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Hyperbolic geometry of the ample cone of a hyperkähler manifold
Let M be a compact hyperkähler manifold with maximal holonomy (IHS). The group H2(M,ℝ) is equipped with a quadratic form of signature (3,b2−3)(3,b2−3), called Bogomolov–Beauville–Fujiki form. This form restricted to the rational Hodge lattice H1,1(M,ℚ)has signature (1, k). This gives a hyperbolic Riemannian metric on the projectivization H of the positive cone in H1,1(M,ℚ). Torelli theorem implies that the Hodge monodromy group Γ acts on H with finite covolume, giving a hyperbolic orbifold X=H/Γ. We show that there are finitely many geodesic hypersurfaces, which cut X into finitely many polyhedral pieces in such a way that each of these pieces is isometric to a quotient P(M′)/Aut(M′), where P(M′) is the projectivization of the ample cone of a birational model M′ of M, and Aut(M′) the group of its holomorphic automorphisms. This is used to prove the existence of nef isotropic line bundles on a hyperkähler birational model of a simple hyperkähler manifold of Picard number at least 5 and also illustrates the fact that an IHS manifold has only finitely many birational models up to isomorphism (cf. Markman and Yoshioka in Int. Math. Res. Not. 2015(24), 13563–13574, 2015).