The Gerstenhaber product $\HH^2(A)\times \HH^2(A)\to \HH^3(A)$ of affine toric varieties

TL;DR

This paper provides a convex geometric interpretation of the Gerstenhaber product in Hochschild cohomology for affine toric varieties, showing it vanishes for Gorenstein toric surfaces and applying this to deformation theory.

Contribution

It introduces a convex geometric perspective on the Gerstenhaber product for affine toric varieties and extends previous results to non-isolated Gorenstein singularities.

Findings

Gerstenhaber product is zero for Gorenstein toric surfaces

Convex geometric interpretation of Hochschild cohomology

Explicit equations for versal base space in certain cases

Abstract

For an affine toric variety $\spec(A)$, we give a convex geometric interpretation of the Gerstenhaber product $\HH^2(A)\times \HH^2(A)\to \HH^3(A)$ between the Hochschild cohomology groups. In the case of Gorenstein toric surfaces we prove that the Gerstenhaber product is the zero map. As an application in commutative deformation theory we find the equations of the versal base space (in special lattice degrees) up to second order for not necessarily isolated toric Gorenstein singularities. Our construction reproves and generalizes results obtained in [1] and [13].