Jordan property for non-linear algebraic groups and projective varieties

Abstract

A century ago, Camille Jordan proved that the complex general linear group $G{L}_n(C)$ has the Jordan property: there is a Jordan constant $C_n$ such that every finite subgroup $H\le G{L}_n(C)$ has an abelian subgroup $H_1$ of index $[H:H_1]\le C_n$. We show that every connected algebraic group $G$ (which is not necessarily linear) has the Jordan property with the Jordan constant depending only on $\dim G$, and that the full automorphism group $Aut(X)$ of every projective variety $X$ has the Jordan property

American Journal of Mathematics (to appear); minor changes

References in corpus (1)

• Bounds for the orders of the finite subgroups of G(k)

Cited by in corpus (16)

• Finite group actions on homology spheres and manifolds with nonzero Euler characteristic
• Automorphisms of pointless surfaces
• Jordan property for automorphism groups of compact spaces in Fujiki's class $C$
• Automorphism groups of compact complex surfaces
• Jordan property for algebraic groups and automorphism groups of projective varieties in arbitrary characteristic
• Invariant subvarieties with small dynamical degree
• Boundedness results for singular Fano varieties and applications to Cremona groups
• Finite $p$-groups of birational automorphisms and characterizations of rational varieties
• Automorphisms of surfaces over fields of positive characteristic
• Automorphism groups of Inoue and Kodaira surfaces
• Geometry and automorphisms of non-Kähler holomorphic symplectic manifolds
• Automorphism groups of $P^1$-bundles over a non-uniruled base
• Bimeromorphic automorphisms groups of certain conic bundles
• A theorem of Tits type for automorphism groups of projective varieties in arbitrary characteristic (with an appendix by Tomohide Terasoma)
• Derived length of zero entropy groups acting on projective varieties in arbitrary characteristic -- A remark to a paper of Dinh-Oguiso-Zhang
• Automorphism groups of Moishezon threefolds