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 , ω) is a closed symplectic parabolic manifold which satisfies the hard Lefschetz property, then its Euler number satisfies the inequality (−1) n χ(M 2n ) ≥ 0. (2000): 53D05. AMS Classification

Based on recent work of T. Draghici, T.-J. Li and W. Zhang, we further investigate properties of the dimension h − J of the J-anti-invariant cohomology subgroup H − J of a closed almost Hermitian 4-manifold (M, g, J, F ) using metric compatible almost complex structures. We prove that h − J = 0 for generic almost complex structures J on M .

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 ∞ -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. (2000): 53C55, 53C22. AMS Classification

In this paper, we calculate the dimension of the J-anti-invariant cohomology subgroup H − J on T 4 . Inspired by the concrete example, T 4 , we get that: On a closed symplectic 4-dimensional manifold (M, ω), h − J = 0 for generic ω-compatible almost complex structures. (2000): 53C55, 53D05. AMS ClassificationKeywords: almost Kähler four-manifold, deformations of almost complex structures, dimension of J-anti-invariant cohomology.