  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.
