$\overline{\partial }$-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 ${\Delta }_{\overline{\partial }}:=\overline{\partial }{\overline{\partial }}^{\ast }+{\overline{\partial }}^{\ast }\overline{\partial }$ the $\overline{\partial }$-Laplacian. If $g$ is \emph{globally conformally Kähler}, respectively \emph{(strictly) locally conformally Kähler}, we prove that the dimension of the space of $\overline{\partial }$-harmonic $(1,1)$-forms on $X$, denoted as $h_{\overline{\partial }}^{1,1}$, 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 $h_{\overline{\partial }}^{1,1}$ depends on the metric, answering to a Kodaira and Spencer's problem. Notice that such almost-complex manifolds admit both almost-Kähler and (strictly) locally conformally Kähler metrics and this fact cannot occur on compact complex manifolds.

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