Bott–Chern and $$\bar{\partial }$$ harmonic forms on almost Hermitian 4-manifolds

Abstract

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