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 K\"ahler 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 K\"ahler manifold, providing an explicit almost K\"ahler 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 K\"ahler, then $h^{1,1}_{BC}=b^-+1$, extending the result by Holt and Zhang 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 K\"ahler metrics. Finally, we relate some spaces of Bott-Chern harmonic forms to the Bott-Chern cohomology groups for almost complex manifolds, recently introduced by Coelho, Placini and Stelzig.