Lagrangian fibrations on hyperkähler fourfolds

Lagrangian fibrations on hyperkähler fourfolds

Daniel Greb Daniel Greb
Institut für Mathematik
Abteilung für Reine Mathematik
Albert-Ludwigs-Universität Freiburg
Eckerstr. 1
79104 Freiburg im Breisgau
Christian Lehn Christian Lehn
Institut de Recherche Mathématique Avancée
Université de Strasbourg
7 rue René Descartes
67084 Strasbourg Cedex
 and  Sönke Rollenske Sönke Rollenske
Fakultät für Mathematik
Universtät Bielefeld
Universitätsstr. 25
33615 Bielefeld

Answering the strong form of a question posed by Beauville, we give a short geometric proof that any hyperkähler fourfold containing a Lagrangian subtorus admits a holomorphic Lagrangian fibration with fibre .

Key words and phrases:
hyperkähler manifold, Lagrangian fibration
2010 Mathematics Subject Classification:
53C26, 14D06, 14E30, 32G10, 32G05.

1. Introduction

Let be a hyperkähler manifold, that is, a compact, simply-connected Kähler manifold such that is spanned by a holomorphic symplectic form . By work of Matsushita it is well-known that the only possible non-trivial holomorphic maps from to a lower-dimensional complex space are Lagrangian fibrations, see section LABEL:sec:_prelim. Moreover, a special version of the so-called Hyperkähler SYZ-conjecture asserts that any hyperkähler manifold can be deformed to a hyperkähler manifold admitting a Lagrangian fibration.

Hence, it is an important problem to find geometric conditions on a given hyperkähler manifold that guarantee the existence of a Lagrangian fibration; here we address a question posed by Beauville [beauville10, Sect. 1.6]:

Question B

Let be a hyperkähler manifold and a Lagrangian torus in . Is a fibre of a (meromorphic) Lagrangian fibration ?

In our previous article [glr11a] it is shown that Question B has a positive answer in case is non-projective. Moreover, for any hyperkähler manifold that admits an almost holomorphic Lagrangian fibration, a further hyperkähler manfold, birational to the first one, is found, on which the Lagrangian fibration becomes holomorphic.

The approach to the projective case of Beauville’s question pursued here is based on a detailed study of the deformation theory of in . For this, consider the component of the Barlet space that contains together with its universal family and the evaluation map to :

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