2022/05/17 by Julio C. Rebelo, Rebelo, Julio C., Helena Reis +1
Mathematics · #Advanced Differential Equations and Dynamical Systems #Classical Analysis and ODEs (math.CA) #Complex Variables (math.CV) #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Meromorphic and Entire Functions
paper · pdf · doi:10.48550/arxiv.2205.08626
openalex publication_date 2022/05/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let 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 ℂ then the restriction of F to any invariant complex 2-dimensional analytic set admits a first integral of Liouvillean type. In particular, on ℂ3, every rational vector fields whose solutions are meromorphic functions defined on ℂ admits a non-empty invariant analytic set of dimension 2 where the restriction of the vector field yields a Liouvillean integrable foliation.