The Dolbeault dga of the formal neighborhood of the diagonal

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_∞- algebra structure on the shifted Dolbeault resolution (A_X^•-1 (TX), ∂) 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 (A_X^0,• (J_X^∞, ∂) of the jet bundle J_X^∞ via the construction of the holomorphic exponential map of the Kähler manifold. In this paper, we show that (A_X^0,• (J_X^∞, ∂) is naturally isomorphic to the Dolbeault dga (A^•(X_{X × X}^(∞)), ∂) associated to the formal neighborhood of the diagonal of X × 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.