Primitive cohomology of real degree two 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 ∞ -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

On second non-HLC degree of closed symplectic manifold

“…where ∆ d is the Riemannian Laplacian with respect to the metric g(•, •) = ω(•, J•) (here we use the convention g(ω, ω) = n). By studying Lejmi's operator P J [9], Tan-Wang-Zhou [13] proved that J is C ∞ -pure and full when dim(ker…”

“…If the space of de Rham harmonic forms is contained in the space of symplectic-Bott-Chern harmonic forms, i.e., H 2 dR (X) ⊂ H 2 d+d Λ (X), then identity b 2 ≤ h 2 d+d Λ naturally holds. In the fifth section of [9], Lejmi introduced the differential operator P J on a closed almost Kähler 4-manifold (X, J) Tan-Wang-Zhou [13] extended the defined to higher dimensions. We can give a sufficient condition for H 2 dR (X) ⊂ H 2 d+d Λ (X).…”

On Tamed Almost Complex Four-Manifolds

“…On 2n-dimensional closed almost Kähler manifolds, Tan, Wang and Zhou define a generalized operator of P in [41]. It is natural to ask the following question: Find a generalized operator of D + J on a 2n-dimensional closed almost Kähler manifolds for n ≥ 3.…”

A Note on the Deformations of Almost Complex Structures on Closed Four-Manifolds

“…In this section, we calculate the dimension of the J-anti-invariant cohomology subgroup on 4-torus under the deformation of ω-compatible almost complex structures. The following construction is a generalization of Example 2.6 in [21] which is constructed by T. Draghici and C. H. Taubes ([6, 23]). These computations provide another example where the dimension of the J-anti-invariant cohomology is not an invariant under the deformation of almost complex structures.…”