Hard Lefschetz condition for locally conformally Kähler manifolds

Prove that every 2n-dimensional compact locally conformally Kähler manifold (M, J, g) with Lee form θ satisfies the hard Lefschetz condition for locally conformally symplectic manifolds, namely that for each j = 1, …, n the Lefschetz maps [L]^{n−j}: H^j_{−(n−j)/2}(M) → H^{2n−j}_{(n−j)/2}(M), induced by wedge multiplication with the fundamental 2-form ω of (J, g) in the twisted de Rham cohomology defined by d_k = d − k θ∧, are isomorphisms.

Background

This paper develops a hard Lefschetz duality for locally conformally almost Kähler manifolds and formulates a hard Lefschetz condition (HLC) for locally conformally symplectic manifolds using twisted de Rham cohomology. In the classical symplectic setting, all compact Kähler manifolds satisfy the hard Lefschetz theorem.

Motivated by the classical result, the authors propose the conjecture that every compact locally conformally Kähler (LCK) manifold should satisfy the newly defined HLC. The HLC here is expressed via the twisted cohomology groups H^j_k(M) defined by the twisted differential d_k = d − k θ∧, where θ is the Lee form, and the Lefschetz operator is induced by wedge with the fundamental 2-form ω. The conjecture asserts that the corresponding Lefschetz maps are isomorphisms in this twisted cohomology for all degrees j up to n.