Hochschild cohomology of dg manifolds associated to integrable distributions

TL;DR

This paper investigates the Hochschild cohomology of dg manifolds linked to integrable distributions, establishing canonical isomorphisms with leaf space function algebras and connecting to classical theorems in complex geometry.

Contribution

It provides a canonical isomorphism between Hochschild cohomology of certain dg manifolds and leaf space function algebras, extending the Duflo-Kontsevich theorem to complex manifolds.

Findings

Hochschild cohomology of dg manifolds associated to integrable distributions is characterized.

For complex manifolds, Hochschild cohomology of the dg manifold matches that of the manifold.

Duflo-Kontsevich theorem extends to the setting of complex manifolds via dg manifolds.

Abstract

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 cohomology $HH^{\bullet}(X)$ of the complex manifold $X$. As an application, we show that the Duflo-Kontsevich type theorem for the dg manifold $(T_X^{0,1}[1],\bar{\partial})$ implies the Duflo-Kontsevich theorem for complex manifolds.