Symplectic solvmanifolds not satisfying the hard-Lefschetz condition
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866908361931358208 |
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| author | Andrada, Adrián Garrone, Agustín |
| author_facet | Andrada, Adrián Garrone, Agustín |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_08113 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Symplectic solvmanifolds not satisfying the hard-Lefschetz condition Andrada, Adrián Garrone, Agustín Differential Geometry 53D05, 22E25, 22E40 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. |
| title | Symplectic solvmanifolds not satisfying the hard-Lefschetz condition |
| topic | Differential Geometry 53D05, 22E25, 22E40 |
| url | https://arxiv.org/abs/2505.08113 |