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arXiv · 2608.09877

Effective Results for Foliations on Smooth Projective Complete Intersection Surfaces

Jorge Olivares·Daniel Posada-Buriticá
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Abstract

We study holomorphic foliations on projective spaces that leave smooth projective complete intersection surfaces $M$ invariant. We determine precisely for which degrees such foliations on $M$ exist. As a consequence, we obtain new bounds for the classical Poincar\'e problem for smooth projective complete intersection surfaces and prove that previously known bounds for smooth hypersurfaces in $\mathbb{P}^3$ are optimal. Furthermore, for a foliation $[s]$ on $M$ with isolated singularities and for degrees beyond an explicit bound that we provide, we show that a section $s'$ has singular scheme containing that of $s$ if and only if $s'=\phi(s)$ for some global endomorphism $\phi$ of the tangent bundle of $M$.

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Jorge Olivares, Daniel Posada-Buriticá. 2026-08-10. Effective Results for Foliations on Smooth Projective Complete Intersection Surfaces. https://arxiv.org/abs/2608.09877

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