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

2022/06/13 by Holt, Tom, Piovani, Riccardo
#32Q60 #53C15 #Differential Geometry (math.DG) #FOS: Mathematics #Symplectic Geometry (math.SG)

paper · doi:10.48550/arxiv.2206.05919

Abstract

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\, we prove that the L^k P⁰ component of ψ∈ H_D^k,k, is a constant multiple of ω^k. Focusing on dimension 8, we give a full description of the spaces H_BC^2,2 and H_A^2,2, from which follows H^2,2_BC\subseteqH^2,2_∂ and H^2,2_A\subseteqH^2,2_∂. We also provide an almost Kähler 8-dimensional example where the previous inclusions are strict and the primitive components of an harmonic form ψ∈ 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.

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