On Compact Affine Quaternionic Curves and Surfaces

This paper is devoted to the study of affine quaternionic manifolds and to a possible classification of all compact affine quater- nionic curves and surfaces. It is established that on an affine quater- nionic manifold there is one and only one affine quaternionic structure. A direct result, based on the celebrated Kodaira Theorem that classifies all compact complex manifolds in complex dimension 2, states that the only compact affine quaternionic curves are the quaternionic tori. As for compact affine quaternionic surfaces, the study of their fundamental groups, together with the inspection of all nilpotent hypercomplex sim- ply connected 8-dimensional Lie Groups, identifies a path towards their classification.