Infinitesimally Equivariant Bundles on Complex Manifolds

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 of order at most the rank of the bundle plus one. Bundles equipped with such a splitting can be thought of as \emph{infinitesimally equivariant} bundles, and our theorem implies these are, in a certain sense, in a categorical formal neighbourhood of vector bundles with a flat connection.