Extension of germs of holomorphic foliations

TL;DR

This paper investigates conditions under which germs of plane holomorphic foliations can be extended to compact surfaces, highlighting the role of meromorphic first integrals and polynomial representations.

Contribution

It demonstrates that germs becoming regular after one blow-up and possessing meromorphic first integrals can be extended to compact surfaces, with some being polynomially definable.

Findings

Germs with meromorphic first integrals can be extended after local modifications.

Simple elements in this class are polynomially definable.

Without meromorphic first integrals, many elements lack polynomial representations.

Abstract

We consider the problem of extending germs of plane holomorphic foliations to foliations of compact surfaces. We show that the germs that become regular after a single blow up and admit meromorphic first integrals can be extended, after local changes of coordinates, to foliations of compact surfaces. We also show that the simplest elements in this class can be defined by polynomial equations. On the other hand we prove that, in the absence of meromorphic first integrals there are uncountably many elements without polynomial representations.