A well-known theorem of Kapranov states that the Atiyah class of the tangentbundle $TX$ 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 showedthat 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 $X$ isK\"ahler. The corresponding Chevalley-Eilenberg complex is isomorphic to theDolbeault resolution of the jet bundle $\mathcal{J}^\infty_X$ via theconstruction of the holomorphic exponential map of the K\"ahler manifold. Inthis paper, we show that the Dolbeault resolution of the jet bundle isnaturally isomorphic to the Dolbeault dga associated to the formal neighborhoodof the diagonal of $X \times X$ which we introduced in a previous paper. Wealso give an alternative proof of Kapranov's theorem by obtaining an explicitformula for the pullback of functions via the holomorphic exponential map,which allows us to study the general case of an arbitrary embedding later.
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机译：Kapranov的一个著名定理指出，复流形$ X $的切线束$ TX $的Atiyah类使移位的切线束$ TX [-1] $成为派生类别$ D（X ）$。此外，他证明了在$ TX [-1] $的Dolbeault分辨率上存在$ L_ \ infty $-代数结构，并在$ X $为K \“ ahler的情况下明确写下了该结构图。通过建立Kahler流形的全纯指数图，Eilenberg复合体与喷气束$ \ mathcal {J} ^ \ infty_X $的Dolbeault分辨率是同构的。在本文中，我们证明了喷气束的Dolbeault分辨率与Dolbeault dga同构，该Dolbeault dga与我们在上一篇论文中介绍的对角线$ X \ times X $的形式邻域相关。我们还通过全纯指数图获得用于函数回调的显式，从而给出了Kapranov定理的另一种证明，这使我们能够在以后研究任意嵌入的一般情况。
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