arXiv subjects
Publications and source records attributed to Alexis Garcia.
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.
We give a geometric condition on a meromorphic affine connection for its Killing vector fields to be single valued. More precisely, this condition relies on the pole of the connection and its geodesics, and defines a subcategory. To this end, we use the equivalence between these objects and meromorphic affine Cartan geometries. The proof of the previous result is then a consequence of a more general result linking the distinguished curves of meromorphic Cartan geometries, their poles and their infinitesimal automorphisms, which is the main purpose of the paper. This enables to extend the classification result from [Biswas I., Dumitrescu S., McKay B., \'Epijournal G\'eom. Alg\`ebrique 3 (2019), 19, 10 pages, arXiv:1804.08949] to the subcategory of meromorphic affine connection described before.