

In-class presentations.
02/13/03, I. Unal: The First Bertini Theorem.

A list of papers for topics for a presentation. Two people should jointly present each topic. A. Grothendieck, La theorie des classes de Chern, Bull. Soc. Math. France, 86, 1958, 137-154. M. Atiyah, Vector bundles over an elliptic curve, Proc. London Math. Soc. (3) 7 1957, 414-452. M. Goresky-R. MacPherson, Intersection Homology Theory, Topology, 19 1980, no.2, 135-162.