A residue formula for meromorphic connections and applications to stable sets of foliations

TL;DR

This paper introduces residue formulae that localize the first Chern class of line bundles to singularities of holomorphic connections, with applications to foliation theory and complex geometry.

Contribution

It provides a new residue formula for meromorphic connections and applies it to prove conjectures about minimal sets of foliations and the nonexistence of certain Levi flat hypersurfaces.

Findings

Proof of Brunella's conjecture on minimal sets of foliations

Nonexistence theorem for Levi flat hypersurfaces in Kähler surfaces

Development of residue formulae for holomorphic connections

Abstract

We discuss residue formulae that localize the first Chern class of a line bundle to the singular locus of a given holomorphic connection. As an application, we explain a proof for Brunella's conjecture about exceptional minimal sets of codimension one holomorphic foliations with ample normal bundle and for a nonexistence theorem of Levi flat hypersurfaces with transversely affine Levi foliation in compact K\"ahler surfaces.