Hyperbolic geometry of the ample cone of a hyperkähler manifold

Research in the Mathematical Sciences. 2016. Vol. 3. No. 7. P. 1–9.

Amerik E., Verbitsky M.

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).

Keywords: hyperkahler manifolds, Kahler cone, кэлеров конус, гиперболическая геометрия, гиперкэлеровы многообразия, hyperbolic geometry, cusp points, каспидальные точки

Publication based on the results of: Алгебраическая геометрия симплектических многообразий (2016)