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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 foli-ations of(More)
In this article we introduce algorithms which compute iterations of Gauss-Manin connections, Picard-Fuchs equations of Abelian integrals and mixed Hodge structure of affine varieties of dimension n in terms of differential forms. In the case n = 1 such computations have many applications in differential equations and counting their limit cycles. For n > 3,(More)
The main objective of this article is to study the topology of the fibers of a generic rational function of the type F p G q in the projective space of dimension two. We will prove that the action of the monodromy group on a single Lefschetz vanishing cycle δ generates the first homology group of a generic fiber of F p G q. In particular, we will prove that(More)