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

Let M be a hyperkähler manifold of maximal holonomy (i.e., an IHS manifold), and let K be its Kähler cone, which is an open, convex subset in the space H1,1(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 Kähler cone. We say that the Kähler cone is round if it is equal to the positive cone. The manifolds with round Kähler cones have a unique bimeromorphic model and correspond to Hausdorff points in the corresponding Teichmüller space. We prove that any maximal holonomy hyperkähler manifold with b2>5 has a deformation with round Kähler cone and the Picard lattice of signature (1,1), admitting two noncollinear integer isotropic classes. This is used to show that all known examples of hyperkähler manifolds admit a deformation with two transversal Lagrangian fibrations, and their Kobayashi metric vanishes, unless the Picard rank is maximal.