From Atiyah Classes to Homotopy Leibniz Algebras

摘要: A celebrated theorem of Kapranov states that the Atiyah class tangent bundle a complex manifold $X$ makes $T_X[-1]$ into Lie algebra object in $D^+(X)$, bounded below derived category coherent sheaves on $X$. Furthermore proved that, for K\"ahler $X$, Dolbeault resolution $\Omega^{\bullet-1}(T_X^{1,0})$ is an $L_\infty$ algebra. In this paper, we prove Kapranov's holds much wider generality vector bundles over pairs. Given pair $(L,A)$, i.e. algebroid $L$ together with subalgebroid $A$, define $\alpha_E$ $A$-module $E$ (relative to $L$) as obstruction existence $A$-compatible $L$-connection $E$. We classes $\alpha_{L/A}$ and respectively make $L/A[-1]$ $E[-1]$ module $D^+(\mathcal{A})$, where $\mathcal{A}$ abelian left $\mathcal{U}(A)$-modules $\mathcal{U}(A)$ universal enveloping $A$. Moreover, produce homotopy Leibniz stemming from $L/A$ $E$, inducing aforesaid structures $D^+(\mathcal{A})$.