Meromorphic vector fields with single-valued solutions on complex surfaces

Abstract

We study ordinary differential equations in the complex domain given by meromorphic vector fields on Kähler 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.

post-referee version

References in corpus (4)

• Foliations on complex projective surfaces
• Invariant sets near singularities of holomorphic foliations
• The geometry of Chazy's homogeneous third-order differential equations
• Completeness is determined by any non-algebraic trajectory

Cited by in corpus (1)

• On the geometry, flows and visualization of singular complex analytic vector fields on Riemann surfaces