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

Abstract

We prove that the dimension $h_{\overline{\partial }}^{1,1}$ of the space of Dolbeault harmonic $(1,1)$-forms is not necessarily always equal to $b^{-}$ on a compact almost complex 4-manifold endowed with an almost Hermitian metric which is not locally conformally almost Kähler. Indeed, we provide examples of non integrable, non locally conformally almost Kähler, almost Hermitian structures on compact 4-manifolds with $h_{\overline{\partial }}^{1,1}=b^{-}+1$. This answers to a question by Holt.

Published version

References in corpus (4)

• Dolbeault Harmonic $(1,1)$-forms on $4$-dimensional compact quotients of Lie Groups with a left invariant almost Hermitian structure
• Bott-Chern harmonic forms and primitive decompositions on compact almost Kähler manifolds
• Primitive decompositions of Dolbeault harmonic forms on compact almost-Kähler manifolds
• Bott-Chern and $\bar{\partial }$ Harmonic forms on Almost Hermitian 4-manifolds

Cited by in corpus (1)

• Dolbeault Harmonic $(1,1)$-forms on $4$-dimensional compact quotients of Lie Groups with a left invariant almost Hermitian structure