Primitive decomposition of Bott-Chern and Dolbeault harmonic $(k,k)$-forms on compact almost K\"ahler manifolds

TL;DR

This paper studies the primitive decomposition of harmonic forms on compact almost Kähler manifolds, revealing structural properties and providing explicit descriptions in dimension 8, with examples showing limitations of primitive decomposition.

Contribution

It proves that primitive components of harmonic forms are multiples of the Kähler form and fully describes harmonic form spaces in dimension 8, highlighting when primitive decomposition fails.

Findings

Primitive components are multiples of the Kähler form.

Complete description of harmonic spaces in dimension 8.

Primitive decomposition does not always preserve harmonicity.

Abstract

We consider the primitive decomposition of $\bar \partial, \partial$, Bott-Chern and Aeppli-harmonic $(k,k)$-forms on compact almost K\"ahler manifolds $(M,J,\omega)$. For any $D \in \{\bar\partial, \partial, BC, A\}$, we prove that the $L^k P^0$ component of $\psi \in \mathcal{H}_{D}^{k,k}$, is a constant multiple of $\omega^k$. Focusing on dimension 8, we give a full description of the spaces $\mathcal{H}_{BC}^{2,2}$ and $\mathcal{H}_{A}^{2,2}$, from which follows $\mathcal{H}^{2,2}_{BC}\subseteq\mathcal{H}^{2,2}_{\partial}$ and $\mathcal{H}^{2,2}_{A}\subseteq\mathcal{H}^{2,2}_{\bar\partial}$. We also provide an almost K\"ahler 8-dimensional example where the previous inclusions are strict and the primitive components of an harmonic form $\psi \in \mathcal{H}_{D}^{k,k}$ are not $D$-harmonic, showing that the primitive decomposition of $(k,k)$-forms in general does not descend to harmonic forms.