Four paths from birational geometry to the elliptic genus

TL;DR

This paper explores four reasons why the elliptic genus uniquely extends to singular spaces and proves its invariance under specific geometric modifications, using equivariant cohomology techniques.

Contribution

It demonstrates that the elliptic characteristic class is essentially the only invariant under certain modifications, extending Totaro's cobordism ring results with a new local calculus approach.

Findings

Elliptic genus is the most general characteristic class for singular spaces.

Elliptic characteristic class is invariant under Atiyah flop, Grassmannian flops, and Bott-Samelson modifications.

The approach uses local calculus in equivariant cohomology.

Abstract

The article presents four reasons why the elliptic genus is the most general characteristic class that admits a generalization to singular spaces. We prove that the elliptic characteristic class (with an additional factor) is essentially the only characteristic class invariant under certain modifications, such as the Atiyah flop, Grassmannian flops, and modifications of Bott-Samelson resolutions.This result confirms and extends Totaro's result concerning the cobordism ring modulo classical flops. However, our approach is based on local calculus in equivariant cohomology.