Uniformizing complex ODEs and applications

Abstract

We introduce a method for estimating the size of the domain of definition of the solutions of a meromorphic vector field on a neighborhood of its pole divisor. The technique relies, in a certain sense, on obtaining a quantitative variant of some well-known results concerning the distance function between complex submanifolds in the presence of metrics with positive curvature. Several applications of these ideas are provided including a type of “confinement theorem” for the solutions of the differential equations associated to complete polynomial vector fields on ${\mathbb{C}}^n$ as well as obstructions to realizing certain germs of vector fields as a singularity of a globally defined holomorphic vector field on a compact Kähler manifold. As a complement, a new approach to certain classical equations is proposed and detailed in the case of Halphen equations.