A measurable stability theorem for holomorphic foliations transverse to fibrations

TL;DR

This paper establishes that certain holomorphic foliations and groups of diffeomorphisms exhibit stability and finiteness properties when the set of special leaves or orbits has positive measure, linking geometric and dynamical stability.

Contribution

It provides a measurable stability theorem for holomorphic foliations transverse to fibrations and a finiteness result for groups of holomorphic diffeomorphisms based on measure conditions.

Findings

Transversely holomorphic foliations with positive measure of compact leaves are Seifert fibrations.

Finitely generated subgroups of holomorphic diffeomorphisms are finite if the set of periodic orbits has positive measure.

Abstract

We prove that a transversely holomorphic foliation which is transverse to the fibers of a fibration, is a Seifert fibration if the set of compact leaves is not of zero measure. Similarly, we prove that a finitely generated subgroup of holomorphic diffeomorphisms of a connected complex manifold, is finite provided that the set of periodic orbits is not of zero measure.