Primitive cohomology of degree 2 on compact symplectic manifolds

Abstract

In this paper, we define the generalized Lejmi's $P_J$ operator on a compact almost Kähler $2n$-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.