Zero cycles on homogeneous varieties

TL;DR

This paper investigates the group of zero cycles of degree zero on projective homogeneous varieties, relating rational equivalence to R-equivalence and moduli spaces, leading to new vanishing results.

Contribution

It introduces a novel approach connecting rational equivalence of zero cycles to R-equivalence on symmetric powers and moduli spaces, extending known results.

Findings

Proves $A_0(X) = 0$ for certain homogeneous varieties.

Establishes a link between zero cycles and moduli spaces of étale subalgebras.

Extends previous results by Swan, Karpenko, Merkurjev, and Panin.

Abstract

In this paper we study the group $A_0(X)$ of zero dimensional cycles of degree 0 modulo rational equivalence on a projective homogeneous algebraic variety $X$. To do this we translate rational equivalence of 0-cycles on a projective variety into R-equivalence on symmetric powers of the variety. For certain homogeneous varieties, we then relate these symmetric powers to moduli spaces of \'etale subalgebras of central simple algebras which we construct. This allows us to show $A_0(X) = 0$ for certain classes of homogeneous varieties, extending previous results of Swan / Karpenko, of Merkurjev, and of Panin.