Some density results for hyperkähler manifolds

Lagrangian fibrations of hyperka¨hler 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 hyperka¨hler 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 hyperka¨hler 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 hyperka¨hler manifolds that admit a nef algebraic isotropic line bundle.