Four paths from birational geometry to the elliptic genus

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.