Uniformizing complex ODEs and applications

  • Julio C. Rebelo

    Université Toulouse 3, France
  • Helena Reis

    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 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.

Cite this article

Julio C. Rebelo, Helena Reis, Uniformizing complex ODEs and applications. Rev. Mat. Iberoam. 30 (2014), no. 3, pp. 799–874

DOI 10.4171/RMI/800