Abelian integrals in holomorphic foliations
Copyright policy )Abelian integrals in holomorphic foliations
The aim of this paper is to introduce the theory of Abelian integrals for holomorphic foliations in a complex manifold of dimension two. We will show the importance of Picard-Lefschetz theory and the classification of relatively exact 1-forms in this theory. As an application we identify some irreducible components of the space of holomorphic foliations of a fixed degree and with a center singularity in the projective space of dimension two. Also we calculate higher Melnikov functions under some generic conditions.
- Max Planck Society Germany
14D99, Ordinary differential equations and connections with real algebraic geometry (fewnomials, desingularization, zeros of abelian integrals, etc.), limit cycles, holomorphic foliations, algebro-geometric approach, Theory of limit cycles of polynomial and analytic vector fields (existence, uniqueness, bounds, Hilbert's 16th problem and ramifications) for ordinary differential equations, Hilbert's 16th problem, Mathematics - Algebraic Geometry, 57R30 - 14D99 - 32G34, Picard-Lefschetz theory, 32G34, Singularities of holomorphic vector fields and foliations, 57R30, FOS: Mathematics, holonomy, Dynamical aspects of holomorphic foliations and vector fields, Algebraic Geometry (math.AG)
14D99, Ordinary differential equations and connections with real algebraic geometry (fewnomials, desingularization, zeros of abelian integrals, etc.), limit cycles, holomorphic foliations, algebro-geometric approach, Theory of limit cycles of polynomial and analytic vector fields (existence, uniqueness, bounds, Hilbert's 16th problem and ramifications) for ordinary differential equations, Hilbert's 16th problem, Mathematics - Algebraic Geometry, 57R30 - 14D99 - 32G34, Picard-Lefschetz theory, 32G34, Singularities of holomorphic vector fields and foliations, 57R30, FOS: Mathematics, holonomy, Dynamical aspects of holomorphic foliations and vector fields, Algebraic Geometry (math.AG)
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