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

par Amerik, Ekaterina;Verbitskiy, Misha
Référence International mathematics research notices, 2020, 1, page (25-38)
Publication Publié, 2020-01

Article révisé par les pairs

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Résumé :

Consider the space M = O(p, q)/O(p) × O(q) of positive p-dimensional subspaces in a pseudo-Euclidean space V of signature (p, q), where p > 0, q > 1 and (p,q)neq (1,2), with integral structure: V = VZ ⊗ Z. Let Γ be an arithmetic subgroup in G = O(VZ), and R VZ a Γ-invariant set of vectors with negative square. Denote by RT the set of all positive p-planes W V such that the orthogonal complement W contains some r R. We prove that either R is dense in M or Γ acts on R with finitely many orbits. This is used to prove that the squares of primitive classes giving the rational boundary of the Kähler cone (i.e., the classes of "negative" minimal rational curves) on a hyperkähler manifold X are bounded by a number which depends only on the deformation class of X. We also state and prove the density of orbits in a more general situation when M is the space of maximal compact subgroups in a simple real Lie group.