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Working paper

Existence of HKT metrics on hypercomplex manifolds of real dimension 8

Verbitsky M., Grantcharov G., Lejmi M.
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 b1(M) is even. We show that a hypercomplex manifold M with Obata holonomy SL(2,H) admits an HKT structure iff H0,1(M)=H1(OM) is even.