Topology of singular algebraic varieties

TL;DR

This paper reviews recent advances in extending topological invariants from manifolds to singular algebraic varieties, highlighting developments like intersection homology, mixed Hodge theory, motivic integration, and the elliptic genus, and exploring their topological significance.

Contribution

It summarizes recent progress and new invariants in the topology of singular algebraic varieties, connecting algebraic, geometric, and topological perspectives.

Findings

Development of intersection homology and mixed Hodge theory for singular spaces

Introduction of motivic integration and elliptic genus as new invariants

Evidence of topological significance of algebraic invariants

Abstract

I will discuss recent progress by many people in the program of extending natural topological invariants from manifolds to singular spaces. Intersection homology theory and mixed Hodge theory are model examples of such invariants. The past 20 years have seen a series of new invariants, partly inspired by string theory, such as motivic integration and the elliptic genus of a singular variety. These theories are not defined in a topological way, but there are intriguing hints of their topological significance.