We study the cohomology $H^*_{ \lambda \omega}(G/\Gamma, { \mathbb C})$ of the deRham complex $\Lambda^*(G/\Gamma)\otimes{ \mathbb C}$ of a compact solvmanifold $G/\Gamma$ with a deformed differential $d_{ \lambda \omega}=d + \lambda\omega$, where $\omega$ is a closed 1-form. This cohomology naturally arises in the Morse-Novikov theory. We show that for a solvable Lie group $G$ with a completely solvable Lie algebra $\mathfrak{ g}$ and a cocompact lattice $\Gamma \subset G$ the cohomology $H^*_{ \lambda \omega}(G/\Gamma, { \mathbb C})$ coincides with the cohomology $H^*_{ \lambda \omega}(\mathfrak{ g})$ of the Lie algebra $\mathfrak{ g}$ associated with the one-dimensional representation $\rho_{ \lambda \omega}: \mathfrak{ g} \to { \mathbb K}, \rho_{ \lambda \omega}(\xi) = \lambda \omega(\xi)$. Moreover $H^*_{ \lambda \omega}(G/\Gamma, { \mathbb C})$ is non-trivial if and only if $-\lambda [\omega]$ belongs to the finite subset $\{ 0\} \cup \tilde \Omega_{ \mathfrak{ g}}$ in $H^1(G/\Gamma, { \mathbb C})$ well defined in terms of $\mathfrak{ g}$.
Categories: math.DG math.RA
