Hochschild Cohomology and Deformation Quantization of Affine Toric Varieties

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 $T_{(i)}^1(A)$, 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.