[Paper Review] On the existence of stable compact leaves for transversely holomorphic foliations

Bruno Scárdua|arXiv (Cornell University)|Mar 31, 2012

Geometry and complex manifolds1 references4 citations

TL;DR

This paper establishes that a transversely holomorphic foliation on a compact complex manifold possesses a stable compact leaf (with finite holonomy) if and only if the set of compact leaves has positive measure. Using measure-theoretic arguments and holonomy group analysis, the authors prove that a positive-measure set of compact leaves implies the existence of a leaf with finite holonomy, extending Reeb’s stability theorem to the holomorphic setting.

ABSTRACT

A transversely holomorphic foliation on a compact complex manifold, exhibits a compact stable leaf if and only if the set of compact leaves is not a zero measure subset of the manifold.

Motivation & Objective

• To determine conditions under which a transversely holomorphic foliation on a compact complex manifold admits a stable compact leaf.
• To investigate the measure-theoretic structure of the set of compact leaves in such foliations.
• To extend Reeb’s local stability theorem to the holomorphic setting by linking finite holonomy to positive measure of compact leaves.
• To analyze the dynamics of holonomy groups and their periodicity properties in relation to compact leaf sets.
• To establish a dichotomy: either the foliation has a compact leaf with finite holonomy, or the set of compact leaves has zero measure.

Proposed method

• Define the set of compact leaves, denoted $\Omega(\mathcal{F})$, and analyze its measure in the ambient manifold $M$.
• Use a finite covering of $M$ by distinguished transverse neighborhoods $T_j$ to characterize compact leaves via finite intersection numbers.
• Apply measure-theoretic arguments: if $\mu(\Omega(\mathcal{F})) > 0$, then some $\Omega(\mathcal{F}, T, n)$ has positive measure for some $n$.
• Analyze holonomy germs $h \in \operatorname{Hol}(\mathcal{F}, L_0, \Sigma_p, p)$ on a transverse disc $\Sigma_p$; show that positive measure of periodic points implies finite order of $h$.
• Use analyticity of holonomy maps: a positive-measure set of periodic points implies the map is identity on a neighborhood, hence finite order.
• Apply Burnside’s theorem and Schur’s result on periodic linear groups to conclude that the holonomy group is finite.

Experimental results

Research questions

• RQ1 Under what conditions does a transversely holomorphic foliation on a compact complex manifold admit a stable compact leaf?
• RQ2 Can the existence of a positive-measure set of compact leaves be used to deduce finite holonomy of some compact leaf?
• RQ3 Is there a dichotomy between the existence of a compact leaf with finite holonomy and the measure of the set of all compact leaves?
• RQ4 How do periodicity properties of holonomy groups relate to the geometry of compact leaves in holomorphic foliations?
• RQ5 To what extent does the measure-theoretic structure of the leaf space constrain the holonomy of compact leaves?

Key findings

• A transversely holomorphic foliation on a compact complex manifold has a stable compact leaf if and only if the set of compact leaves has positive measure.
• If the set of compact leaves has positive measure, then there exists a compact leaf $L_0$ with a fundamental system of saturated neighborhoods where all nearby leaves are compact and have finite holonomy.
• The holonomy group of such a leaf $L_0$ is finite, as shown by the fact that each holonomy germ acts as a periodic map of bounded order.
• The proof relies on the analyticity of holonomy maps: a positive-measure set of periodic points forces the germ to be the identity, implying finite order.
• The group of diffeomorphisms $G \subset \operatorname{Diff}(F)$ acting on a complex manifold $F$ is either finite or the set of periodic orbits has zero measure.
• The result generalizes Reeb’s stability theorem to the holomorphic setting, replacing the codimension-one smooth assumption with transverse holomorphicity and measure-theoretic conditions.