On Complex Solvmanifolds and Affine Structures

Let G be a complex, connected, simply connected, solvable Lie group and U a discrete uniform subgroup. The authors give a sufficient condition, in terms of structure constants of the Lie algebra of G, under which the complex solvmanifold M = G/U has an analytic affine structure (i.e., there is an atlas on M whose transition functions are affine). The condition is fulfilled for all solvmanifolds M = G/U of dimension ≤ 5.