$\overline\partial$-Harmonic forms on $4$-dimensional almost-Hermitian manifolds

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 \emph{globally conformally K\"ahler}, respectively \emph{(strictly) locally conformally K\"ahler}, we prove that the dimension of the space of $\overline\partial$-harmonic $(1,1)$-forms on $X$, denoted as $h^{1,1}_{\overline\partial}$, 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. Furthermore, the previous example shows that $h^{1,1}_{\overline\partial}$ depends on the metric, answering to a Kodaira and Spencer's problem. Notice that such almost-complex manifolds admit both almost-K\"ahler and (strictly) locally conformally K\"ahler metrics and this fact cannot occur on compact complex manifolds.