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.
Cite this article
Hossein Movasati, Abelian integrals in holomorphic foliations. Rev. Mat. Iberoam. 20 (2004), no. 1, pp. 183–204DOI 10.4171/RMI/385