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

Abstract

Let $$(X,J,\omega )$$ ( X , J , ω ) be a compact 2n-dimensional almost Kähler manifold. We prove primitive decompositions for Bott–Chern and Aeppli harmonic forms in special bidegrees and show that such bidegrees are optimal. We also show how the spaces of primitive Bott–Chern, Aeppli, Dolbeault and $$\partial $$ ∂ -harmonic forms on $$(X,J,\omega )$$ ( X , J , ω ) are related.

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

“…More precisely, in Propositions 3.1 and 3.2, Theorem 3.4 and Corollary 3.5 we prove, on compact almost-Kähler 2n-dimensional manifolds, primitive decompositions for ∂and ∂-harmonic forms in bidegrees (p, 0), (0, q), (1, 1), (n, n−p), (n−q, n) and (n−1, n−1), with p, q ≤ n. One cannot hope to have such decompositions for any bidegree as shown in Example 5.3. For similar results in the case of Bott-Chern harmonic forms, we refer the reader to [12].…”