Jordan property for non-linear algebraic groups and projective varieties

Abstract

A century ago, Camille Jordan proved that the complex general linear group ${\rm GL}_n(\Bbb{C})$ has the Jordan property: there is a Jordan constant ${\rm C}_n$ such that every finite subgroup $H\le{\rm GL}_n(\Bbb{C})$ has an abelian subgroup $H_1$ of index $[H:H_1]\le{\rm C}_n$. 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 ${\rm Aut}(X)$ of every projective variety $X$ has the Jordan property.