Introduction to intersection cohomology

Summary

Introduction

Natural examples of singular varieties often arise in the study of algebraic groups. For example, if G is a connected reductive group with a Borel subgroup B, then B acts by left multiplication on the flag variety G/B with orbits BwB/B indexed by the elements w ∈ W of the Weyl group. Although the variety BwB/B (called a Bruhat cell) is nonsingular, its closure in G/B (called a Schubert cell) is usually a singular variety.

Homology and cohomology have long been powerful tools for the study of complex algebraic varieties (and other topological spaces), and when ℓ-adic cohomology was introduced by Grothendieck to tackle the Weil conjectures, this provided a corresponding tool for the study of algebraic varieties over fields of prime characteristic.

However, ordinary (co)homology of manifolds and algebraic varieties works better when they are nonsingular. For example, many theorems and techniques such as Poincaré duality and Hodge theory do not work for singular varieties. In the early 1980's Goresky and MacPherson defined a new kind of homology, called intersection homology, which is identical to ordinary homology for nonsingular varieties, but is better for singular varieties since it does have desirable properties such as Poincaré duality. Since then this new tool, and developments of it such as ℓ-adic intersection cohomology, have been used to great effect in the study of algebraic groups, most notably in the work of Lusztig.