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

Abstract

We study Dolbeault harmonic $(1,1)$-forms on compact quotients $M=\Gamma \backslash G$ 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 $i{d}^c\gamma =d\omega$. Otherwise, its dimension is $b^{-}$. In this way, we answer to a question by Zhang.

20 pages, published version

References in corpus (1)

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

Cited by in corpus (1)

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