Twistor spaces of hypercomplex manifolds are balanced

TL;DR

This paper proves that the twistor spaces of all compact hypercomplex manifolds admit balanced metrics, extending previous results known for hyperkaehler manifolds.

Contribution

It generalizes the known balanced metric property of twistor spaces from hyperkaehler to all compact hypercomplex manifolds.

Findings

Twistor spaces of hyperkaehler manifolds are balanced.

Twistor spaces of general compact hypercomplex manifolds are balanced.

Abstract

A hypercomplex structure on a differentiable manifold consists of three integrable almost complex structures that satisfy quaternionic relations. If, in addition, there exists a metric on the manifold which is Hermitian with respect to the three structures, and such that the corresponding Hermitian forms are closed, the manifold is said to be hyperkaehler. In the paper "Non-Hermitian Yang-Mills connections", Kaledin and Verbitsky proved that the twistor space of a hyperkaehler manifold admits a balanced metric; these were first studied in the article "On the existence of special metrics in complex geometry" by Michelsohn. In the present article, we review the proof of this result and then generalize it and show that twistor spaces of general compact hypercomplex manifolds are balanced.