Hard Lefschetz property of symplectic structures on compact Kähler manifolds

Abstract

In this paper, we give a new method to construct a compact symplectic manifold which does not satisfy the hard Lefschetz property. Using our method, we construct a simply connected compact Kähler manifold (M, ω, J)and a symplectic form σ on M which does not satisfy the hard Lefschetz property, but is symplectically deformation equivalent to the Kähler form ω. As a consequence, we can give an answer to the question posed by Khesin and Mc-Duff as follows. According to symplectic Hodge theory, any symplectic form ω on a smooth manifold M defines symplectic harmonic forms on M. In a paperby D. Yan (1996), Khesin and McDuff posed a question whether there exists a path of symplectic forms {ωt} such that the dimension hk hr(M,ω) of the spaceof symplectic harmonic k-forms varies along t. By Yan and O. Mathieu, the hard Lefschetz property holds for (M,ω) if and only if hk hr(M, ω) is equal to the Betti number bk(M) for all k > 0. Thus our result gives an answer tothe question. Also, our construction provides an example of a compact Kähler manifold whose Kähler cone is properly contained in the symplectic cone. © 2016 American Mathematical Society