Partial holomorphic connections and extension of foliations

Abstract

This paper stresses the strong link between the existence of partial holomorphic connections on the normal bundle of a foliation seen as a quotient of the ambient tangent bundle and the extendability of a foliation to an infinitesimal neighborhood of a submanifold. We find the obstructions to extendability, and thanks to the theory developed we obtain some new Khanedani–Lehmann–Suwa type index theorems.