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.
"Partial holomorphic connections and extension of foliations." Kyoto J. Math. 52 (3) 517 - 555, Fall 2012. https://doi.org/10.1215/21562261-1625190