Hyperbolic geometry of the ample cone of a hyperkähler manifold

Abstract

Let M be a compact hyperkahler manifold with maximal holonomy (IHS). The group $$H^2(M, {\mathbb {R}})$$ H 2 ( M , R ) is equipped with a quadratic form of signature $$(3, b_2-3)$$ ( 3 , b 2 - 3 ) , called Bogomolov–Beauville–Fujiki form. This form restricted to the rational Hodge lattice $$H^{1,1}(M,{\mathbb {Q}})$$ H 1 , 1 ( M , Q ) has signature (1, k). This gives a hyperbolic Riemannian metric on the projectivization H of the positive cone in $$H^{1,1}(M,{\mathbb {Q}})$$ H 1 , 1 ( M , Q ) . Torelli theorem implies that the Hodge monodromy group $$\varGamma $$ Γ acts on H with finite covolume, giving a hyperbolic orbifold $$X=H/\varGamma $$ 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 $$P(M')/{\text {Aut}}(M')$$ P ( M ′ ) / Aut ( M ′ ) , where $$P(M')$$ P ( M ′ ) is the projectivization of the ample cone of a birational model $$M'$$ M ′ of M, and $${\text {Aut}}(M')$$ Aut ( M ′ ) the group of its holomorphic automorphisms. This is used to prove the existence of nef isotropic line bundles on a hyperkahler birational model of a simple hyperkahler 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).