Hochschild cohomology and Atiyah classes

Abstract

In this paper we prove that on a smooth algebraic variety the HKR-morphism twisted by the square root of the Todd genus gives an isomorphism between the sheaf of poly-vector fields and the sheaf of poly-differential operators, both considered as derived Gerstenhaber algebras. In particular we obtain an isomorphism between Hochschild cohomology and the cohomology of poly-vector fields which is compatible with the Lie bracket and the cupproduct. The latter compatibility is an unpublished result by Kontsevich. Our proof is set in the framework of Lie algebroids and so applies without modification in much more general settings as well.Comment: Reference to work of Cattaneo, Felder and Willwacher adde

Compatibility with Cap-Products in Tsygan’s Formality and Homological Duflo Isomorphism

“…The main argument, of homotopical nature, was sketched by Kontsevich in [20], later clarified by Manchon and Torossian in [21] in the framework of deformation quantization, and finally adapted to the case of Q-manifolds in [3]. The globalisation of the compatibility between cup products was first seriously considered in [5], and is addressed in Section 8.…”

Weak quantization of Poisson structures

“…A formality theorem. There exists several results [23,4,22,5] about the extension of Kontsevich formality theorem for algebraic varietes and/or complex manifolds. The one we will use in the paper is taken from [5] (Theorem 6.4.1) that we translate as follows in the language we are using here: …”

“…Recall also that if u is a L-polyvector field then one can define 5 In the case when O = O X and L = T X are the structure and tangent sheaf of a smooth algebraic variety X, elements ofh are called normalized poly-differential operators in [23].…”

“…Recall from [2,3,4,5] that there are two sheaves of quantum type DG-Lie algebras g and h associated to L: g is the sheaf of L-poly-vector fields equipped with zero differential and Schouten type bracket [−, −], and h is the sheaf of Lpoly-differential operators (a-k-a Hochschild cochains associated to L 4 ) equipped with Hochschild differential d H and Gerstenhaber bracket [−, −] G (we refer to [2,4] for details). This is a well-known fact that g is the cohomology sheaf of (h, d H ), and that the Lie bracket on g induced from…”

“…This formality theorem is generalized to a large class of sheaves of Lie algebroids in [5] (see also [2,3,4] for the particular cases of C ∞ and holomorphic Lie algebroids). In this paper, we prove that any Poisson structure admits a weak deformation quantization (Theorem 3.5).…”