Balanced HKT metrics and strong HKT metrics on hypercomplex manifolds

TL;DR

This paper explores the properties of balanced and strong HKT metrics on hypercomplex manifolds, establishing their uniqueness, existence conditions, and cohomological characteristics, including a hyperkähler Hodge-Riemann relation.

Contribution

It characterizes balanced HKT metrics as quaternionic Calabi-Yau metrics, proves their uniqueness, and investigates the cohomology of strong HKT metrics, including a hyperkähler Hodge-Riemann relation.

Findings

Balanced HKT metrics are quaternionic Calabi-Yau metrics.

Balanced HKT metrics are unique in their cohomology class.

Manifolds with balanced HKT metrics do not admit strong HKT metrics in high dimensions.

Abstract

A manifold (M,I,J,K) is called hypercomplex if I,J,K are complex structures satisfying quaternionic relations. A quaternionic Hermitian metric is called HKT (hyperkaehler with torsion) if $Id\omega_I = Jd \omega_J=Kd\omega_K$, where $\omega_I,\omega_J, \omega_K$ are Hermitian forms associated with I, J, K. A Hermitian metric $\omega$ on a complex manifold is called balanced if $d^*\omega=0$. We show that balanced HKT metrics are precisely the quaternionic Calabi-Yau metrics defined in terms of the quaternionic Monge-Ampere equation. In particular, a balanced HKT-metric is unique in its cohomology class, and it always exists if the quaternionic Calabi-Yau theorem is true. We investigate the cohomological properties of strong HKT metrics (the quaternionic Hermitian metrics, satisfying, in addition to the HKT condition, the relation $dd^c \omega=0$), and show that the space of strong HKT metrics is finite-dimensional. Using Howe's duality for representations of Sp(n), we prove a hyperkaehler version of Hodge-Riemann bilinear relations. We use it to show that a manifold admitting a balanced HKT-metric never admits a strong HKT-metric, if $\dim_\R M \geq 12$.