标题： Kapranov $L_{\infty}[1]$ 代数 显示英文标题

标题： Kapranov $L_{\infty}[1]$ algebras

摘要： 给定任何凯勒流形$X$，卡普兰诺夫在$\Omega^{0,\bullet}_X(T^{1,0}_X)$上发现了 $L_\infty[1]$代数结构。 受此结果的启发，我们引入了一个概念，即$L_\infty[1]$ $\mathfrak{R}$ -代数，这是$L_\infty[1]$代数的一种推广，其中$\mathfrak{R}$是一个带有单位的微分分次交换代数。 我们证明了标准概念（如拟同构和线性化）和结果（包括同伦传递定理）可以扩展到这种上下文中。 例如，我们提供了线性化定理。 作为应用，我们证明，给定任何DG李代数丛$(\mathcal{L},Q_{\mathcal{L}})$在DG流形$(\mathcal{M},Q)$上，存在一个在$\Gamma(\mathcal{L})$上的诱导$L_\infty[1]$ $\mathfrak{R}$ -代数结构，其中$\mathfrak{R}$是DG交换代数$(C^\infty(\mathcal{M}),Q)$-- 其一元括号是$Q_{\mathcal{L}}$而其二元括号是DG李代数丛的Atiyah类的上循环代表。 这个$L_\infty[1]$ $\mathfrak{R}$ -代数$\Gamma(\mathcal{L})$是可线性化的当且仅当 DG 李代数的 Atiyah 类为零。 然而，由这个 $L_\infty[1]$ $\mathfrak{R}$ -代数诱导的 $L_\infty[1]$ ($\mathbb{K}$-)代数 $\Gamma(\mathcal{L})$必然是同伦交换的。 作为特殊情况，我们证明了，给定任何复流形$X$，Kapranov$L_\infty[1]$ $\mathfrak{R}$ -代数$\Omega^{0,\bullet}_X(T^{1,0}_X)$，其中$\mathfrak{R}$是DG交换代数$(\Omega^{0,\bullet}_X,\bar{\partial})$，当且仅当全纯切丛$T_X$的Atiyah类消失时，它是可线性化的。 尽管在$\Omega^{0,\bullet}_X(T^{1,0}_X)$上诱导的$L_\infty[1]$ $\mathbb{C}$ -代数结构一定是同伦交换的。 显示英文摘要

摘要： 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.