Vari\'et\'es homog\`enes sous $\PGL_n$

TL;DR

This paper explicitly constructs a bundle and an isomorphism for projective homogeneous varieties under automorphism groups of Azumaya algebras, enabling computation of their Chow groups via associated Severi-Brauer varieties.

Contribution

It provides an explicit construction linking projective homogeneous varieties to flag bundles over Severi-Brauer varieties, facilitating Chow group calculations.

Findings

Explicit bundle construction for homogeneous varieties

Canonical isomorphism to flag bundles

Chow groups computed via Severi-Brauer varieties

Abstract

Let $A$ be an Azumaya algebra over a field. If $G$ is the group of automorphisms of $A$ and $X$ denotes a projective homogeneous variety under $G$, we construct in a very explicit way and under suitable hypotheses a bundle $\mathcal{V}$ on $S$, where $S$ is a (generalized) Severi-Brauer variety associated to $A$, and a canonical isomorphism between $X$ and a flag bundle on $\mathcal{V}$. This allows to explicitely compute Chow groups of $X$ in terms of the Chow groups of $S$.