Meromorphic vector fields with single-valued solutions on complex surfaces

TL;DR

This paper investigates meromorphic vector fields on complex surfaces, showing that if they admit a maximal single-valued solution with dense image, then the vector field is either holomorphic or preserves a fibration, up to bimeromorphic transformations.

Contribution

It characterizes the structure of meromorphic vector fields with dense solutions on complex surfaces, revealing conditions under which they are holomorphic or preserve fibrations.

Findings

If a meromorphic vector field has a dense single-valued solution, it is either holomorphic or preserves a fibration.

The result applies to solutions with Zariski-dense images, including entire solutions.

The classification is up to bimeromorphic transformations.

Abstract

We study ordinary differential equations in the complex domain given by meromorphic vector fields on K\"ahler compact complex surfaces. We prove that if such an equation has a maximal single valued solution with Zariski-dense image (in particular, if it has an entire one) then, up to a bimeromorphic transformation, either the vector field is holomorphic or it preserves a fibration.