HKT manifolds with holonomy SL(n,H)

TL;DR

This paper characterizes when the Obata connection's holonomy on an HKT manifold is contained in SL(n,H), linking it to the Lee form, and explores implications for compact manifolds and metric existence.

Contribution

It provides a new criterion connecting the Lee form to holonomy containment and offers insights into the existence of HKT metrics on hypercomplex manifolds.

Findings

Holonomy of Obata connection in SL(n,H) iff Lee form is exact

Existence of compact HKT manifolds with trivial canonical bundle that are not balanced

Criterion for non-existence of HKT metrics based on Ricci-type tensors

Abstract

We show that on an HKT manifold the holonomy of the Obata connection is contained in SL(n,H) if and only if the Lee form is an exact one form. As an application, we show compact HKT manifolds with holomorphically trivial canonical bundle which are not balanced. A simple criterion for non-existence of HKT metric on hypercomplex manifold is given in terms of the Ricci-type tensors of the Obata connection.