Higher School of Economics

The phase portrait of a polynomial equation in \(\mathbb{C}^2\) with complex time is a
holomorphic foliation of \(\mathbb{C}^2\). The talk is devoted to the topology of (generic
or some specific) leaves of (generic or some specific) polynomial foliations.

For a generic *analytic* foliation of \(\mathbb{C}^2\), at most countable number of
leaves are topological cylinders, all other leaves are topological discs. In
the space \(A_n\) of foliations of \(\mathbb{C}^2\) given by polynomials of degree n, the
analogous result is not known. However, there are some partial results.

Limit cycles (Ilyashenko, 1978; improved by others):

For a generic vector field from \(A_n\), there is a countable number of
(homologically independent) non-contractible loops on the leaves.

Thus either a countable number of leaves are at least cylinders, or
there is a leaf of infinite genus.

Separatrix connections (Volk, 2006):

For a dense subset in \(A_n\), there is a separatrix connection, i.e. two
different fixed points of the vector field have a common separatrix.
This result is not directly related to the topology of the leaves, but the
ideas of the proof are used to prove the following statement.

One handle (NG&YK):
For a dense subset in \(A_n\), there is a leaf with at least one handle.

Infinite genus (NG&YK):
There are examples (with positive dimension in \(A_n\)) of foliations having
the leaves of infinite genus.

I will outline the proofs of these four results.