Primitive decomposition of Bott-Chern and Dolbeault harmonic $(k,k)$-forms on compact almost Kähler manifolds

Abstract

We consider the primitive decomposition of ∂, ∂, Bott-Chern and Aeppli-harmonic (k, k)-forms on compact almost Kähler manifolds (M, J, ω). For any D ∈ {∂, ∂, BC, A}, we prove that theis a constant multiple of ω k . Focusing on dimension 8, we give a full description of the spaces H 2,2 BC and H 2,2 A , from which follows H 2,2 BC ⊆ H 2,2 ∂ and H 2,2 A ⊆ H 2,2 ∂. We also provide an almost Kähler 8-dimensional example where the previous inclusions are strict and the primitive components of an harmonic form ψ ∈ H k,k D are not D-harmonic, showing that the primitive decomposition of (k, k)-forms in general does not descend to harmonic forms.