Lagrangian fibrations on hyperkähler fourfolds
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.
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]:
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 :