Symplectic Parabolicity and L^2 Symplectic Harmonic Forms

TL;DR

This paper investigates the relationship between symplectic cohomologies, harmonic forms, and topological invariants of compact symplectic manifolds, establishing an inequality involving the Euler number under certain conditions.

Contribution

It proves that for compact symplectic parabolic manifolds satisfying the hard Lefschetz property, the Euler number meets a specific inequality, linking symplectic geometry and topology.

Findings

Euler number inequality for symplectic parabolic manifolds

Connection between symplectic harmonic forms and topological invariants

Extension of Tseng and Yau's symplectic cohomology concepts

Abstract

In this paper, we study the symplectic cohomologies and symplectic harmonic forms which introduced by Tseng and Yau. Based on this, we get if $(M^{2n},\omega)$ is a compact symplectic parabolic manifold which satisfies the hard Lefschetz property, then its Euler number satisfies the inequality $(-1)^n\chi(M)\geq 0$.