Bott-Chern Laplacian on almost Hermitian manifolds

Let (M,J, g,ω) be a 2n-dimensional almost Hermitian manifold. We extend the definition of the Bott-Chern Laplacian on (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,ω) is a compact almost Kähler manifold, providing an explicit almost Kähler structure on the Kodaira-Thurston manifold. Furthermore, if (M,J, g,ω) is a connected compact almost Hermitian 4-manifold, denoting by h BC the dimension of the space of Bott-Chern harmonic (1,1)-forms, we prove that either h BC = b− or h BC = b− + 1. In particular, if g is almost Kähler, then h BC = b− + 1, extending the result by Holt and Zhang [11] 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 [5].