Moscow Mathematical Journal

Volume 24, Issue 2, April–June 2024 pp. 181–199.

Monodromy Problem and Tangential Center-Focus Problem for Products of Lines in General Position in $\mathbb{P}^2$

Authors: Daniel López García (1)
Author institution:(1) Instituto de Matemática e Estatística da Universidade de São Paulo (IME-USP), Rua do Matão, 1010, São Paulo 05508-090, SP, Brazil

Summary:

We consider a rational map $F$ defined by a quotient of products of lines in general position and we study the monodromy problem and the tangential center-focus problem for the fibration associated with $F$ . Thus, we study the submodule of the 1-homology group of a regular fiber of $F$ generated by the orbit of the monodromy action on a vanishing cycle. Moreover, we characterize the meromorphic 1-forms $\omega$ in $\mathbb{P}^2$ such that the Abelian integral $\int_{\delta_t}\omega$ vanishes on a family of cycles $\delta_t$ around a center singularity.

2020 Math. Subj. Class. 34C07, 32S65, 14D05, 34C08.

Keywords: Holomorphic foliations, center problem, monodromy action, Abelian integral.

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