Balanced HKT metrics and strong HKT metrics on hypercomplex manifolds

A manifold (M, I, J,K) is called hypercomplex if I, J,K are complex structures satisfying quaternionic relations. A quaternionic Hermitian hypercomplex manifold is called HKT (hyperkähler with torsion) if the (2,0)-form Ω associated with the corresponding Sp(n)-structure satisfies ∂Ω = 0. A Hermitian metric ω on a complex manifold is called balanced if d∗ω = 0. We show that balanced HKT metrics are precisely the quaternionic Calabi-Yau metrics defined in terms of the quaternionic Monge-Ampère 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 ddω = 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 hyperkähler 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 dimR M > 12.