Existence of HKT metrics on hypercomplex manifolds of real dimension 8

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 $H^{0,1}(M)=H^1({\mathcal{O}}_M)$ is even.

30 pages. arXiv admin note: text overlap with arXiv:0808.3218, arXiv:1009.1178

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Cited by in corpus (7)

• The $C^0$ estimate for the quaternionic Calabi conjecture
• Hypercomplex almost abelian solvmanifolds
• On the canonical bundle of complex solvmanifolds and applications to hypercomplex geometry
• Applications of the quaternionic Jordan form to hypercomplex geometry
• Quaternionic Bott-Chern cohomology and existence of HKT metrics
• Poisson structures on twistor spaces of hyperkaehler and HKT manifolds
• Special Hermitian structures on suspensions