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 $h_{BC}^{2,1}$ and $h_{BC}^{1,2}$ on the Kodaira-Thurston manifold, thereby providing a full account of when $h_{BC}^{p,q}$ is or is not invariant of the choice of almost Hermitian metric. Finally, we introduce a decomposition of the space of $L^2$ functions on all torus bundles over $S^1$, which has proven useful for solving linear PDEs, and we demonstrate its use in the calculation of $h_{\bar{\partial }}^{p,q}$.

27 pages

Cited by in corpus (1)

• On the dimension of Dolbeault harmonic (1,1)-forms on almost Hermitian 4-manifolds