The Dolbeault dga of the formal neighborhood of the diagonal

TL;DR

This paper connects the Dolbeault dga of the formal neighborhood of the diagonal in a complex manifold to Kapranov's theorem, providing explicit formulas and a new proof for the Lie algebra structure on the shifted tangent bundle.

Contribution

It establishes a natural isomorphism between the Dolbeault resolution of the jet bundle and the Dolbeault dga of the formal neighborhood, offering an alternative proof of Kapranov's theorem.

Findings

Isomorphism between Dolbeault resolution of jet bundle and Dolbeault dga of formal neighborhood

Explicit formula for pullback of functions via holomorphic exponential map

New proof of Kapranov's theorem using these formulas

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 Dolbeault resolution of $TX[-1]$ and wrote down the structure maps explicitly in the case when $X$ is K\"ahler. The corresponding Chevalley-Eilenberg complex is isomorphic to the Dolbeault resolution of the jet bundle $\mathcal{J}^\infty_X$ via the construction of the holomorphic exponential map of the K\"ahler manifold. In this paper, we show that the Dolbeault resolution of the jet bundle is naturally isomorphic to the Dolbeault dga associated to the formal neighborhood of the diagonal of $X \times X$ which we introduced in a previous paper. 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.