de Rham and Dolbeault Cohomology of solvmanifolds with local systems

TL;DR

This paper generalizes the relationship between Lie algebra cohomology and solvmanifold cohomology to include local systems, providing explicit complexes for computing cohomology even when classical isomorphisms fail.

Contribution

It introduces a new isomorphism involving characters and flat line bundles, extending known results from nilpotent to solvable Lie groups, and constructs explicit cochain complexes for cohomology calculations.

Findings

Established a generalized isomorphism for cohomology with local systems on solvmanifolds.

Constructed explicit finite-dimensional cochain complexes for cohomology computations.

Extended Dolbeault cohomology calculations to complex parallelizable solvmanifolds.

Abstract

Let $G$ be a simply connected solvable Lie group with a lattice $\Gamma$ and the Lie algebra $\g$ and a representation $\rho:G\to GL(V_{\rho})$ whose restriction on the nilradical is unipotent. Consider the flat bundle $E_{\rho}$ given by $\rho$. By using "many" characters $\{\alpha\}$ of $G$ and "many" flat line bundles $\{E_{\alpha}\}$ over $G/\Gamma$, we show that an isomorphism \[\bigoplus_{\{\alpha\}} H^{\ast}(\g, V_{\alpha}\otimes V_{\rho})\cong \bigoplus_{\{E_{\alpha}\}} H^{\ast}(G/\Gamma, E_{\alpha}\otimes E_{\rho})\] holds. This isomorphism is a generalization of the well-known fact:"If $G$ is nilpotent and $\rho$ is unipotent then, the isomorphism $H^{\ast}(\g, V_{\rho})\cong H^{\ast}(G/\Gamma, E_{\rho})$ holds". By this result, we construct an explicit finite dimensional cochain complex which compute the cohomology $H^{\ast}(G/\Gamma, E_{\rho})$ of solvmanifolds even if the isomorphism $H^{\ast}(\g, V_{\rho})\cong H^{\ast}(G/\Gamma, E_{\rho})$ 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.