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 $M=Γ\backslash G$ 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 $id^cγ=dω$. Otherwise, its dimension is $b^-$. In this way, we answer to a question by Zhang.

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