Bott-Chern Laplacian on almost Hermitian manifolds

TL;DR

This paper extends the Bott-Chern Laplacian to almost Hermitian manifolds, analyzes its properties, and explores its harmonic forms, revealing differences from the classical Kähler case and connections to cohomology.

Contribution

It introduces an elliptic Bott-Chern Laplacian on almost Hermitian manifolds and studies its harmonic forms, highlighting differences from Kähler geometry and linking to recent cohomology theories.

Findings

On compact Kähler manifolds, kernels of Dolbeault and Bott-Chern Laplacians coincide.

In almost Kähler manifolds, the kernel property does not hold, with explicit examples provided.

For 4-manifolds, the dimension of Bott-Chern harmonic forms is either b^- or b^-+1.

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.