Zero cycles on Severi--Brauer flag varieties

TL;DR

This paper investigates the zero-cycle Chow groups of Severi--Brauer flag varieties, establishing torsion bounds and conditions for triviality, especially over local and global fields, extending to stably birational varieties.

Contribution

It proves new torsion bounds and triviality results for zero cycles on Severi--Brauer flag varieties, reducing to prime power cases and extending to stably birational varieties.

Findings

A_0( ext{SB}_r(A)) is ( ext{d}, n/d)-torsion where d=(r,n).

A_0( ext{SB}_r(A))=0 over local or global fields.

Results extend to varieties stably birational to Severi--Brauer varieties.

Abstract

Let \(A\) be a central simple algebra over a field \(F\) with index \(n\) and let \(\mathrm{SB}_r(A)\) denote the \(r\)-th generalized Severi--Brauer variety associated with \(A\). We prove that the Chow group of zero cycles of degree zero \(\mathrm{A_0}(\mathrm{SB}_r(A))\) is \((d, n/d)\)-torsion where \(d = (r,n)\). Our approach reduces the general case to division algebras of prime power index and yields several new instances in which \(\mathrm{A_0}\) is trivial, together with sharper torsion bounds in general.\\ We also show that if \(F\) is a local or global field, then \(\mathrm{A_0}(\mathrm{SB}_r(A))=0\). Since Severi--Brauer flag varieties are stably birational to generalized Severi--Brauer varieties, these results extend to them, yielding corresponding torsion bounds and vanishing results for \(\mathrm{A_0}(X)\), where \(X\) is stably birational to \(\mathrm{SB}_r(A)\).