Moscow Mathematical Journal

Volume 18, Issue 1, January–March 2018 pp. 63–83.

Genera of Non-Algebraic Leaves of Polynomial Foliations of ℂ²

Authors: Nataliya Goncharuk (1) and Yury Kudryashov (1)
Author institution:(1) Higher School of Economics, Department of Mathematics, 20 Myasnitskaya street, Moscow 101000, Russia
Cornell University, College of Arts and Sciences, Department of Mathematics, 310 Mallot Hall, Ithaca, NY, 14853, US

Summary:

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 (n+1)(n+2)/2−4 handles. Next, we prove that for a generic foliation invariant under the map (x,y) ↦ (x, −y) all leaves (except for a finite set of algebraic leaves) have infinitely many handles.

2010 Math. Subj. Class. Primary: 37F75; Secondary: 32M25.

Keywords: Riemann surfaces, complex foliations, polynomial foliations, complex limit cycles.

Contents Full-Text PDF