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

TL;DR

This paper investigates the relationship between the dimension of Dolbeault harmonic (1,1)-forms and the Betti number on almost Hermitian 4-manifolds, showing they are not always equal and providing counterexamples.

Contribution

It demonstrates that on certain almost Hermitian 4-manifolds, the dimension of Dolbeault harmonic forms can differ from the Betti number, answering a question by Holt.

Findings

Dimension $h^{1,1}_{\overline\partial}$ can exceed $b^-$ by 1.

Counterexamples on non integrable, non locally conformally almost Kähler structures.

The relationship between harmonic form dimensions and topology is more nuanced than previously thought.

Abstract

We prove that the dimension $h^{1,1}_{\overline\partial}$ 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 K\"ahler. Indeed, we provide examples of non integrable, non locally conformally almost K\"ahler, almost Hermitian structures on compact 4-manifolds with $h^{1,1}_{\overline\partial}=b^-+1$. This answers to a question by Holt.