Bott–Chern Laplacian on almost Hermitian manifolds

Abstract

Let $$(M,J,g,\omega )$$ ( M , J , g , ω ) be a 2n-dimensional almost Hermitian manifold. We extend the definition of the Bott–Chern Laplacian on $$(M,J,g,\omega )$$ ( M , J , g , ω ) , proving that it is still elliptic. On a compact Kähler manifold, the kernels of the Dolbeault Laplacian and of the Bott–Chern Laplacian coincide. We show that such a property does not hold when $$(M,J,g,\omega )$$ ( M , J , g , ω ) is a compact almost Kähler manifold, providing an explicit almost Kähler structure on the Kodaira–Thurston manifold. Furthermore, if $$(M,J,g,\omega )$$ ( M , J , g , ω ) is a connected compact almost Hermitian 4-manifold, denoting by $$h^{1,1}_{BC}$$ h BC 1 , 1 the dimension of the space of Bott–Chern harmonic (1, 1)-forms, we prove that either $$h^{1,1}_{BC}=b^-$$ h BC 1 , 1 = b - or $$h^{1,1}_{BC}=b^-+1$$ h BC 1 , 1 = b - + 1 . In particular, if g is almost Kähler, then $$h^{1,1}_{BC}=b^-+1$$ h BC 1 , 1 = b - + 1 , extending the result by Holt and Zhang (Harmonic forms on the Kodaira–Thurston manifold. arXiv:2001.10962, 2020) for the kernel of Dolbeault Laplacian. We also show that the dimensions of the spaces of Bott–Chern and Dolbeault harmonic (1, 1)-forms behave differently on almost complex 4-manifolds endowed with strictly locally conformally almost Kähler metrics. Finally, we relate some spaces of Bott-Chern harmonic forms to the Bott–Chern cohomology groups for almost complex manifolds, recently introduced in Coelho et al. (Maximally non-integrable almost complex structures: an h-principle and cohomological properties, arXiv:2105.12113, 2021).

Dolbeault Harmonic $(1,1)$-forms on $4$-dimensional compact quotients of Lie Groups with a left invariant almost Hermitian structure

“…See [16] for a self contained survey of the just mentioned results. See [11,17] for similar results about Bott-Chern harmonic forms on compact almost Hermitian 4manifolds. See also [4,15,20] for other interesting results concerning Dolbeault and Bott-Chern harmonic forms on compact almost Hermitian manifolds of higher dimension.…”

“…We endow G and M with the left invariant almost complex structure J β given by ϕ 1 = e 4 + ie 1 , ϕ 2 = e 2 − iβe 3 , with β ∈ R ∖ {0}, being a global coframe of the vector bundle of (1, 0) forms T 1,0 G. The associated structure equations aredϕ 1 = 0, dϕ 2 ϕ 12 + ϕ 21 − ϕ 12 ,therefore the almost complex structure J β is non integrable. This is the same almost complex structure considered in[5, Section 6],[12, Section 2] and[17, Section 5]. From the structure equations we derive , J β ) with a left invariant almost Hermitian metricω β = ir 2 ϕ 11 + is 2 ϕ 22 + uϕ 12 − uϕ 21 , with u ∈ C, r, s ∈ R, r, s > 0 and r 2 s 2 > u 2 .…”

“…Summing ( 16) and (17), subtracting ( 16) from (17), summing (18) and (19), subtracting (18) from (19), we obtain…”

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

“…∂ is either equal to b − or b − + 1, depending on the choice of metric, see [4,5,12]. Similarly, for Bott-Chern harmonic forms, it yields h 1,1 BC ∶= dim C H 1,1 BC = b − + 1 for all metrics, see [4,10]. See [8,11,13] for other related results and [7,15] for two surveys on the subject.…”

“…We also remark that the almost Kähler assumption is necessary for this kind of primitive harmonic decomposition. To see that this is the case in dimension 4 we refer the reader to [12,10].…”