de Rham and Dolbeault Cohomology of solvmanifolds with local systems

Abstract

Let $G$ be a simply connected solvable Lie group with a lattice $Γ$ and the Lie algebra $\g$ and a representation $ρ:G\to GL(V_ρ)$ whose restriction on the nilradical is unipotent. Consider the flat bundle $E_ρ$ given by $ρ$. By using "many" characters $\{α\}$ of $G$ and "many" flat line bundles $\{E_α\}$ over $G/Γ$, we show that an isomorphism \[\bigoplus_{\{α\}} H^{\ast}(\g, V_α\otimes V_ρ)\cong \bigoplus_{\{E_α\}} H^{\ast}(G/Γ, E_α\otimes E_ρ)\] holds. This isomorphism is a generalization of the well-known fact:"If $G$ is nilpotent and $ρ$ is unipotent then, the isomorphism $H^{\ast}(\g, V_ρ)\cong H^{\ast}(G/Γ, E_ρ)$ holds". By this result, we construct an explicit finite dimensional cochain complex which compute the cohomology $H^{\ast}(G/Γ, E_ρ)$ of solvmanifolds even if the isomorphism $H^{\ast}(\g, V_ρ)\cong H^{\ast}(G/Γ, E_ρ)$ does not hold. For Dolbeault cohomology of complex parallelizable solvmanifolds, we also prove an analogue of the above isomorphism result which is a generalization of computations of Dolbeault cohomology of complex parallelizable nilmanifolds. By this isomorphism, we construct an explicit finite dimensional cochain complex which compute the Dolbeault cohomology of complex parallelizable solvmanifolds.