On the dimension of Dolbeault harmonic (1, 1)-forms on almost Hermitian 4-manifolds

We prove that the dimension h1,1 of the space of Dol-beault harmonic (1, 1)-forms is not necessarily always equal to b− ∂ on a compact almost complex 4-manifold endowed with an almost Hermitian metric which is not locally conformally almost Kähler. Indeed, we provide examples of non integrable, non locally con-formally almost Kähler, almost Hermitian structures on compact 4-manifolds with h1,1 = b−+1. This gives an answer to [6, Question ∂ 3.3] by Holt.