Base manifolds for Lagrangian fibrations on hyperkähler manifolds
A compact Kähler manifold \(X\) is called hyper-Kähler or irreducible holomorphic symplectic if it is simply-connected, and if \(H^0(X, \Omega^2_X)=\mathbb{C} \sigma \) where \(\sigma\) is an everywhere nondegenerate 2-form. A fibration on \(X\) is a (proper) surjective holomorphic map \( f : X \to B,\, 0<\dim B<\dim X\). It is called Lagrangian if every irreducible component of every fibre is a Lagrangian subvariety with respect to the holomorphic symplectic form \(\sigma\). The following theorem is proven.NEWLINENEWLINELet \(f:X \to B\) be a holomorphic fibration of a compact hyper-Kähler manifold onto a smooth complex manifold. Then the fibration is Lagrangian and \(B\) is biholomorphic to the projective space \(P^n\).NEWLINENEWLINEThis is a generalization of a result by J-M. Hwang, who assumed moreover that \(X\) is a projective manifold.
scientific article; zbMATH DE number 6369079
scientific article