Existence of HKT metrics on hypercomplex manifolds of real dimension 8

摘要

AbstractA hypercomplex manifoldMis a manifold equipped with three complex structuresI,J,Ksatisfying quaternionic relations. Such a manifold admits a canonical torsion-free connection preserving the quaternion action, called the Obata connection. A quaternionic Hermitian metric is a Riemannian metric which is invariant with respect to unitary quaternions. Such a metric is called hyperk?hler with torsion (HKT for short) if it is locally obtained as the Hessian of a function averaged with quaternions. An HKT metric is a natural analogue of a K?hler metric on a complex manifold. We push this analogy further, proving a quaternionic analogue of the result of Buchdahl and of Lamari that a compact complex surface M admits a K?hler structure if and only ifb1(M)is even. We show that a hypercomplex manifold M with the Obata holonomy contained inSL(2,H)admits an HKT structure if and onl