Collections of parabolic orbits in homogeneous spaces, homogeneous dynamics and hyperkahler geometry

TL;DR

This paper proves that certain geometric classes on hyperk"ahler manifolds have bounded invariants, using ergodic theory on homogeneous spaces, advancing the understanding of the Morrison-Kawamata cone conjecture.

Contribution

It establishes a uniform bound on the Beauville-Bogomolov squares of primitive MBM classes for hyperk"ahler manifolds, improving previous results.

Findings

Bound on the Beauville-Bogomolov square depending only on deformation class

Application of ergodic theory to hyperk"ahler geometry

Advancement in the Morrison-Kawamata cone conjecture

Abstract

Let $M$ be a hyperk\"ahler manifold with $b_2(M)\geq 5$. We improve our earlier results on the Morrison-Kawamata cone conjecture by showing that the Beauville-Bogomolov square of the primitive MBM classes (i.e. the classes whose orthogonal hyperplanes bound the K\"ahler cone in the positive cone, or, in other words, the classes of negative extremal rational curves on deformations of $M$) is bounded in absolute value by a number depending only on the deformation class of $M$. The proof uses ergodic theory on homogeneous spaces.