$\bar{\mathrm{\partial }}$-Harmonic forms on $4$-dimensional almost-Hermitian manifolds

Let $(X,J)$ be a $4$-dimensional compact almost-complex manifold and let $g$ be a Hermitian metric on $(X,J)$. Denote by ${\mathrm{\Delta }}_{\bar{\mathrm{\partial }}}:=\bar{\mathrm{\partial }}{\bar{\mathrm{\partial }}}^{\ast }+{\bar{\mathrm{\partial }}}^{\ast }\bar{\mathrm{\partial }}$ the $\bar{\mathrm{\partial }}$-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 $\bar{\mathrm{\partial }}$-harmonic $(1,1)$-forms on $X$, denoted as $h_{\bar{\mathrm{\partial }}}^{1,1}$, 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 $h_{\bar{\mathrm{\partial }}}^{1,1}$ 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.