Moduli spaces of germs of holomorphic foliations in the plane.

*(English)*Zbl 1054.32018Roughly speaking, the moduli space of germs of singular holomorphic foliations in the complex plane around the origin is the set of holomorphic foliations which are topologically the same, up to equivalence under local holomorphic changes of coordinates around the origin. As is usual in moduli problems there are some starting topological data (for the foliations), the question is about the shape and natural complex structure in the moduli space.

The main goal of the paper is the study of moduli spaces for foliations whose vanishing order at the origin is two or three. The first result is that two topological conjugated general foliations have topologically conjugated holonomy representations. “General” is a technical condition on the lower homogeneous part of the 1-form defining the foliation. Also a topological classification is given for generic homogeneous foliations. The author introduces a strong hypothesis; a general foliation is in addition “non abelian” if the holonomy is non abelian and the residues, of a suitable 1-form associated to the foliation, generates a dense additive subgroup of the complex line.

The author performs the computation of the moduli space of general and non abelian foliations, having vanishing order two; the moduli space is just one point, i.e., such a foliation is rigid. For vanishing of order three, the moduli space from one of these foliations, is a connected covering of the complex line with two points deleted, whose fundamental group is either trivial or isomorphic to the integers, and it is completely determined by the holonomy representation of the original foliation. Also, an explicit criterion for topological conjugation between germs of general non abelian foliations, and some technical moduli results for foliations having vanishing of higher order, are given.

The main goal of the paper is the study of moduli spaces for foliations whose vanishing order at the origin is two or three. The first result is that two topological conjugated general foliations have topologically conjugated holonomy representations. “General” is a technical condition on the lower homogeneous part of the 1-form defining the foliation. Also a topological classification is given for generic homogeneous foliations. The author introduces a strong hypothesis; a general foliation is in addition “non abelian” if the holonomy is non abelian and the residues, of a suitable 1-form associated to the foliation, generates a dense additive subgroup of the complex line.

The author performs the computation of the moduli space of general and non abelian foliations, having vanishing order two; the moduli space is just one point, i.e., such a foliation is rigid. For vanishing of order three, the moduli space from one of these foliations, is a connected covering of the complex line with two points deleted, whose fundamental group is either trivial or isomorphic to the integers, and it is completely determined by the holonomy representation of the original foliation. Also, an explicit criterion for topological conjugation between germs of general non abelian foliations, and some technical moduli results for foliations having vanishing of higher order, are given.

Reviewer: Jesus Muciño-Raymundo (Morelia)