Hochschild cohomology and deformation quantization of affine toric varieties

Abstract

For an affine toric variety 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 André–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.