Primitive cohomology of degree 2 on compact symplectic manifolds

TL;DR

This paper introduces a new operator on compact almost Kähler manifolds, explores conditions for certain cohomological properties, and investigates relationships between different symplectic cohomologies on 4-manifolds.

Contribution

It defines the generalized Lejmi's $P_J$ operator and studies its implications for the purity and fullness of the almost complex structure, as well as relationships between various symplectic cohomologies.

Findings

$J$ is $C^$-pure and full if $\, ext{dim}\, ext{ker}\,P_J=b^2-1$

Established connections between $J$-anti-invariant cohomology and new symplectic cohomologies

Provided conditions under which certain cohomological decompositions occur

Abstract

In this paper, we define the generalized Lejmi's $P_J$ operator on a compact almost K\"{a}hler $2n$-manifold. We get that $J$ is $C^\infty$-pure and full if $\dim\ker P_J=b^2-1$. Additionally, we investigate the relationship between $J$-anti-invariant cohomology introduced by T.-J. Li and W. Zhang and new symplectic cohomologies introduced by L.-S. Tseng and S.-T. Yau on a closed symplectic $4$-manifold.