Hard Lefschetz property of symplectic structures on compact Kaehler manifolds

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 $(M,J,ω)$ 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 McDuff as follows. According to symplectic Hodge theory, any symplectic form $ω$ on a smooth manifold $M$ defines \textit{symplectic harmonic forms} on $M$. In \cite{Yan}, Khesin and McDuff posed a question whether there exists a path of symplectic forms $\{ω_t \}$ such that the dimension $h^k_{hr}(M,ω)$ of the space of \textit{symplectic harmonic $k$-forms} varies along $t$. By \cite{Yan} and \cite{Ma}, the hard Lefschetz property holds for $(M,ω)$ if and only if $h^k_{hr}(M,ω)$ is equal to the Betti number $b_k(M)$ for all $k>0$. Thus our result gives an answer to the question. Also, our construction provides an example of compact Kähler manifold whose Kähler cone is properly contained in the symplectic cone (c.f. \cite{Dr}).

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