2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/99948We 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 G/L where G is a simply connected solvable Lie group and L is a lattice of G, and every complex structure J on S is the canonical complex structure induced from a left-invariant complex structure on G. We also give a complete list of all the complex structures on four-dimensional compact homogeneous spaces, referring to their corresponding complex surfaces.27 pagesComplex VariablesDifferential GeometrySymplectic Geometry32Q55; 53C30; 14J80text