Recognizing flag varieties and reductive groups

Abstract

Fix a flat and projective morphism $X\rightarrow\Sigma$ of schemes. We show, first, that any set of $\mathbb{P}^1$-fibrations on $X$ defines a set of simple roots, a set of simple coroots and a Cartan matrix $C$. Second, $X$ is an \'etale $F$-bundle over some projective $\Sigma$-scheme, where $F$ is the flag variety of the adjoint Chevalley group over the integers defined by $C$. In particular, if the simple roots generate the N\'eron--Severi group of $X$ relative to $\Sigma$ and $X$ is cohomologically flat in degree zero over $\Sigma$ then $X$ is a form of $F$. When $X$ is a smooth Fano variety over the complex numbers all of whose extremal rays are accounted for by these fibrations this is due to Occhetta, Sol\'a-Conde, Watanabe and Wi\'sniewski. Third, we recover, in a uniform way, the isomorphism and isogeny theorems of Chevalley and Demazure: over any base a pinned reductive group is determined by its pinned root datum, and a $p$-morphism of pinned root data determines a unique homomorphism of the corresponding groups.