ON DE RHAM AND DOLBEAULT COHOMOLOGY OF SOLVMANIFOLDS

Abstract

For a simply connected (non-nilpotent) solvable Lie group G with a lattice Γ the de Rham and Dolbeault cohomologies of the solvmanifold G/Γ are not in general isomorphic to the cohomologies of the Lie algebra g of G. In this paper we construct, up to a finite group, a new Lie algebra eg whose cohomology is isomorphic to the de Rham cohomology of G/Γ by using a modification of G associated with an 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/Γ with holomorphic Mostow bundle and we give a construction of a new Lie algebra $$ \overset{\smile }{\mathfrak{g}} $$ with a complex structure whose cohomology is isomorphic to the Dolbeault cohomology of G/Γ.