Dolbeault harmonic (1,1)-forms on 4-dimensional compact quotients of Lie groups with a left invariant almost Hermitian structure

We study Dolbeault harmonic (1,1)-forms on compact quotients M=Γ﹨G of 4-dimensional Lie groups G admitting a left invariant almost Hermitian structure (J,ω). In this case, we prove that the space of Dolbeault harmonic (1,1)-forms on (M,J,ω) has dimension b−+1 if and only if there exists a left invariant anti self dual (1,1)-form γ on (G,J) satisfying idcγ=dω. Otherwise, its dimension is b−. In this way, we answer to a question by Zhang.