Some density results for hyperkähler manifolds

Mathematical Research Letters, vol. 32, pp. 1465–1479

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.

Authors 4

1. Yajnaseni Dutta Aachen Mathematisch Instituut

   Leiden University

   Affiliation as printed

   Mathematisch Instituut, Universiteit Leiden, Leiden, Netherlands

2. Elham Izadi

   University of California San Diego

   Affiliation as printed

   Department of Mathematics, University of California San Diego, La Jolla, CA, USA

3. Ljudmila Kamenova

   Stony Brook University

   Affiliation as printed

   Department of Mathematics, Stony Brook University, Stony Brook, NY, USA

4. Lisa Marquand

   Courant Institute of Mathematical Sciences · New York University

   Affiliation as printed

   Courant Institute of Mathematical Sciences, New York University, NY, USA