∂-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 ∆∂ := ∂∂∗ + ∂∗∂ the ∂-Laplacian. If g is globally conformally Kähler, respectively (strictly) locally conformally Kähler, we prove that the dimension of the space of ∂-harmonic (1, 1)-forms on X, denoted as h∂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 h1∂,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.