On de Rham and Dolbeault Cohomology of Solvmanifolds

TL;DR

This paper develops methods to compute de Rham and Dolbeault cohomologies of solvmanifolds by constructing modified Lie algebras whose cohomologies match those of the manifolds, extending previous techniques.

Contribution

It introduces a new algebraic modification technique for solvmanifolds to compute their cohomologies, including a Dolbeault version for complex structures.

Findings

Constructed a Lie algebra $ ilde{rak g}$ matching the de Rham cohomology

Developed a Dolbeault cohomology computation method for complex solvmanifolds

Provided explicit finite-dimensional cochain complexes for cohomology calculations

Abstract

For a simply connected (non-nilpotent) solvable Lie group $G$ with a lattice $\Gamma$ the de Rham and Dolbeault cohomologies of the solvmanifold $G/\Gamma$ are not in general isomorphic to the cohomologies of the Lie algebra $\mathfrak g$ of $G$. In this paper we construct, up to a finite group, a new Lie algebra $\tilde{\mathfrak g}$ whose cohomology is isomorphic to the de Rham cohomology of $G/\Gamma$ 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 $G/\Gamma$ 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 $G/\Gamma$.