*Accepted Paper*

**Inserted:** 30 apr 2021

**Last Updated:** 28 oct 2021

**Journal:** Math. Res. Lett.

**Year:** 2021

**Abstract:**

Let $(X,J)$ be a $4$-dimensional compact almost-complex manifold and let $g$ be a Hermitian metric on $(X,J)$. Denote by $\Delta_{\overline\partial}:=\overline\partial\overline\partial^*+\overline\partial^*\overline\partial$ the $\overline\partial$-Laplacian. If $g$ is globally conformally K\"ahler, respectively (strictly) locally conformally K\"ahler, we prove that the dimension of the space of $\overline\partial$-harmonic $(1,1)$-forms on $X$ is a topological invariant given by $b_-+1$, respectively $b_-$. As an application, we provide a one-parameter family of almost-Hermitian structures on the Kodaira-Thurston manifold for which such a dimension is $b_-$. This gives a positive answer to a question raised by T. Holt and W. Zhang.