Global Stability for Holomorphic Foliations in Kaehler Manifolds

TL;DR

This paper proves a global stability theorem for holomorphic foliations in compact Kähler manifolds, showing that the existence of a compact leaf with finite holonomy implies all leaves are compact with finite holonomy, extending previous stability results.

Contribution

It establishes a new global stability criterion for holomorphic foliations in Kähler manifolds, generalizing earlier local stability theorems.

Findings

If a compact leaf with finite holonomy exists, then all leaves are compact with finite holonomy.

Reobtains classical stability theorems for compact foliations in Kähler manifolds.

Extends stability results to a broader class of holomorphic foliations.

Abstract

We prove the following theorem for Holomorphic Foliations in compact complex kaehler manifolds: if there is a compact leaf with finite holonomy, then every leaf is compact with finite holonomy. As corollary we reobtain stability theorems for compact foliations in Kaehler manifolds of Edwards-Millett-Sullivan and Hollman.