# Construction of almost-Kähler solvmanifolds satisfying the hard-Lefschetz condition

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Agustín Nicolás Garrone

created by daniele on 08 Sep 2024

16 sep 2024
-- 14:30

Aula Tricerri, DIMaI, Firenze

Seminari di Geometria del Dini

**Abstract.**

A compact symplectic manifold (M,ω) is said to satisfy the hard-Lefschetz condition if, loosely speaking, there exists a notion of harmonicity of differential forms on M, depending solely on ω, such that every de Rham cohomology class has an ω-harmonic representative.

In this talk, we study two non-equivalent families of diagonal almost-abelian Lie algebras that admit a distinguished almost-Kähler structure, from which one can construct solvmanifolds that satisfy the hard Lefschetz condition with respect to any symplectic form. We accomplish this through the following three-step program: First, we compute the cohomology of these Lie algebras explicitly. Second, we use a combinatorial feature involving Kneser graphs and the aforementioned almost-Kähler structure to establish that the hard Lefschetz property holds for every symplectic form. Finally, we show that, for some choice of parameters, the simply connected, completely solvable Lie groups associated with these Lie algebras admit lattices.

This is joint work with Adrián Andrada.