Holomorphic Lagrangian fibrations on hypercomplex manifolds

A hypercomplex manifold is a smooth manifold with a triple of integrable almost-complex structures that satisfy the quaternionic relations. A holomorphic Lagrangian variety on a hypercomplex manifold with trivial canonical bundle is a complex subvariety which is Lagrangian with respect to a (2,0)-form associated with a hyperhermitian metric. We will consider a special class of metrics - HKT (hyperkaehler with torsion) metrics - on hypercomplex manifolds. We will discuss the following theorem: the base of a holomorphic Lagrangian fibration is always Kaehler, if its total space admits an HKT metric. This can be used to construct hypercomplex manifolds which do not admit an HKT structure. This is joint work with Misha Verbitsky