  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.

Algebraic Geometry;Differential Geometry