2010/04/15 by S. Ivashkovich, Sergey Ivashkovich, Ivashkovich, Sergey · 1 citation
Mathematics · Physics and Astronomy · #32H04 #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry (math.AG) #Complex Variables (math.CV) #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Primary - 37F10 #Quantum chaos and dynamical systems #Secondary - 32D20 #math.AG #math.CV #math.DS #msc:32D20 #msc:32H04 #msc:37F10
paper · pdf · doi:10.48550/arxiv.1004.2618
49 pages, 2 figures
arxiv created 2010/04/15 · openalex publication_date 2010/04/15 · arxiv updated 2010/04/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We prove that every trajectory of a polynomial vector field on the complex projective plane accumulates to the singular locus of the vector field. This statement represents a holomorphic version of the Poincare-Bendixson theorem and solves the complex analytic counterpart of Hilbert's 16th problem. The main result can be also reformulated as the nonexistence of "exceptional minimals" of holomorphic foliations on \pp2 and, in particular, implies the nonexistence of real analytic Levi flat hypersurfaces in the complex projective plane. Finally, we describe (in the first approximation) the way a minimal complex trajectory approaches the singular locus of the vector field.