# Hochschild Cohomology and Deformation Quantization of Affine Toric Varieties

## TL;DR

This paper provides a convex geometric description of Hochschild cohomology for affine toric varieties, computes dimensions of Hodge summands, and proves the quantizability of Poisson structures on such varieties.

## Contribution

It introduces a geometric framework for Hochschild cohomology of affine toric varieties and establishes the quantization of Poisson structures in this setting.

## Key findings

- Convex geometric description of Hochschild cohomology.
- Explicit computation of Hodge summand dimensions.
- Proof of deformation quantization for Poisson structures.

## Abstract

For an affine toric variety $\mathrm{Spec}(A)$, we give a convex geometric description of the Hodge decomposition of its Hochschild cohomology. Under certain assumptions we compute the dimensions of the Hodge summands $T^1_{(i)}(A)$, generalizing the existing results about the Andre-Quillen cohomology group $T^1_{(1)}(A)$. We prove that every Poisson structure on a possibly singular affine toric variety can be quantized in the sense of deformation quantization.