Abstract:
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.
Key words and phrases:holomorphic foliations, center problem, monodromy action, Abelian integral.
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