On closed currents invariant by holomorphic foliations, I

TL;DR

This paper investigates the properties of closed positive currents invariant under singular holomorphic foliations on algebraic surfaces, demonstrating conditions under which the foliation must preserve an algebraic curve.

Contribution

It establishes new conditions linking invariant currents to algebraic curves in the context of singular holomorphic foliations on algebraic surfaces.

Findings

Invariant currents imply the existence of invariant algebraic curves under certain conditions.

The paper provides criteria for when a foliation must leave an algebraic curve invariant.

Results contribute to understanding the structure of singular holomorphic foliations.

Abstract

We consider closed positive currents invariant by a singular holomorphic foliation on an algebraic surface. We show that under some conditions the foliation must leave invariant an algebraic curve.