Holomorphic Lagrangian fibrations on hypercomplex manifolds

Abstract

A hypercomplex manifold is a manifold equipped with a triple of complex structures satisfying the quaternionic relations. A holomorphic Lagrangian variety on a hypercomplex manifold with trivial canonical bundle is a holomorphic subvariety which is calibrated by a form associated with the holomorphic volume form; this notion is a generalization of the usual holomorphic Lagrangian subvarieties known in hyperkaehler geometry. An HKT (hyperkaehler with torsion) metric on a hypercomplex manifold is a metric determined by a local potential, in a similar way to the Kaehler metric. We prove that a base of a holomorphic Lagrangian fibration is always Kaehler, if its total space is HKT. This is used to construct new examples of hypercomplex manifolds which do not admit an HKT structure.