Kapranov $L_{\infty}[1]$ algebras
Ruggero Bandiera, Seokbong Seol, Mathieu Stiénon, Ping Xu
TL;DR
The paper generalizes Kapranov's $L_ ablafty[1]$-structure from geometry to a DG commutative algebra base $\mathfrak{R}$, introducing the notion of $L_ ablafty[1]\mathfrak{R}$-algebras and establishing a linearization theorem governed by the Atiyah class. It develops the theory with a robust base-change framework, a homotopy-transfer toolkit, and higher derived brackets, enabling a unified treatment of DG Lie algebroids and Lie pairs. A central result is that Kapranov $L_ ablafty[1]\mathfrak{R}$-algebras on sections of a DG Lie algebroid encode the Atiyah class via the binary bracket, with linearization equivalent to Atiyah class vanishing; over the ground field these structures are always homotopy abelian. For Lie pairs and complex-geometric settings, the paper shows quasi-isomorphisms between different Kapranov realizations and links Atiyah classes across constructions, providing a cohesive picture of how curvature-type obstructions govern linearization in these higher-algebraic contexts. Overall, the work extends Rozansky–Witten-type structures, clarifies when nontrivial invariants appear, and formalizes a suite of functorial base-change and transfer results that connect geometry, derived brackets, and $L_ ablafty[1]$-theory.
Abstract
Given any Kähler manifold $X$, Kapranov discovered an $L_\infty[1]$ algebra structure on $Ω^{0,\bullet}_X(T^{1,0}_X)$. 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 $Γ(\mathcal{L})$, 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 $Γ(\mathcal{L})$ is linearizable if and only if the Atiyah class of the DG Lie algebroid vanishes. However, the $L_\infty[1]$ ($\mathbb{K}$-)algebra $Γ(\mathcal{L})$ 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 $Ω^{0,\bullet}_X(T^{1,0}_X)$, where $\mathfrak{R}$ is the DG commutative algebra $(Ω^{0,\bullet}_X,\bar{\partial})$, 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 $Ω^{0,\bullet}_X(T^{1,0}_X)$ is necessarily homotopy abelian.