The Dolbeault dga of the formal neighborhood of the diagonal

Abstract

A well-known theorem of Kapranov states that the Atiyah class of the tangent bundle $TX$ of a complex manifold $X$ makes the shifted tangent bundle $TX[-1]$ into a Lie algebra object in the derived category $D(X)$. Moreover, he showed that there is an $L_{\infty }$-algebra structure on the shifted Dolbeault resolution $({\mathcal{A}}_X^{\bullet -1}(TX),\overline{\partial })$ of $TX$ and wrote down the structure maps explicitly in the case when $X$ is Kähler. The corresponding Chevalley-Eilenberg complex is isomorphic to the Dolbeault resolution $({\mathcal{A}}_X^{0,\bullet }({\mathcal{J}}_X^{\infty }),\overline{\partial })$ of the jet bundle ${\mathcal{J}}_X^{\infty }$ via the construction of the holomorphic exponential map of the Kähler manifold. In this paper, we show that $({\mathcal{A}}_X^{0,\bullet }({\mathcal{J}}_X^{\infty }),\overline{\partial })$ is naturally isomorphic to the Dolbeault dga $({\mathcal{A}}^{\bullet }(X_{X\times X}(\infty ),\overline{\partial })$ associated to the formal neighborhood of the diagonal of $X\times X$ which we introduced in [15]. We also give an alternative proof of Kapranov's theorem by obtaining an explicit formula for the pullback of functions via the holomorphic exponential map, which allows us to study the general case of an arbitrary embedding later.