Zero cycles on Severi--Brauer flag varieties

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 $\mathrm{SB}_r(A)$.