  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.
