R-equivalence on three-dimensional tori and zero-cycles (Q1005854)

Let \(X\) be a variety over the field \(F\). Two \(F\)-points in \(X\) are \(R\)-equivalent if they can be connected by a chain of maps \(\mathbb{P}^1\to X\) over \(F\). In the article under review, the author proves the following: Let \(X\) be a smooth proper geometrically irreducible \(F\)-variety containing an algebraic torus \(T\) of dimension at most 3 as an open subset. Then the natural map \[ T(F)/R \to A_0(X) \] from the group of \(R\)-equivalence classes of \(F\)-points on \(T\) to the group of classes of zero cycles on \(X\) of degree zero is an isomorphism. Results of \textit{J.-L. Colliot-Thélène} and \textit{J.-J. Sansuc} [Ann. Sci. Éc. Norm. Supér. (4) 10, 175--229 (1977; Zbl 0356.14007)] then imply finiteness of \(A_0(X)\) if \(F\) is finitely generated over its prime field, the complex numbers, or a \(p\)-adic field