Stiefel-Whitney Numbers for Singular Varieties

TL;DR

This paper explores which Stiefel-Whitney numbers can be consistently defined for singular varieties, establishing invariance under flops and proposing a potential new elliptic genus for unoriented manifolds.

Contribution

It identifies the space of invariant Stiefel-Whitney numbers under classical flops and defines them for certain singular varieties, suggesting a new elliptic genus for unoriented manifolds.

Findings

Identified the F_2-vector space of invariant Stiefel-Whitney numbers.

Defined Stiefel-Whitney numbers for real projective normal Gorenstein varieties.

Proposed a connection to a new elliptic genus for unoriented manifolds.

Abstract

This paper determines which Stiefel-Whitney numbers can be defined for singular varieties compatibly with small resolutions. First an upper bound is found by identifying the F_2-vector space of Stiefel-Whitney numbers invariant under classical flops, equivalently by computing the quotient of the unoriented bordism ring by the total spaces of RP^3 bundles. These Stiefel-Whitney numbers are then defined for any real projective normal Gorenstein variety and shown to be compatible with small resolutions whenever they exist. In light of Totaro's result [Tot00] equating the complex elliptic genus with complex bordism modulo flops, equivalently complex bordism modulo the total spaces of twisted(CP^3) bundles, these findings can be seen as hinting at a new elliptic genus, one for unoriented manifolds.