Roundness of the ample cone and existence of double Lagrangian fibrations on hyperkahler manifolds

TL;DR

This paper investigates the geometric structure of hyperkahler manifolds, proving that they can be deformed to have a 'round' Kahler cone, which implies the existence of double Lagrangian fibrations and specific metric properties.

Contribution

It demonstrates that any maximal holonomy hyperkahler manifold with second Betti number greater than four can be deformed to have a round Kahler cone with particular lattice properties.

Findings

Hyperkahler manifolds can be deformed to have round Kahler cones.

All known hyperkahler examples admit deformations with two transversal Lagrangian fibrations.

The Kobayashi metric vanishes unless the Picard rank is maximal.

Abstract

Let $M$ be a hyperkahler manifold of maximal holonomy (that is, an IHS manifold), and let $K$ be its Kahler cone, which is an open, convex subset in the space $H^{1,1}(M, R)$ of real (1,1)-forms. This space is equipped with a canonical bilinear symmetric form of signature $(1,n)$ obtained as a restriction of the Bogomolov-Beauville-Fujiki form. The set of vectors of positive square in the space of signature $(1,n)$ is a disconnected union of two convex cones. The "positive cone" is the component which contains the Kahler cone. We say that the Kahler cone is "round" if it is equal to the positive cone. The manifolds with round Kahler cones have unique bimeromorphic model and correspond to Hausdorff points in the corresponding Teichmuller space. We prove thay any maximal holonomy hyperkahler manifold with $b_2 > 4$ has a deformation with round Kahler cone and the Picard lattice of signature (1,1), admitting two non-collinear integer isotropic classes. This is used to show that all known examples of hyperkahler manifolds admit a deformation with two transversal Lagrangian fibrations, and the Kobayashi metric vanishes unless the Picard rank is maximal.