Infinitesimally equivariant bundles on complex manifolds

We study holomorphic vector bundles equipped with a compatible action of vector field by Lie derivatives. We will show that the dependence of the Lie derivative on a vector field is almost $$\mathcal {O}$$-linear. More precisely, after an algebraic reformulation, we show that any continuous $$\mathbf {C}$$-linear Lie algebra splitting of the symbol map from the Atiyah algebra of a vector bundle on a complex manifold is given by a differential operator, which is further of order at most the rank of the bundle plus one. The proof is quite elementary. When the differential operator we obtain has order 0 we have simply a vector bundle with flat connection, so in a sense, our theorem says that we are always a uniformly bounded order away from this simplest case.