Finiteness of Klein actions and real structures on compact hyperkähler manifolds

Abstract

One central problem in real algebraic geometry is to classify the real structures of a given complex manifold. We address this problem for compact hyperkähler manifolds by showing that any such manifold admits only finitely many real structures up to equivalence. We actually prove more generally that there are only finitely many, up to conjugacy, faithful finite group actions by holomorphic or anti-holomorphic automorphisms (the so-called Klein actions). In other words, the automorphism group and the Klein automorphism group of a compact hyperkähler manifold contain only finitely many conjugacy classes of finite subgroups. We furthermore answer a question of Oguiso by showing that the automorphism group of a compact hyperkähler manifold is finitely presented.

Final version, published in Mathematische Annalen

References in corpus (15)

• A survey of Torelli and monodromy results for holomorphic-symplectic varieties
• Mapping class group and global Torelli theorem for hyperkahler manifolds: an erratum
• A global Torelli theorem for hyperkahler manifolds
• The cone conjecture for Calabi-Yau pairs in dimension two
• Coxeter groups, Lorentzian lattices, and K3 surfaces
• Discrete automorphism groups of convex cones of finite type
• Cremona transformations, surface automorphisms and plane cubics
• Real structures on rational surfaces and automorphisms acting trivially on Picard groups
• A surface with discrete and non-finitely generated automorphism group
• Collections of parabolic orbits in homogeneous spaces, homogeneous dynamics and hyperkahler geometry
• A proof of the Kawamata-Morrison Cone Conjecture for holomorphic symplectic varieties of K3^[n] or generalized Kummer deformation type
• Algebraic surfaces and hyperbolic geometry
• Moduli spaces of bundles over non-projective K3 surfaces
• The Morrison-Kawamata Cone Conjecture and Abundance on Ricci flat manifolds
• Automorphisms of hyperkähler manifolds and groups acting on CAT(0) spaces

Cited by in corpus (2)

• Geometry and automorphisms of non-Kähler holomorphic symplectic manifolds
• Holomorphic motions, natural families of entire maps, and multiplier-like objects for wandering domains