Morrison-Kawamata cone conjecture for hyperkahler manifolds

Abstract

Let M be a simple holomorphically symplectic manifold, that is, a simply connected compact holomorphically symplectic manifold of Kähler type with h 2,0 = 1. Assuming b2(M ) = 5, we prove that the group of holomorphic automorphisms of M acts on the set of faces of its Kähler cone with finitely many orbits. This statement is known as Morrison-Kawamata cone conjecture for hyperkähler manifolds. As an implication, we show that any hyperkähler manifold has only finitely many non-equivalent birational models. The proof is based on the following observation, proven with ergodic theory. Let M be a complete Riemannian orbifold of dimension at least three, constant negative curvature and finite volume, and {Si} an infinite set of complete, locally geodesic hypersurfaces. Then the union of Si is dense in M .

Collections of Orbits of Hyperplane Type in Homogeneous Spaces, Homogeneous Dynamics, and Hyperkähler Geometry

“…We deduce Theorem 1.7 from the general formalism of Ratner, Mozes-Shah and Eskin-Mozes-Shah used in [AV2] to prove Theorem 1.1. In order to be able to apply this machinery we first prove a simple statement on Lie groups which replaces a set of orbits of hyperplane type of conjugated parabolic subgroups on G/K by a set of orbits of a single one on a suitable fibration over G/K, which is a G-homogeneous space "in between" G/K and G itself.…”

“…One may ask whether there are only finitely many of them up to the action of automorphism group (this is a version of the "cone conjecture" by Kawamata and Morrison). Our theorem from [AV2] implies this as soon as the Picard rank is greater than three. The group Γ of the theorem is the Hodge monodromy group (see subsection Definition 3.9) rather than the automorphism group, but this is handled using the global Torelli theorem ( [Ma], [V1]).…”

“…Remark 1.5: This corollary also removes a technical assumption b 2 = 5 needed in [AV2] to prove that the set of faces of the Kähler cone is finite up to automorphism group action; see section 3.…”

“…We have seen that the collection of S i lifts to a Γ-invariant collection of P 0 -orbits R i in G. It suffices to prove that this collection is dense or finite up to the action of Γ, or, in other words, that the corresponding orbits in Γ\G are finitely many or dense. To this end, we apply the same argument as in [AV2]. The P 0 -orbits R i give rise to probability measures µ i on Γ\G, supported on those orbits; those are simply the translates of µ P0 , the "pushforward" of the Haar measure on P 0 .…”

Construction of automorphisms of hyperkähler manifolds

“…The following theorem has been proved in [AV2]. This result is a version of Morrison-Kawamata cone conjecture for hyperkähler manifolds.…”