Four-dimensional compact solvmanifolds with and without complex analytic structures

We classify four-dimensional compact solvmanifolds up to diffeomorphism, while determining which of them have complex analytic structures. In particular, we shall see that a four-dimensional compact solvmanifold S can be written, up to double covering, as Γ\G where G is a simply connected solvable Lie group and Γ is a lattice of G, and every complex structure J on S is the canonical complex structure induced from a leftinvariant complex structure on G. We are thus led to conjecture that complex analytic structures on compact solvmanifolds are all canonical.