Existence of HKT metrics on hypercomplex manifolds of real dimension 8

TL;DR

This paper investigates the existence of HKT metrics on 8-dimensional hypercomplex manifolds, establishing a criterion based on the parity of certain cohomology groups, thus extending complex geometric results to quaternionic geometry.

Contribution

It proves a quaternionic analogue of Buchdahl-Lamari's theorem, linking HKT metric existence to the evenness of specific cohomology groups on hypercomplex manifolds.

Findings

HKT metrics exist iff $H^{0,1}(M)$ is even.

Established a quaternionic analogue of Buchdahl-Lamari's theorem.

Connected HKT metric existence to topological invariants.

Abstract

A hypercomplex manifold $M$ is a manifold equipped with three complex structures satisfying quaternionic relations. Such a manifold admits a canonical torsion-free connection preserving the quaternion action, called Obata connection. A quaternionic Hermitian metric is a Riemannian metric on which is invariant with respect to unitary quaternions. Such a metric is called HKT if it is locally obtained as a Hessian of a function averaged with quaternions. HKT metric is a natural analogue of a Kahler metric on a complex manifold. We push this analogy further, proving a quaternionic analogue of Buchdahl-Lamari's theorem for complex surfaces. Buchdahl and Lamari have shown that a complex surface M admits a Kahler structure iff $b_1(M)$ is even. We show that a hypercomplex manifold M with Obata holonomy $SL(2,{\mathbb H})$ admits an HKT structure iff $H^{0,1}(M)=H^1({\cal O}_M)$ is even.