Infinitesimally equivariant bundles on complex manifolds

Emile Bouaziz

Mathematische Nachrichten, 2025, vol. 298, issue 3, 1076-1081

Abstract: 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 O$\mathcal {O}$‐linear. More precisely, after an algebraic reformulation, we show that any continuous C$\mathbf {C}$‐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.

Date: 2025
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