Recognising flag varieties and reductive groups

Abstract

Fix a flat and projective morphism X→Σ$X\rightarrow \Sigma$ of schemes. We show, first, that any set of P1${\mathbb {P}}^1$ ‐fibrations on X$X$ defines a set of simple roots, a set of simple coroots and a Cartan matrix C$C$ . Second, X$X$ is an étale F${\mathcal {F}}$ ‐bundle over some projective Σ$\Sigma$ ‐scheme, where F${\mathcal {F}}$ is the flag variety of the adjoint Chevalley group over SpecZ${\mathop {\bf Spec\hspace{1.5pt}}\nolimits }{\mathbb {Z}}$ defined by C$C$ ; our approach works uniformly over any base and for any Cartan matrix. In particular, if the simple roots generate NS(X/Σ)Q$\operatorname{NS}(X/\Sigma)_{\mathbb {Q}}$ and X$X$ is cohomologically flat in degree zero over Σ$\Sigma$ then X$X$ is a form of F${\mathcal {F}}$ . When X$X$ is a smooth Fano variety over SpecC${\mathop {\bf Spec\hspace{1.5pt}}\nolimits }{\mathbb {C}}$ all of whose extremal rays are accounted for by these fibrations this is due to Occhetta, Solá Conde, Watanabe and Wiśniewski. Third, we recover, also 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$p$ ‐morphism of pinned root data determines a unique homomorphism of the corresponding groups.