$L^{2}$-hard Lefschetz complete symplectic manifolds

TL;DR

This paper introduces the $L^{2}$-hard Lefschetz property for complete symplectic manifolds, characterizes it via harmonic forms, and derives inequalities for the Euler characteristic in certain cases.

Contribution

It defines the $L^{2}$-hard Lefschetz property for complete symplectic manifolds and establishes an equivalence with harmonic form classes, providing new insights into symplectic geometry.

Findings

$L^{2}$-hard Lefschetz property characterized by harmonic forms

Equivalence between $L^{2}$-hard Lefschetz property and harmonic form classes

Euler characteristic inequality for closed symplectic parabolic 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 $(-1)^{n}\chi(M^{2n})\geq0$.