Morrison-Kawamata cone conjecture for hyperkahler manifolds

TL;DR

This paper proves a version of the Morrison-Kawamata cone conjecture for hyperkähler manifolds, showing that automorphisms act with finitely many orbits on the faces of the Kähler cone, using ergodic theory techniques.

Contribution

It establishes the conjecture for hyperkähler manifolds with second Betti number not equal to five, connecting geometric automorphism actions with ergodic theory.

Findings

Automorphism groups have finitely many orbits on Kähler cone faces.

The union of infinitely many geodesic hypersurfaces is dense in certain orbifolds.

The result applies to simple holomorphically symplectic manifolds with specific topological conditions.

Abstract

Let $M$ be a simple holomorphically symplectic manifold, that is, a simply connected holomorphically symplectic manifold of Kahler type with $h^{2,0}=1$. We prove that the group of holomorphic automorphisms of $M$ acts on the set of faces of its Kahler cone with finitely many orbits, whenever $b_2(M)\neq 5$. This is a version of the Morrison-Kawamata cone conjecture for hyperkahler manifolds. 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 $\{S_i\}$ an infinite set of locally geodesic hypersurfaces. Then the union of $S_i$ is dense in $M$.