R-equivalence on three-dimensional tori and zero-cycles

Abstract

We prove that the natural map $T\left(F\right)∕R\to A_0\left(X\right)$, where $T$ is an algebraic torus over a field $F$ of dimension at most $3$, $X$ a smooth proper geometrically irreducible variety over $F$ containing $T$ as an open subset and $A_0\left(X\right)$ is the group of classes of zero-dimensional cycles on $X$ of degree zero, is an isomorphism. In particular, the group $A_0\left(X\right)$ is finite if $F$ is finitely generated over the prime subfield, over the complex field, or over a $p$-adic field.