Genera of non-algebraic leaves of polynomial foliations of $\mathbb C^2$

TL;DR

This paper investigates the topology of leaves in polynomial foliations of complex plane, showing that generic foliations have leaves with infinitely many handles, and constructing dense subsets with leaves having many handles.

Contribution

It introduces new results on the topology of leaves in polynomial foliations, including the existence of leaves with many handles and the generic property of leaves having infinitely many handles.

Findings

Dense subset of polynomial foliations with leaves having at least a specified number of handles.

Generic foliations invariant under a symmetry have leaves with infinitely many handles.

Abstract

In this article, we prove two results. First, we construct a dense subset in the space of polynomial foliations of degree $n$ such that each foliation from this subset has a leaf with at least $\frac{(n+1)(n+2)}2-4$ handles. Next, we prove that for a generic foliation invariant under the map $(x, y)\mapsto (x, -y)$ all leaves have infinitely many handles.