Hard Lefschetz property of symplectic structures on compact Kaehler manifolds

TL;DR

This paper introduces a new method to construct compact symplectic manifolds that do not satisfy the hard Lefschetz property, providing counterexamples and answering a question about symplectic harmonic forms.

Contribution

It presents a novel construction of simply connected compact Kähler manifolds with symplectic forms lacking the hard Lefschetz property, and addresses a question on the variation of symplectic harmonic form dimensions.

Findings

Constructed examples of symplectic forms without the hard Lefschetz property.

Provided an answer to the question on the variation of symplectic harmonic form dimensions.

Showed the existence of Kähler manifolds with a proper inclusion of Kähler and symplectic cones.

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 K\"ahler 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 K\"ahler 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 $h^k_{hr}(M,\omega)$ 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 $h^k_{hr}(M,\omega)$ 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 K\"ahler manifold whose K\"ahler cone is properly contained in the symplectic cone (c.f. \cite{Dr}).