Revista Matemática Iberoamericana

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Volume 30, Issue 3, 2014, pp. 799–874
DOI: 10.4171/RMI/800

Published online: 2014-08-27

Uniformizing complex ODEs and applications

Julio C. Rebelo[1] and Helena Reis[2]

(1) Université Toulouse 3, France
(2) Universidade do Porto, Portugal

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.

Keywords: Complex ODE, maximal domain of solutions, entire solutions, Halphen equations

Rebelo Julio, Reis Helena: Uniformizing complex ODEs and applications. Rev. Mat. Iberoamericana 30 (2014), 799-874. doi: 10.4171/RMI/800