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 $h_{BC}^{1,1}$ 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 Kähler, then $h_{BC}^{1,1}=b^{-}+1$, 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 Kähler 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.

final version

References in corpus (4)

• Autour de la cohomologie de Bott-Chern
• Kodaira dimensions of almost complex manifolds II
• Maximally non-integrable almost complex structures: an $h$-principle and cohomological properties
• Almost Kähler Kodaira-Spencer problem