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 = Gamma\G of 4-dimensional Lie groups G admitting a left invariant almost Hermitian structure (J, omega). In this case, we prove that the space of Dolbeault harmonic (1, 1)-forms on (M, J, omega) has dimension b(-) + 1 if and only if there exists a left invariant anti self dual (1, 1)-form gamma on (G, J) satisfying id(c)gamma =d omega. Otherwise, its dimension is b(-). In this way, we answer to a question by Zhang. (C) 2022 Elsevier B.V. All rights reserved.