Invariant Hochschild cohomology of smooth functions

Hochschild cohomology of dg manifolds associated to integrable distributions

For the field $\mathbb{K} = \mathbb{R}$ or $\mathbb{C}$, and an integrable distribution $F \subseteq T_M \otimes_{\mathbb{R}} \mathbb{K}$ on a smooth manifold $M$, we study the Hochschild cohomology of the dg manifold $(F[1],d_F)$ and establish a canonical isomorphism with the Hochschild cohomology of the algebra of functions on leaf space in terms of transversal polydifferential operators of $F$. In particular, for the dg manifold $(T_X^{0,1}[1],\bar{\partial})$ associated with a complex manifold $X$, we prove that its Hochschild cohomology is canonically isomorphic to the Hochschild cohomolo…

Hochschild Cohomology versus De Rham Cohomology without Formality Theorems

We exploit the Fedosov-Weinstein-Xu (FWX) resolution proposed in q-alg/9709043 to establish an isomorphism between the ring of Hochschild cohomology of the quantum algebra of functions on a symplectic manifold M and the ring H(M, C((h))) of De Rham cohomology of M with the coefficient field C((h)) without making use of any version of the formality theorem. We also show that the Gerstenhaber bracket induced on H(M,C((h))) via the isomorphism is vanishing. We discuss equivariant properties of the isomorphism and propose an analogue of this statement in an algebraic geometric setting.

广义格拉斯曼流形的Hochschild上同调

Hochschild cohomology of generalised Grassmannians

本文计算广义格拉斯曼流形的Hochschild上同调的HKR分解，证明其在（余）极小型和（余）伴随型情形集中于整体截面，并给出Gerstenhaber代数结构的显式部分描述。