Integrable deformations of foliations: a generalization of Ilyashenko's\n result

2018/11/10 by Dominique Cerveau, Cerveau, Dominique, Bruno Scárdua +1
Mathematics · Physics and Astronomy · #32S65 #57R30 #Advanced Differential Equations and Dynamical Systems #Advanced Differential Geometry Research #Algebraic Geometry (math.AG) #Complex Variables (math.CV) #FOS: Mathematics #Nonlinear Waves and Solitons #Primary 37F75 #Secondary 32M25

paper · pdf · doi:10.48550/arxiv.1811.04222

openalex publication_date 2018/11/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study analytic deformations of holomorphic differential 1-forms. The\ninitial 1-form is exact homogeneous and the deformation is by polynomial\nintegrable 1-forms. We investigate under which conditions the elements of the\ndeformation are still exact or, more generally, exhibit a first integral. Our\nresults are related to natural extensions of classical results of Ilyashenko on\nlimit cycles of perturbations of hamiltonian systems in two complex variables.\n

Related

• Schlesinger foliation for deformations of foliations
• Rapid evolution of complex limit cycles
• Analytic varieties invariant by foliations and Pfaff systems
• Universal Deformations of a Curve and a Differential
• On a theorem of Lyapunov-Poincaré in higher dimension