The hard Lefschetz duality for locally conformally almost K\"ahler manifolds

We prove the hard Lefschetz duality for locally conformally almost K\"ahler manifolds. This is a generalization of that for almost K\"ahler manifolds studied by Cirici and Wilson. We generalize the K\"ahler identities to prove the duality. Based on the result, we introduce the hard Lefschetz condition for locally conformally symplectic manifolds. As examples, we give solvmanifolds which do not satisfy the hard Lefschetz condition.