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

Abstract

AbstractWe consider the primitive decomposition of $$\overline{\partial }, \partial $$ ∂ ¯ , ∂ , Bott–Chern and Aeppli-harmonic (k, k)-forms on compact almost Kähler manifolds $$(M,J,\omega )$$ ( M , J , ω ) . For any $$D \in \{\overline{\partial }, \partial , \textrm{BC}, \textrm{A}\}$$ D ∈ { ∂ ¯ , ∂ , BC , A } , it is known that the $$L^k P^{0,0}$$ L k P 0 , 0 component of "Equation missing" is a constant multiple of $$\omega ^k$$ ω k up to real dimension 6. In this paper we generalise this result to every dimension. We also deduce information on the components $$L^{k-1} P^{1,1}$$ L k - 1 P 1 , 1 and $$L^{k-2} P^{2,2}$$ L k - 2 P 2 , 2 of the primitive decomposition. Focusing on dimension 8, we give a full description of the spaces "Equation missing" and "Equation missing", from which follows "Equation missing" and "Equation missing". We also provide an almost Kähler 8-dimensional example where the previous inclusions are strict and the primitive components of a harmonic form "Equation missing" are not D-harmonic, showing that the primitive decomposition of (k, k)-forms in general does not descend to harmonic forms.