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 1 (i) (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.

Hodge theory and deformations of affine cones of subcanonical projective varieties

“…, see, e.g., [17,Lemma 3.2]. Since X is also arithmetically Cohen-Macaulay, we have depth 0 A X ≥ n and this, following SGA 2 Exposè VI, [22], implies that the inclusion U X ֒→ A X induces an isomorphism Ext…”

A differential graded Lie algebra controlling the Poisson deformations of an affine Poisson variety

“…where the later is the degree R part of the (higher) André-Quillen cohomology group T n−i,R (i) (A) (see [3,Section 4]). We will not use general André-Quillen cohomology theory, we will only use the well-known isomorphism T n−i (i) (A) ∼ = H n (i) (A) for n ≥ i (see e.g.…”

“…where f 0 is skew-symmetric and bi-additive with f 0 (S 1 , S 3 ) = −(n + 1) (see [3,Example 4]). Thus we see that π g ∈ H 2,−S2…”

“…In [3] we obtained a convex geometric description of T 1 (i) (A) for i ≥ 1, which we recall now. Let the cone σ = a 1 , ..., a N represent an n-dimensional toric variety X σ = Spec(A), n ≥ 3.…”

“…Note that in the case i = 1 our formulas agree with the ones given in [1, Theorem 4.1], which were obtained by different methods. (2) (A) and T 0 (3) (A) were given in [3]. The module T 2 (1) (A) was analysed in [2, Corollary 5.4].…”