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Stability of closed leaves holomorphic foliations via torsion behavior of groups of germs
Abstract
Consider a compact leaf ℒ of a holomorphic foliation ℱ such that all the leaves of the restriction of ℱ to some neighborhood U of ℒ are closed in U. We show that there exists an invariant germ of analytic set V in a neighborhood of ℒ, of dimension higher than dim (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.
Related
- Opers on transversely holomorphic foliations
- Bounded Leaf Volumes and Analytic Quotients of Singular Holomorphic Foliations
- Holomorphic realizations of pairs of foliations on Riemann surfaces
- Singular Levi-flats and holomorphic foliations on projective spaces
- Superconnections, descent, and monodromy on transversely holomorphic foliations
BibTeX
@misc{ribon2026stability,
title = {{Stability of closed leaves holomorphic foliations via torsion behavior of groups of germs}},
author = {Javier Ribón},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.16234},
eprint = {2609.16234},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.16234},
}
RIS
TY - UNPB
TI - Stability of closed leaves holomorphic foliations via torsion behavior of groups of germs
AU - Javier Ribón
PY - 2026
DA - 2026/09/14
PB - arXiv
DO - 10.48550/arxiv.2609.16234
UR - https://arxiv.org/abs/2609.16234
AB - Consider a compact leaf ℒ of a holomorphic foliation ℱ such that all the leaves of the restriction of ℱ to some neighborhood U of ℒ are closed in U. We show that there exists an invariant germ of analytic set V in a neighborhood of ℒ, of dimension higher than dim (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.
ER -