The hard Lefschetz duality for locally conformally almost Kähler manifolds

We prove the hard Lefschetz duality for locally conformally almost K & auml;hler manifolds. This is a generalization of that for almost K & auml;hler manifolds studied by Cirici and Wilson. We generalize the K & auml;hler 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. (c) 2025 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).