Year: 2026
Type: preprint
Abstract: Consider a compact leaf $\mathcal{L}$ of a holomorphic foliation $\mathcal{F}$ such that all the leaves of the restriction of $\mathcal{F}$ to some neighborhood $U$ of $\mathcal{L}$ are closed in $U$. We show that there exists an invariant germ of analytic set $V$ in a neighborhood of $\mathcal{L}$, of dimension higher than $\dim (\mathcal{F})$, consisting of compact leaves, such that the volume of the leaves in $V$ is uniformly bounded by above. We also provide a globalization of this result if the ambient manifold is projective. In order to study closed leaves foliations, and via the holonomy representation, we introduce a new concept, of independent interest, for subgroups $G$ of germs of holomorphic diffeomorphisms, the so called {\it torsion locus}. We show that it is non-trivial if $G$ has finite orbits.
Source: arXiv (Cornell University)
Author: Ribón, Javier
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7213423212