Kapranov $L_{\infty }[1]$ algebras

Abstract

Given any Kähler manifold $X$, Kapranov discovered an $L_{\infty }[1]$ algebra structure on $\hat{I}{\text{c◯}}_X^{0,\bullet }(T_X^{1,0})$. Motivated by this result, we introduce, as a generalization of $L_{\infty }[1]$ algebras, a notion of $L_{\infty }[1]$ $\mathfrak{R}$-algebra, where $\mathfrak{R}$ is a differential graded commutative algebra with unit. We show that standard notions (such as quasi-isomorphism and linearization) and results (including homotopy transfer theorems) can be extended to this context. For instance, we provide a linearization theorem. As an application, we prove that, given any DG Lie algebroid $(\mathcal{L},Q_{\mathcal{L}})$ over a DG manifold $(\mathcal{M},Q)$, there exists an induced $L_{\infty }[1]$ $\mathfrak{R}$-algebra structure on $\hat{I}“(\mathcal{L})$, where $\mathfrak{R}$ is the DG commutative algebra $(C^{\infty }(\mathcal{M}),Q)$ -- its unary bracket is $Q_{\mathcal{L}}$ while its binary bracket is a cocycle representative of the Atiyah class of the DG Lie algebroid. This $L_{\infty }[1]$ $\mathfrak{R}$-algebra $\hat{I}“(\mathcal{L})$ is linearizable if and only if the Atiyah class of the DG Lie algebroid vanishes. However, the $L_{\infty }[1]$ ($\mathbb{K}$-)algebra $\hat{I}“(\mathcal{L})$ induced by this $L_{\infty }[1]$ $\mathfrak{R}$-algebra is necessarily homotopy abelian. As a special case, we prove that, given any complex manifold $X$, the Kapranov $L_{\infty }[1]$ $\mathfrak{R}$-algebra $\hat{I}{\text{c◯}}_X^{0,\bullet }(T_X^{1,0})$, where $\mathfrak{R}$ is the DG commutative algebra $(\hat{I}{\text{c◯}}_X^{0,\bullet },\bar{\partial })$, is linearizable if and only if the Atiyah class of the holomorphic tangent bundle $T_X$ vanishes. Nevertheless, the induced $L_{\infty }[1]$ $\mathbb{C}$-algebra structure on $\hat{I}{\text{c◯}}_X^{0,\bullet }(T_X^{1,0})$ is necessarily homotopy abelian.

(32 pages) Comments are welcome