The positivity of local equivariant Hirzebruch class for toric varieties

TL;DR

This paper proves the positivity of local equivariant Hirzebruch classes for all toric varieties, clarifying their geometric properties and correcting previous misconceptions about potential counterexamples.

Contribution

It establishes the positivity of local equivariant Hirzebruch classes for all toric varieties, resolving a previously claimed counterexample.

Findings

Proves positivity of local Hirzebruch classes for all toric varieties.

Provides a formula for local classes in terms of the fan.

Refutes Weber's alleged counterexample.

Abstract

The central object of investigation of this paper is the Hirzebruch class, a deformation of the Todd class, given by Hirzebruch (for smooth varieties) in his celebrated book "Topological Methods in Algebraic Geometry". The generalization for singular varieties is due to Brasselet-Sch\"urmann-Yokura. Following the work of Weber, we investigate its equivariant version for (possibly singular) toric varieties. The local decomposition of the Hirzebruch class to the fixed points of the torus action and a formula for the local class in terms of the defining fan are mentioned. After this review part, we prove the positivity of local Hirzebruch classes for all toric varieties, thus proving false the alleged counterexample given by Weber.