Singular Levi-flats and holomorphic foliations on projective spaces
Abstract
We study codimension-one holomorphic foliations admitting invariant singular real-analytic Levi-flat varieties. Our main problem is to determine when the local meromorphic first integrals furnished by the Levi-flat geometry can be continued coherently and globalized. We introduce a pencil-singularity condition and a projective compatibility condition for local meromorphic parameters. In the projective plane these yield a characterization of foliations admitting a primitive rational first integral, while in higher-dimensional projective space they give a rational integrability criterion under a generic local-product hypothesis along the dicritical locus. We also obtain semi-global and Stein versions of these results. In a different direction, we study foliations admitting infinitely many invariant Levi-flat varieties and prove a Darboux-type alternative: either the foliation admits a rational first integral, or infinitely many Levi-flats contain a common invariant algebraic curve whose virtual holonomy is abelian. Under additional assumptions on the singularities, this leads to the existence of a closed rational, and in the absence of saddle-nodes a closed logarithmic, defining 1-form.
Related
- Bounded Leaf Volumes and Analytic Quotients of Singular Holomorphic Foliations
- Limit cycles of singular algebraic foliations on the real projective plane
- Stability and Logarithmic Foliations on Complex Projective Spaces
- An arithmetic integrability result for codimension-one foliations on complex projective spaces
- Opers on transversely holomorphic foliations
BibTeX
@misc{scardua2026singular,
title = {{Singular Levi-flats and holomorphic foliations on projective spaces}},
author = {Bruno Scárdua},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.12497},
eprint = {2610.12497},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.12497},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Singular Levi-flats and holomorphic foliations on projective spaces
AU - Bruno Scárdua
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.12497
UR - https://arxiv.org/abs/2610.12497
AB - We study codimension-one holomorphic foliations admitting invariant singular real-analytic Levi-flat varieties. Our main problem is to determine when the local meromorphic first integrals furnished by the Levi-flat geometry can be continued coherently and globalized. We introduce a pencil-singularity condition and a projective compatibility condition for local meromorphic parameters. In the projective plane these yield a characterization of foliations admitting a primitive rational first integral, while in higher-dimensional projective space they give a rational integrability criterion under a generic local-product hypothesis along the dicritical locus. We also obtain semi-global and Stein versions of these results. In a different direction, we study foliations admitting infinitely many invariant Levi-flat varieties and prove a Darboux-type alternative: either the foliation admits a rational first integral, or infinitely many Levi-flats contain a common invariant algebraic curve whose virtual holonomy is abelian. Under additional assumptions on the singularities, this leads to the existence of a closed rational, and in the absence of saddle-nodes a closed logarithmic, defining 1-form.
ER -