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 $\Omega$ associated with the corresponding $Sp(n)$-structure satisfies $\6\Omega=0$. 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-Amp\`ere 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 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 $\dim_\R M \geq 12$.

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Published 1 January 2009