An arithmetic integrability result for codimension-one foliations on complex projective spaces
Abstract
Let F be a codimension-one holomorphic foliation of degree d on PPn, n≥3, admitting an invariant hyperplane H. We study the extremal situation in which S=(H∩Sing(F))red is an irreducible hypersurface of H of degree d+1. When d+1 is a power of a prime, we prove that, in suitable homogeneous coordinates with H=(t=0), Ω=Q dt−t/(d+1) dQ , where Q is homogeneous of degree d+1. Thus Q/td+1 is a rational first integral. The proof reduces the Frobenius equation to a twisted closedness equation on a plane section and uses Zariski's theorem on the Alexander polynomial of an irreducible plane curve. We also prove a complementary rigidity theorem for an arbitrary smooth invariant hypersurface D⊂PPn: if the reduced singular divisor on D is smooth, irreducible, and of maximal degree, then the same normal-form phenomenon holds, with no arithmetic hypothesis on its degree; in the low-weight range the smoothness assumption on the singular divisor can be dropped. Finally, we show that the principal hypotheses are sharp. Dropping the maximal-degree condition yields a family with irreducible reduced singular support and no non-constant rational first integral. For every d+1 which is not a prime power we construct a global counterexample with irreducible maximal-degree singular support, and a final family shows that irreducibility of the reduced support is also genuinely necessary.
Related
- Stability and Logarithmic Foliations on Complex Projective Spaces
- Hyperpolarity of polar foliations of codimension two
- Stability of Diophantine Minimal Foliations in Arbitrary Codimension
- Minimal foliations, codimension-one stable norms, and a question of Bangert
- Lp integrability of functions with Fourier support on High codimension manifolds
BibTeX
@misc{leon2026arithmetic,
title = {{An arithmetic integrability result for codimension-one foliations on complex projective spaces}},
author = {Víctor León and Bruno Scárdua},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.14807},
eprint = {2609.14807},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.14807},
}
RIS
TY - UNPB
TI - An arithmetic integrability result for codimension-one foliations on complex projective spaces
AU - Víctor León
AU - Bruno Scárdua
PY - 2026
DA - 2026/09/13
PB - arXiv
DO - 10.48550/arxiv.2609.14807
UR - https://arxiv.org/abs/2609.14807
AB - Let F be a codimension-one holomorphic foliation of degree d on PP^n, n≥3, admitting an invariant hyperplane H. We study the extremal situation in which S=(H∩Sing(F))_red is an irreducible hypersurface of H of degree d+1. When d+1 is a power of a prime, we prove that, in suitable homogeneous coordinates with H=(t=0), Ω=Q dt−t/(d+1) dQ , where Q is homogeneous of degree d+1. Thus Q/t^d+1 is a rational first integral. The proof reduces the Frobenius equation to a twisted closedness equation on a plane section and uses Zariski's theorem on the Alexander polynomial of an irreducible plane curve. We also prove a complementary rigidity theorem for an arbitrary smooth invariant hypersurface D⊂PP^n: if the reduced singular divisor on D is smooth, irreducible, and of maximal degree, then the same normal-form phenomenon holds, with no arithmetic hypothesis on its degree; in the low-weight range the smoothness assumption on the singular divisor can be dropped. Finally, we show that the principal hypotheses are sharp. Dropping the maximal-degree condition yields a family with irreducible reduced singular support and no non-constant rational first integral. For every d+1 which is not a prime power we construct a global counterexample with irreducible maximal-degree singular support, and a final family shows that irreducibility of the reduced support is also genuinely necessary.
ER -