On the classification of Meromorphic Affine Connections on Complex Compact Surfaces

We classify meromorphic affine connections on compact complex surfaces with algebraic dimension one, extending the work of Inoue,Kobayashi and Ochiai (1981) in the holomorphic case. The motivation is to investigate possible extension of the principle of uniformization of complex compact curves by means of geometric structures, in higher dimensions and allowing singularities to the geometric structures.