On de Rham and Dolbeault Cohomology of Solvmanifolds

For a simply connected (non-nilpotent) solvable Lie group $G$ with a lattice $\mathrm{\Gamma }$ the de Rham and Dolbeault cohomologies of the solvmanifold $G\mathrm{/}\mathrm{\Gamma }$ are not in general isomorphic to the cohomologies of the Lie algebra $\mathfrak g$of$G$\mathrm{.}Inthispaperweconstruct,uptoafinitegroup,anewLiealgebra$\tilde{\mathfrak g}$ whose cohomology is isomorphic to the de Rham cohomology of $G\mathrm{/}\mathrm{\Gamma }$ by using a modification of $G$ associated with a algebraic sub-torus of the Zariski-closure of the image of the adjoint representation. This technique includes the construction due to Guan and developed by the first two authors. In this paper, we also give a Dolbeault version of such technique for complex solvmanifolds, i.e. for solvmanifolds endowed with an invariant complex structure. We construct a finite dimensional cochain complex which computes the Dolbeault cohomology of a complex solvmanifold $G\mathrm{/}\mathrm{\Gamma }$ with holomorphic Mostow bundle and we give a construction of a new Lie algebra $\breve{\mathfrak{g}}$ with a complex structure whose cohomology is isomorphic to the Dolbeault cohomology of $G\mathrm{/}\mathrm{\Gamma }$.