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

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 } , it is known that the LkP0 , 0 component of [InlineEquation not available: see fulltext.] is a constant multiple of ωk up to real dimension 6. In this paper we generalise this result to every dimension. We also deduce information on the components Lk-1P1 , 1 and Lk-2P2 , 2 of the primitive decomposition. Focusing on dimension 8, we give a full description of the spaces [InlineEquation not available: see fulltext.] and [InlineEquation not available: see fulltext.], from which follows [InlineEquation not available: see fulltext.] and [InlineEquation not available: see fulltext.]. We also provide an almost Kähler 8-dimensional example where the previous inclusions are strict and the primitive components of a harmonic form [InlineEquation not available: see fulltext.] are not D-harmonic, showing that the primitive decomposition of (k, k)-forms in general does not descend to harmonic forms.