Hard Lefschetz property of symplectic structures on compact Kaehler 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 $(M,J,\omega )$ and a symplectic form $\sigma$ on $M$ which does not satisfy the hard Lefschetz property, but is symplectically deformation equivalent to the Kähler form $\omega$. 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 $\omega$ 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 $\{{\omega }_t\}$ such that the dimension $h_{hr}^k(M,\omega )$ of the space of \textit{symplectic harmonic $k$-forms} varies along $t$. By \cite{Yan} and \cite{Ma}, the hard Lefschetz property holds for $(M,\omega )$ if and only if $h_{hr}^k(M,\omega )$ 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}).

28 pages. To appear in Transactions of the AMS

References in corpus (2)

• Hyperbolic geometry and non-Kahler manifolds with trivial canonical bundle
• Symplectic resolutions, Lefschetz property and formality

Cited by in corpus (1)

• Unimodality of the Betti numbers for Hamiltonian circle actions with index-increasing moment maps