The Dolbeault dga of the formal neighborhood of the diagonal

摘要

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-infinity-algebra structure on the shifted Dolbeault resolution (A(X)(center dot-1) (TX), (partial derivative) over bar) of TX and wrote down the structure maps explicitly in the case when X is Kahler. The corresponding Chevalley-Eilenberg complex is isomorphic to the Dolbeault resolution (A(X)(0,center dot) (J(X)(infinity)), (partial derivative) over bar) of the jet bundle J(X)(infinity) via the construction of the holomorphic exponential map of the Kahler manifold. In this paper, we show that (A(X)(0,center dot) (J(X)(infinity)), (partial derivative) over bar) is naturally isomorphic to the Dolbeault dga (A(center dot) (X-XxX((infinity))), (partial derivative) over bar) associated to the formal neighborhood of the diagonal of X 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.