Primitive decompositions of Dolbeault harmonic forms on compact almost-Kähler manifolds

Abstract

Let (X, J, g, ω) be a compact 2n-dimensional almost-Kähler manifold. We prove primitive decompositions of ∂-, ∂-harmonic forms on X in bidegree (1, 1) and (n − 1, n − 1) (such bidegrees appear to be optimal). We provide examples showing that in bidegree (1, 1) the ∂-and ∂-decompositions differ.

Bott-Chern harmonic forms and primitive decompositions on compact almost Kähler manifolds

“…Example 4.5. We recall the following construction of [1]. Let X = T 6 = Z 6 \R 6 be the 6-dimensional torus with (x 1 , x 2 , x 3 , y 1 , y 2 , y 3 ) coordinates on R 6 .…”

“…If 2n = 6, then the only bidegrees for which we do not still have primitive decompositions of the spaces of Bott-Chern and Aeppli harmonic forms are (2, 1), (1,2). Let us focus on the bidegree (2, 1).…”

“…A k-form α ∈ Λ k X, for k ≤ n, is said to be primitive if Λα = 0, or equivalently if L n−k+1 α = 0. Given that, the following vector bundle decomposition holds (1) Λ k X = r≥max(k−n,0)…”

“…Given that, the study of primitive decompositions of Dolbeault, Bott-Chern, Aeppli harmonic forms can be seen as a generalisation of the just mentioned results in higher dimension 2n ≥ 4 and for every bidegree (p, q). The case of primitive decompositions of Dolbeault harmonic forms is studied in [1]. In this paper, we are interested in studying primitive decompositions of Bott-Chern and Aeppli harmonic forms on a given 2n-dimensional compact almost Kähler manifold (X, J, ω).…”

“…A . Considered that the spaces of primitive Bott-Chern and Aeppli harmonic (1, 1)forms turned out to be useful for the above decompositions, and that the same holds for ∂ and Dolbeault harmonic (1, 1)-forms by the results of [1], we study the relations among all these spaces of primitive harmonic forms on compact almost Kähler manifolds. In particular, we analyze inclusions and non inclusions between these spaces.…”