Cohomology of solvable lie algebras and solvmanifolds

The cohomology $H^*_{\lambda\omega}(G/\Gamma,\mathbb C)$ of the de Rham complex $\Lambda^*(G/\Gamma)\otimes\mathbb C$ of a compact solvmanifold $G/\Gamma$ with deformed differential $d_{\lambda\omega}=d+\lambda\omega$, where $\omega$ is a closed 1-form, is studied. Such cohomologies naturally arise in Morse–Novikov theory. It is shown that, for any completely solvable Lie group $G$ containing a cocompact lattice $\Gamma\subset G$, the cohomology $H^*_{\lambda\omega}(G/\Gamma,\mathbb C)$ is isomorphic to the cohomology $H^*_{\lambda\omega}(\mathfrak g)$ of the tangent Lie algebra $\mathfrak g$ of the group $G$ with coefficients in the one-dimensional representation $\rho_{\lambda\omega}\colon\mathfrak g\to\mathbb K$ defined by $\rho_{\lambda\omega}(\xi)=\lambda\omega(\xi)$. Moreover, the cohomology $H^*_{\lambda\omega}(G/\Gamma,\mathbb C)$ is nontrivial if and only if $-\lambda[\omega]$ belongs to a finite subset $\widetilde\Omega_{\mathfrak g}$ of $H^1(G/\Gamma,\mathbb C)$ defined in terms of the Lie algebra $\mathfrak g$.