Holomorphic Lagrangian Fibrations on Hypercomplex Manifolds

Andrey Soldatenkov,Misha Verbitsky

doi:10.1093/imrn/rnt218

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 hyperK¨ ahler geometry. An

Highlights

1.1 HKT metrics and SL(n, H)-structuresLet I, J, K be complex structures on a manifold M satisfying the quaternionic relation I ◦ J = −J ◦ I = K
The theory of hypercomplex manifolds is much richer in examples and still not well understood
In [Sol], the first named author provided an example of such a manifold, by proving that Hol(∇) = GL(H, 2) for the homogeneous hypercomplex structure on SU(3), constructed by D

Summary

Introduction

We obtain a result in a similar vein, constructing a Kahler metric on a base of holomorphic Lagrangian fibration on an HKT manifold. The notion of “holomorphic Lagrangian fibration” is itself highly non-trivial, because an HKT manifold is not necessarily holomorphically symplectic It was developed in [GV], using the theory of calibrations and the holonomy of the Obata connection. In [Sol], the first named author provided an example of such a manifold, by proving that Hol(∇) = GL(H, 2) for the homogeneous hypercomplex structure on SU(3), constructed by D. This result is obtained in [V3] using the Hodge theory for HKT manifolds developed in [V1]. The Hodge-theoretic constructions of [V1] work for SL(n, H)-manifolds with HKT-structure especially well For such manifolds, one obtains Hodgetype decomposition on the holomorphic cohomology bundle H∗(O(M,I))

Calibrations on manifolds and Lagrangian fibrations

Hypercomplex manifolds and quaternionic Dolbeault complex

HKT structures and Lagrangian fibrations

Findings

Examples of Lagrangian fibrations