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
Germany
Christian Lehn Christian Lehn
Institut de Recherche Mathématique Avancée
Université de Strasbourg
7 rue René Descartes
67084 Strasbourg Cedex
France
and  Sönke Rollenske Sönke Rollenske
Fakultät für Mathematik
Universtät Bielefeld
Universitätsstr. 25
33615 Bielefeld
Germany
###### Abstract.

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 :

You are adding the first comment!
How to quickly get a good reply:
• Give credit where it’s due by listing out the positive aspects of a paper before getting into which changes should be made.
• Be specific in your critique, and provide supporting evidence with appropriate references to substantiate general statements.
• Your comment should inspire ideas to flow and help the author improves the paper.

The better we are at sharing our knowledge with each other, the faster we move forward.
The feedback must be of minimum 40 characters and the title a minimum of 5 characters