Symplectic

A classical result of Mikhail Kapranov states that the holomorphic tangent bundle of a complex manifold possesses a natural structure of an $L_\infty[1]$-algebra object in the category of coherent sheaves on the complex manifold, and that the second bracket defining the $L_\infty[1]$-structure on that object is a cocycle representing none other than the Atiyah class of the holomorphic tangent bundle — a class obstructing the existence of a holomorphic affine connection on the complex manifold. Moreover, the result includes also the statement that every holomorphic vector bundle on the manifold is naturally a module object over the $L_\infty[1]$-algebra object that is the tangent bundle, and the second bracket defining this action is a cocycle representing the Atiyah class of the vector bundle in question. In particular, on the level of cohomology, the third bracket defining the module structure becomes a homotopy identifying some cohomology classes, and this yields a "cohomological Bianchi identity" on the Atiyah classes of the tangent bundle and the vector bundle.

I will explain Kapranov's construction(s) and proceed to mention how that result can be carried over from the setting of the pair (holomorphic tangent bundle, holomorphic vector bundle) to the setting of an arbitrary holomorphic VB-algebroid, which is a minute generalisation (but there is more work in progress).