  For a complete symplectic manifold $M^{2n}$, we define the $L^{2}$-hard
Lefschetz property on $M^{2n}$. We also prove that the complete symplectic
manifold $M^{2n}$ satisfies $L^{2}$-hard Lefschetz property if and only if
every class of $L^{2}$-harmonic forms contains a $L^{2}$ symplectic harmonic
form. As an application, we get if $M^{2n}$ is a closed symplectic parabolic
manifold which satisfies the hard Lefschetz property, then its Euler
characteristic satisfies the inequality $(-1)^{n}\chi(M^{2n})\geq0$.
