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arXiv · 2205.08626

Integrability of vector fields and meromorphic solutions

Julio C. Rebelo·Helena Reis
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Abstract

Let $\mathcal{F}$ be a foliation defined on a complex projective manifold $M$ of dimension $n$ and admitting a holomorphic vector field $X$ tangent to it along some non-empty Zariski-open set. In this paper we prove that if $X$ has sufficiently many integral curves that are given by meromorphic functions defined on $\mathbb{C}$ then the restriction of $\mathcal{F}$ to any invariant complex $2$-dimensional analytic set admits a first integral of Liouvillean type. In particular, on $\mathbb{C}^3$, every rational vector fields whose solutions are meromorphic functions defined on $\mathbb{C}$ admits a non-empty invariant analytic set of dimension $2$ where the restriction of the vector field yields a Liouvillean integrable foliation.

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Julio C. Rebelo, Helena Reis. 2022-05-17. Integrability of vector fields and meromorphic solutions. https://arxiv.org/abs/2205.08626

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