Bott-Chern and $\bar\partial$ Harmonic forms on Almost Hermitian 4-manifolds

Abstract: We prove that on a compact almost Hermitian 4-manifold the space of $\bar\partial$-harmonic $(1,1)$-forms always has dimension $h_{\bar\partial}^{1,1} = b_- +1$ or $b_-$, whilst the space of Bott-Chern harmonic $(1,1)$-forms always has dimension $h_{BC}^{1,1} = b_- +1$. We also perform calculations of $h<sup>{2,1}_{BC}$ and $h<sup>{1,2}_{BC}$ on the Kodaira-Thurston manifold, thereby providing a full account of when $h<sup>{p,q}_{BC}$ is or is not invariant of the choice of almost Hermitian metric. Finally, we introduce a decomposition of the space of $L<sup>2$ functions on all torus bundles over $S<sup>1$, which has proven useful for solving linear PDEs, and we demonstrate its use in the calculation of $h<sup>{p,q}_{\bar\partial}$.