Hard Lefschetz Condition on symplectic non-K\"ahler solvmanifolds

TL;DR

This paper constructs new examples of compact complex manifolds that are non-Kähler but admit symplectic structures satisfying the Hard Lefschetz Condition, expanding understanding of symplectic geometry on solvmanifolds.

Contribution

It introduces explicit families of non-Kähler solvmanifolds with symplectic structures meeting the Hard Lefschetz Condition, including detailed cohomological analysis and lattice constructions.

Findings

Existence of non-Kähler manifolds with Hard Lefschetz symplectic structures

Explicit lattice constructions for solvable Lie groups

Cohomological computations of de Rham and Dolbeault cohomologies

Abstract

We provide new families of compact complex manifolds with no K\"ahler structure carrying symplectic structures satisfying the \textit{Hard Lefschetz Condition}. These examples are obtained as compact quotients of the solvable Lie group $\mathbb{C}^{2n} \ltimes_{\rho} \mathbb{C}^{2m}$, for which we construct explicit lattices. By cohomological computations we prove that such manifolds carry symplectic structures satisfying the \textit{Hard Lefschetz Condition}. Furthermore, we compute the Kodaira dimension of an almost-K\"ahler structure and generators for the de Rham and Dolbeault cohomologies.