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

Abstract

Given any K\"ahler manifold $X$, Kapranov discovered an $L_\infty[1]$ algebra structure on $\Omega^{0,\bullet}_X(T^{1,0}_X)$. 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 $\Gamma(\mathcal{L})$, 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 $\Gamma(\mathcal{L})$ is linearizable if and only if the Atiyah class of the DG Lie algebroid vanishes. However, the $L_\infty[1]$ ($\mathbb{K}$-)algebra $\Gamma(\mathcal{L})$ 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 $\Omega^{0,\bullet}_X(T^{1,0}_X)$, where $\mathfrak{R}$ is the DG commutative algebra $(\Omega^{0,\bullet}_X,\bar{\partial})$, 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 $\Omega^{0,\bullet}_X(T^{1,0}_X)$ is necessarily homotopy abelian.