Relative Cohomology and Deformations of Foliations with Holomorphic First Integrals

• Victor León ^[1] ; Bruno Scárdua ^[2]
  1. [1] Universidade Federal da Integração Latino-Americana

     Universidade Federal da Integração Latino-Americana

     Brasil

     
  2. [2] Universidade Federal do Rio de Janeiro

     Universidade Federal do Rio de Janeiro

     Brasil

     
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 25, Nº 5, 2026
• Idioma: inglés
• Enlaces
  • Texto completo
• Resumen

  • This work is devoted to the study and computation of relative cohomology for 1-forms defining foliations with holomorphic first integrals.We study both the local and global cases. In the local setting, we study solutions h of the cohomological equation η ∧ ω0 = dh ∧ ω0, where ω0 = py dx + qx dy admits the holomorphic first integral φ = x p yq , with p, q ∈ N, in local coordinates (x, y) near the origin of C2. We prove the existence of a local canonical form for a given meromorphic 1-form η in terms of ω0 and φ, provided that η has only simple poles contained in the coordinate axes (xy = 0). We then characterize relative exactness by the vanishing of the periods of η over the first homology of the fibers {φ = c}. In the global setting, we prove polynomial versions of the local results on C2 for fibrations φ = f pgq associated with (p, q)-Morse type pairs. The hypotheses include connected regular fibers, nondegenerate affine critical points away from ( f g = 0), and specified local models at infinity. We also obtain variants of Ilyashenko’s integration theorem concerning the exactness of polynomial 1-forms whose periods vanish on the fibers of a polynomial. In the homogeneous polynomial case, we prove an integration lemma and a rigidity result for fibrations of the form φ = f pgq .

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