Symplectic solvmanifolds not satisfying the hard-Lefschetz condition

TL;DR

This paper characterizes when symplectic solvmanifolds fail the hard-Lefschetz condition, showing failure occurs precisely when the associated action is not semisimple, with detailed cohomological analysis and explicit examples.

Contribution

It proves the converse of Kasuya's result for certain solvable Lie groups, linking non-semisimple actions to failure of the hard-Lefschetz condition and providing explicit cohomology representatives and examples.

Findings

Failure of hard-Lefschetz occurs at degree 1 or 2 depending on the spectrum.

Non-semisimple actions lead to failure of the hard-Lefschetz condition.

Explicit cohomology representatives and examples of such solvmanifolds are constructed.

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.