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

TL;DR

This paper investigates Dolbeault harmonic (1,1)-forms on 4-dimensional compact quotients of Lie groups with invariant almost Hermitian structures, establishing conditions for their dimension related to anti self dual forms and answering a specific question by Zhang.

Contribution

It characterizes the dimension of Dolbeault harmonic (1,1)-forms on these quotients, linking it to the existence of special invariant forms and providing a partial answer to Zhang's question.

Findings

Dimension of harmonic forms is b^-+1 if a certain invariant form exists.

Otherwise, the dimension equals b^-.

The result connects harmonic form dimensions to geometric structures on Lie group quotients.

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 $id^c\gamma=d\omega$. Otherwise, its dimension is $b^-$. In this way, we answer to a question by Zhang.