Symplectic solvmanifolds not satisfying the hard-Lefschetz condition

Abstract

For Lie groups $G$ of the form $G = \R^k \ltimes_{\phi} \R^m$, with $k + m$ even, a result of H. Kasuya shows that if the action $\phi:\R^k \to \mathrm{Aut}(\R^m)$ is semisimple then any symplectic solvmanifold $(\Gamma \backslash G, \omega)$ satisfies the hard-Lefschetz condition for any symplectic form. In this article, we prove the converse in the case $k = 1$ and $G$ completely solvable: no symplectic form on such a solvmanifold satisfies the hard-Lefschetz condition if $\phi$ is not semisimple; moreover, we show that the failure occurs either at degree $1$ or at degree $2$ in cohomology, depending on the spectrum of the differential of the action $\phi$. This result is achieved through a detailed analysis of the cohomology groups $H^1(\g)$, $H^2(\g)$, $H^{2n-2}(\g)$, $H^{2n-1}(\g)$ of the Lie algebra $\g$ of such Lie groups. Among other things, this analysis yields useful representatives for each cohomology class corresponding to any symplectic form on $\g$, allowing the most delicate cases to be reduced to a straightforward computation. We also construct lattices for many of the Lie groups under consideration, thereby exhibiting examples of symplectic solvmanifolds of completely solvable Lie groups failing to have the hard-Lefschetz property for any symplectic form.