Bounded Leaf Volumes and Analytic Quotients of Singular Holomorphic Foliations
Abstract
Let ℱ be a singular holomorphic foliation of dimension k on a compact Kähler manifold M of dimension n. Suppose that the volumes of its regular leaves are uniformly bounded, dim(Singℱ)<k, and the union of the leaf closures meeting Singℱ has finite 2k-dimensional Hausdorff measure. We prove that the quotient by chains of intersecting leaf closures is a normal compact complex analytic space of dimension n−k. The quotient map is proper and holomorphic, with connected fibers. The proof uses Bishop's theorems, Remmert and Stein extension, Barlet's cycle space, and Stein factorization. Away from the images of separatrices, the quotient has local charts given by finite holonomy groups. Examples show that these images need not coincide with the singular set of the quotient, and that bounded leaf volumes alone do not imply Hausdorffness. In codimension one, Malgrange's theorem makes the separatrix measure assumption automatic when codim(Singℱ)≥ 3.
Related
- Singular Levi-flats and holomorphic foliations on projective spaces
- Opers on transversely holomorphic foliations
- Stability of closed leaves holomorphic foliations via torsion behavior of groups of germs
- Holomorphic realizations of pairs of foliations on Riemann surfaces
- Superconnections, descent, and monodromy on transversely holomorphic foliations
BibTeX
@misc{martinez2026bounded,
title = {{Bounded Leaf Volumes and Analytic Quotients of Singular Holomorphic Foliations}},
author = {Carlos Eduardo Martínez and Alberto Verjovsky},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.22920},
eprint = {2610.22920},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.22920},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Bounded Leaf Volumes and Analytic Quotients of Singular Holomorphic Foliations
AU - Carlos Eduardo Martínez
AU - Alberto Verjovsky
PY - 2026
DA - 2026/10/16
PB - arXiv
DO - 10.48550/arxiv.2610.22920
UR - https://arxiv.org/abs/2610.22920
AB - Let ℱ be a singular holomorphic foliation of dimension k on a compact Kähler manifold M of dimension n. Suppose that the volumes of its regular leaves are uniformly bounded, dim(Singℱ)<k, and the union of the leaf closures meeting Singℱ has finite 2k-dimensional Hausdorff measure. We prove that the quotient by chains of intersecting leaf closures is a normal compact complex analytic space of dimension n−k. The quotient map is proper and holomorphic, with connected fibers. The proof uses Bishop's theorems, Remmert and Stein extension, Barlet's cycle space, and Stein factorization. Away from the images of separatrices, the quotient has local charts given by finite holonomy groups. Examples show that these images need not coincide with the singular set of the quotient, and that bounded leaf volumes alone do not imply Hausdorffness. In codimension one, Malgrange's theorem makes the separatrix measure assumption automatic when codim(Singℱ)≥ 3.
ER -