Symplectic solvmanifolds not satisfying the hard-Lefschetz condition

2025/05/12 by Andrada, Adrián, Garrone, Agustín
#22E25 #22E40 #53D05 #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2505.08113

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 → 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 H¹(\g), H²(\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.

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