On Cohomology of Almost Complex 4-Manifolds

Abstract

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 .

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

“…In particular, they have confirmed their conjecture for 4-manifolds with b + = 1 ([8, Theorem 3.1]). Fortunately, in [20], Qiang Tan, Hongyu Wang, Ying Zhang and Peng Zhu confirmed the conjecture completely. A symplectic structure on a differentiable manifold is a nondegenerate closed 2-form ω ∈ Ω 2 .…”

“…Lemma 1.6. ( [20]) Let (M, g) be a closed Riemannian 4-manifold. If α ∈ Ω + g and α = α h + dθ + δ g ψ is its Hodge decomposition, then P + g (dθ) = P + g (δ g ψ) and P − g (dθ) = −P − g (δ g ψ), where P ± g : Ω 2 → Ω ± g are the projections.…”

“…Proof. In the proof of Theorem 1.1 in [20], we have gotten that the set of almost complex structures J on M with h − J = 0 is an open subset of J in the C ∞ -topology. Note that J c ω is a subspace of J in the C ∞ -topology.…”

Primitive cohomology of real degree two on compact symplectic manifolds

“…Thus b 2 = b + + b − . It is easy to see that, for a closed almost Kähler 4-manifold (M, g, J, ω), there hold (see [9,10,21]): [22,24]). Here L is the Lefschetz operator (see [4,22,24]) which is defined acting on a k-form…”

“…It is easy to see that, for a closed almost Kähler 4-manifold (M, g, J, ω), there hold (see [9,10,21]):…”