Foliations with small singular set in arbitrary characteristic
Abstract
This paper investigates the geometry of foliations on smooth algebraic varieties over an algebraically closed field of arbitrary characteristic p ≥ 0. We address several specific features of foliations in positive characteristic, aiming to highlight both similarities and differences with the characteristic zero case. First, we provide a complete classification of regular foliations on minimal rational and del Pezzo surfaces, establishing that in positive characteristic, most of them are p-closed. However, there are regular foliations on Hirzebruch surfaces over arbitrary characteristic p>0, and on weak del Pezzo surfaces, in characteristic p=2, which are not p-closed. We extend the Camacho-Sad index and its associated sum formula to arbitrary characteristic, using it to study the behavior of foliations and distributions with small singular set. For foliations on projective spaces, we prove that the existence of an invariant hypersurface with a sufficiently small singular set forces the foliation to be p-closed and imposes strict divisibility conditions on the degree of its normal bundle. Finally, we establish versions of the Bott vanishing theorem in both Hodge cohomology and the Chow ring for p-closed foliations, providing a positive characteristic analogue to classical vanishing results in complex geometry.
Related
- Foliations on abelian surfaces in positive characteristic
- Stability of Diophantine Minimal Foliations in Arbitrary Codimension
- Finite-Rank Lie Algebroids for Singular Foliations of Prescribed Vanishing Order
- Bounded Leaf Volumes and Analytic Quotients of Singular Holomorphic Foliations
- Singular Levi-flats and holomorphic foliations on projective spaces
BibTeX
@misc{fassarella2026foliations,
title = {{Foliations with small singular set in arbitrary characteristic}},
author = {Thiago Fassarella and Wodson Mendson and João Pedro dos Santos and Frédéric Touzet},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2607.18571},
eprint = {2607.18571},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.18571},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Foliations with small singular set in arbitrary characteristic
AU - Thiago Fassarella
AU - Wodson Mendson
AU - João Pedro dos Santos
AU - Frédéric Touzet
PY - 2026
DA - 2026/09/18
PB - arXiv
DO - 10.48550/arxiv.2607.18571
UR - https://arxiv.org/abs/2607.18571
AB - This paper investigates the geometry of foliations on smooth algebraic varieties over an algebraically closed field of arbitrary characteristic p ≥ 0. We address several specific features of foliations in positive characteristic, aiming to highlight both similarities and differences with the characteristic zero case. First, we provide a complete classification of regular foliations on minimal rational and del Pezzo surfaces, establishing that in positive characteristic, most of them are p-closed. However, there are regular foliations on Hirzebruch surfaces over arbitrary characteristic p>0, and on weak del Pezzo surfaces, in characteristic p=2, which are not p-closed. We extend the Camacho-Sad index and its associated sum formula to arbitrary characteristic, using it to study the behavior of foliations and distributions with small singular set. For foliations on projective spaces, we prove that the existence of an invariant hypersurface with a sufficiently small singular set forces the foliation to be p-closed and imposes strict divisibility conditions on the degree of its normal bundle. Finally, we establish versions of the Bott vanishing theorem in both Hodge cohomology and the Chow ring for p-closed foliations, providing a positive characteristic analogue to classical vanishing results in complex geometry.
ER -