Compatible almost complex structures on the Hard Lefschetz condition

Abstract

For a compact K\"ahler manifold, it is well-established that its de Rham cohomology satisfies the Hard Lefschetz condition, which is reflected in the equality between the Betti numbers and the Hodge numbers. A special subclass of symplectic manifolds also adheres to this condition. Cirici and Wilson \cite{CW20} employ the variant Hodge number to propose a sufficient criterion for compact almost K\"ahler manifolds to satisfy this condition. In this paper, we show that this condition is only sufficient by presenting examples of compact almost K\"ahler manifolds that fulfill the Hard Lefschetz condition while violating the equality between the variant Hodge numbers and Betti numbers, that is, \[b^1>2h^{1,0}.\] This phenomenon contrasts with the behavior observed in compact K\"ahler manifolds.