$L^2$-hard Lefschetz complete symplectic manifolds

Abstract

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 $(-1{)}^n\chi (M^{2n})\ge 0$.

Published in Ann. Mat. Pura Appl