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Hyperbolic geometry of the ample cone of a hyperkähler manifold

par Amerik, Ekaterina;Verbitskiy, Misha
Référence Research in Mathematical Sciences, 3, 1, 7
Publication Publié, 2016-12

Article révisé par les pairs

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Résumé :

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) , 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).