On compact affine quaternionic curves and surfaces

This paper is devoted to the study of affne quaternionic manifolds and to a possible classication of all compact affne quaternionic curves and surfaces. It is established that on an affne quaternionic manifold there is one and only one affne quaternionic structure. A direct result, based on the celebrated Kodaira Theorem that studies compact complex manifolds in complex dimension 2, states that the only compact affne quaternionic curves are the quaternionic tori and the primary Hopf surface S^3 x S^1. As for compact affne quaternionic surfaces, we restrict to the complete ones: the study of their fundamental groups, together with the inspection of all nilpotent hypercomplex simply connected 8-dimensional Lie Groups, identies a path towards their classication.