Some density results for hyperkähler manifolds

Abstract

Lagrangian fibrations of hyperkähler manifolds are induced by semi-ample line bundles which are isotropic with respect to the Beauville-Bogomolov-Fujiki form. For a non-isotrivial family of hyperkähler manifolds over a complex manifold $S$ of positive dimension, we prove that the set of points in $S$, for which there is an isotropic class in the Picard lattice of the corresponding hyperkähler manifold represented as a fiber over that point, is analytically dense in $S$. We also prove the expected openness and density of the locus of polarised hyperkähler manifolds that admit a nef algebraic isotropic line bundle.

12 pages, comments welcome. Final version - accepted for publication in Math. Res. Lett. Corrected minor typos