Symplectic solvmanifolds not satisfying the hard-Lefschetz condition
Adrián Andrada, Agustín Garrone
TL;DR
The paper addresses whether a symplectic solvmanifold $\Gamma\backslash G$ with $G=\mathbb{R}^k \ltimes_{\phi} \mathbb{R}^m$ satisfies the hard-Lefschetz condition, building on Kasuya’s result that semisimple actions guarantee HL. It proves, for the case $k=1$ and completely solvable $G$, that non-semisimple actions $\phi$ preclude HL for any invariant symplectic form, with the first failure occurring in degree 1 or 2 depending on the spectrum of the action; this is established by a thorough Lie algebra cohomology analysis of almost abelian algebras, including an explicit decomposition into double generalized eigenspaces and circuit-based representatives of cohomology classes. The authors also construct lattices in many of these groups to yield explicit compact counterexamples, reinforcing a sharp dichotomy: HL holds for semisimple actions and fails for non-semisimple ones in this setting. The results illuminate how Kähler-like cohomological properties break down in non-semisimple solvmanifolds and provide concrete tools to identify HL failure degrees. Overall, the work clarifies the structure of cohomology and symplectic forms on almost abelian solvmanifolds and highlights a path to generating HL-obstructed examples via lattices.
Abstract
For Lie groups $G$ of the form $G = \R^k \ltimes_φ \R^m$, with $k + m$ even, a result of H. Kasuya shows that if the action $φ:\R^k \to \mathrm{Aut}(\R^m)$ is semisimple then any symplectic solvmanifold $(Γ\backslash G, ω)$ 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 $φ$ 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 $φ$. 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.