Holomorphic Lagrangian fibrations on hypercomplex manifolds

TL;DR

This paper explores holomorphic Lagrangian fibrations on hypercomplex manifolds, demonstrating that bases of such fibrations are Kaehler under HKT conditions and constructing examples of hypercomplex manifolds without HKT structures.

Contribution

It establishes that bases of holomorphic Lagrangian fibrations are Kaehler if the total space has an HKT metric and constructs new hypercomplex manifolds lacking HKT structures.

Findings

Bases are Kaehler under HKT conditions

Construction of hypercomplex manifolds without HKT structures

Generalization of holomorphic Lagrangian subvarieties

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.