Bott–Chern Laplacian on almost Hermitian manifolds

Let (M, J, g, omega) be a 2n-dimensional almostHermitianmanifold. We extend the definition of theBott-Chern Laplacian on (M, J, g, omega), proving that it is still elliptic. On a compact Kahler 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 Kahler manifold, providing an explicit almost Kahler structure on the Kodaira-Thurston manifold. Furthermore, if (M, J, g, omega) is a connected compact almost Hermitian 4-manifold, denoting by h(BC)(1,1) the dimension of the space of Bott-Chern harmonic (1, 1)-forms, we prove that either h(BC)(1,1) = b(-) or h(BC)(1,1) = b(-) + 1. In particular, if g is almost Kahler, then 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 Kahler metrics. Finally, we relate some spaces of Bott-Chern harmonic forms to the BottChern 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).