2019-01-23  03:33 hrs.

### The topology of solutions of polynomial vector fields in $\mathbb{C}^2$

#### Nataliya GoncharukHigher 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.

#### Palapa Guillermo Torres -- Jueves 22 de mayo de 2014, 16:00 horas

Unidad Cuernavaca del Instituto de Matemáticas UNAM