Infinitesimally Equivariant Bundles on Complex Manifolds

TL;DR

This paper proves that continuous complex-linear Lie algebra splittings of the Atiyah algebra's symbol map correspond to differential operators of bounded order, characterizing infinitesimally equivariant bundles as near-flat connections.

Contribution

It establishes a bound on the order of differential operators representing Lie algebra splittings, linking infinitesimally equivariant bundles to bundles with flat connections.

Findings

Lie algebra splittings correspond to differential operators of order ≤ rank + 1

Infinitesimally equivariant bundles are close to flat connection bundles

Provides a categorical framework for understanding these bundles

Abstract

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.