Construction of symplectic solvmanifolds satisfying the hard-Lefschetz condition

TL;DR

This paper constructs explicit examples of symplectic solvmanifolds that satisfy the hard-Lefschetz condition, using Lie algebra cohomology and graph theory, expanding the class of known manifolds with this property.

Contribution

It introduces new families of almost-K"ahler solvmanifolds satisfying the hard-Lefschetz condition and computes their cohomology explicitly, linking Lie algebra structures with combinatorial graph theory.

Findings

Explicit cohomology computations for the constructed manifolds

Verification of the hard-Lefschetz condition on these examples

Connection between Lie algebra cohomology and Kneser graphs

Abstract

A compact symplectic manifold $(M, \omega)$ is said to satisfy the hard-Lefschetz condition if it is possible to develop an analogue of Hodge theory for $(M, \omega)$. This loosely means that there is a notion of harmonicity of differential forms in $M$, depending on $\omega$ alone, such that every de Rham cohomology class in has a $\omega$-harmonic representative. In this article, we study two non-equivalent families of diagonal almost-abelian Lie algebras that admit a distinguished almost-K\"ahler structure and compute their cohomology explicitly. We show that they satisfy the hard-Lefschetz condition with respect to any left-invariant symplectic structure by exploiting an unforeseen connection with Kneser graphs. We also show that for some choice of parameters their associated simply connected, completely solvable Lie groups admit lattices, thereby constructing examples of almost-K\"ahler solvmanifolds satisfying the hard-Lefschetz condition, in such a way that their de Rham cohomology is fully known.