Title: Abelian integrals in holomorphic foliations
Abstract: 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.