To any $\mathfrak{g}$-manifold $M$ are associated two dglas $\operatorname{tot}\big(Λ^{\bullet} \mathfrak{g}^\vee \otimes_{\Bbbk} T_{\operatorname{poly}}^{\bullet} \big)$ and $\operatorname{tot} \big(Λ^{\bullet} \mathfrak{g}^\vee\otimes_{\Bbbk} D_{\operatorname{poly}}^{\bullet} \big)$, whose cohomologies $H_{\operatorname{CE}}(\mathfrak{g}, T_{\operatorname{poly}}^{\bullet} \xrightarrow{0} T_{\operatorname{poly}}^{\bullet+1})$ and $H_{\operatorname{CE}}(\mathfrak{g}, D_{\operatorname{poly}}^{\bullet} \xrightarrow{0} D_{\operatorname{poly}}^{\bullet+1})$ are Gerstenhaber algebras. We establish a formality theorem for $\mathfrak{g}$-manifolds: there exists an $L_\infty$ quasi-isomorphism $Φ: \operatorname{tot}\big(Λ^{\bullet} \mathfrak{g}^\vee \otimes_{\Bbbk} T_{\operatorname{poly}}^{\bullet} \big) \to \operatorname{tot} \big(Λ^{\bullet} \mathfrak{g}^\vee\otimes_{\Bbbk} D_{\operatorname{poly}}^{\bullet} \big)$ whose first `Taylor coefficient' (1) is equal to the Hochschild-Kostant-Rosenberg map twisted by the square root of the Todd cocycle of the $\mathfrak{g}$-manifold $M$ and (2) induces an isomorphism of Gerstenhaber algebras on the level of cohomology. Consequently, the Hochschild-Kostant-Rosenberg map twisted by the square root of the Todd class of the $\mathfrak{g}$-manifold $M$ is an isomorphism of Gerstenhaber algebras from $H_{\operatorname{CE}}(\mathfrak{g}, T_{\operatorname{poly}}^{\bullet} \xrightarrow{0} T_{\operatorname{poly}}^{\bullet+1})$ to $H_{\operatorname{CE}}(\mathfrak{g}, D_{\operatorname{poly}}^{\bullet} \xrightarrow{0} D_{\operatorname{poly}}^{\bullet+1})$.
$\newcommand{\poly}{_{\operatorname{poly}}^{\bullet}}\newcommand{\td}{(\operatorname{td}_{L/A}^{\nabla})^{\frac{1}{2}}}\newcommand{\cx}[1]{\operatorname{tot}\big(Γ(Λ^\bullet A^\vee)\otimes_R\mathcal{#1}\poly\big)}\newcommand{\cy}[1]{\mathbb{H}^\bullet_{\operatorname{CE}}(A,\mathcal{#1}\poly)}$Kontsevich's formality theorem states that there exists an $L_\infty$ quasi-isomorphism from the dgla $T\poly(M)$ of polyvector fields on a smooth manifold $M$ to the dgla $D\poly(M)$ of polydifferential operators on $M$, which extends the classical Hochschild--Kostant--Rosenberg map. In this paper, we ex…
Formality of hypercommutative algebras of Kähler and Calabi-Yau manifolds
Universitat de Barcelona（巴塞罗那大学）；Centre de Recerca Matemàtica（数学研究中心）；Université Sorbonne Paris Nord, Laboratoire Analyse, Géométrie et Applications, CNRS (UMR 7539)（索邦巴黎北大学，分析、几何与应用实验室，法国国家科学研究中心）
