Bott–Chern Laplacian on almost Hermitian manifolds

Abstract

Let (M,J,g,omega ) be a 2n-dimensional almost Hermitian manifold. We extend the definition of the Bott–Chern Laplacian on (M,J,g,omega ), 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 ) is a compact almost Kähler manifold, providing an explicit almost Kähler structure on the Kodaira–Thurston manifold. Furthermore, if (M,J,g,omega ) is a connected compact almost Hermitian 4-manifold, denoting by h^{1,1}_{BC} the dimension of the space of Bott–Chern harmonic (1, 1)-forms, we prove that either h^{1,1}_{BC}=b^- or h^{1,1}_{BC}=b^-+1. In particular, if g is almost Kähler, then h^{1,1}_{BC}=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).