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Let $M$ be a hyperkahler manifold of maximal holonomy (that is, an IHSmanifold), and let $K$ be its Kahler cone, which is an open, convex subset inthe space $H^{1,1}(M, R)$ of real (1,1)-forms. This space is equipped with acanonical bilinear symmetric form of signature $(1,n)$ obtained as arestriction of the Bogomolov-Beauville-Fujiki form. The set of vectors ofpositive square in the space of signature $(1,n)$ is a disconnected union oftwo convex cones. The "positive cone" is the component which contains theKahler cone. We say that the Kahler cone is "round" if it is equal to thepositive cone. The manifolds with round Kahler cones have unique bimeromorphicmodel and correspond to Hausdorff points in the corresponding Teichmullerspace. 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 toshow that all known examples of hyperkahler manifolds admit a deformation withtwo transversal Lagrangian fibrations, and the Kobayashi metric vanishes unlessthe Picard rank is maximal.

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