Complex varieties for which the Chow group\(\mod n\) is not finite (Q2778308)

Let \(V\) be a smooth complete geometrically integral variety of dimension \(d\) over a field \(k\) of any characteristic \(p\geq 0\). Let \(k\subset L\subset L'\) be field extensions, with \(L\) and \(L'\) both separably closed. A result of \textit{F. Lecomte} [Duke Math. J. 53, 405--426 (1986; Zbl 0632.14007)] shows that the natural map \(\varepsilon: \text{CH}^i(V_L)/n \text{CH}^i(V_L)\to \text{CH}^i(V_{L'})/n \text{CH}^i(V_{L'})\) is an isomorphism for all \(p\) and for all integers \(n\) which are prime to \(p\). From the Kummer sequence it follows that these two quotient groups are finite if \(i=1\). Some results of Bloch imply that these groups are also finite for \(i=d\). Colliot-Thélène has asked whether these groups ar finite for every \(i\).NEWLINENEWLINEThe aim of the paper under review is to produce examples of varieties \(V\) and integers \(i\) and \(n\) for which \(\text{CH}^i(V_L)/n \text{CH}^i(V_L)\) is not finite. The examples in question are threefolds which are obtained by desingularizing self-fiber-products of elliptic surfaces.