Partial holomorphic connections and extension of foliations

TL;DR

This paper explores the relationship between partial holomorphic connections on the normal bundle of a foliation and the extendability of the foliation to infinitesimal neighborhoods, providing new index theorems and obstruction criteria.

Contribution

It establishes a link between holomorphic connections and foliation extendability, introducing new obstructions and index theorems in the process.

Findings

Identified obstructions to foliation extendability.

Developed new index theorems of Khanedani-Lehmann-Suwa type.

Connected partial holomorphic connections with foliation extension properties.

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.