Rationality of integral cycles
Let \(k\) be a field of characteristic \(0\) and let \(Y\) be a smooth quasi-projective variety defined over \(k\). In this article, the author considers the natural question when an element \(\overline{y}\) in \(\text{CH}^m(Y_{\overline{k}})\), the Chow group of \(Y\) over an algebraic closure \(\overline{k}\) of \(k\), is actually defined over \(k\), i.e. if \(\overline{y}\) is in the image of the map \(\text{CH}^m(Y)\to \text{CH}^m(Y_{\overline{k}})\). In an earlier paper [Manuscr. Math. 122, No. 3, 365--374 (2007; Zbl 1154.14003)], the author has shown in the situation of Chow groups with \(\mathbb{Z}/2\)-coefficients, that it suffices to check that the element is defined over the function field of a quadric of dimension \(>2m\) which the author has applied with great success to show the existence of fields whose \(u\)-invariants can take values of the form \(2^r+1\) for any prescribed \(r\), a very significant result in quadratic form theory, see [Progr. Math. 270, 661--685 (2009; Zbl 1236.11037)]. Similar results for \(\mathbb{Z}/p\)-coefficients have later been obtained by \textit{K. Zainoulline} [J. Reine Angew. Math. 628, 195--204 (2009; Zbl 1171.14004)]. The purpose of the present paper is to derive analogous results for the integral Chow groups. The main result in this paper states that if \(Q\) is an anisotropic projective quadric over \(k\) whose corresponding quadratic form has Witt index \(i_1(Q)>1\) over its own function field \(k(Q)\), and if \(m<\dim(Q)/2\), then \(\overline{y}\) is defined over \(k\) if and only if \(\overline{y}|_{\overline{k(Q)}}\) is defined over \(k(Q)\). If one makes the stronger assumption that \(i_1(Q)>m\), then it is shown that \(\text{CH}^m(Y)\to \text{CH}^m(Y_{k(Q)})\) is surjective. This holds in particular in the case where \(Q\) is the quadric belonging to an \(r\)-fold Pfister quadratic form and \(m<2^{r-1}-1\). Using an iterated function field construction (`Merkurjev's method'), one can now construct field extensions \(F/k\) such that for the Milnor \(K\)-groups, one has \(K^M_r(F)/2=0\), and such that \(\text{CH}^m(Y)\to \text{CH}^m(Y_{F})\) is surjective for all \(m<2^{r-1}-1\). A crucial tool in the proofs is given by the theory of symmetric operations in algebraic cobordism developed earlier by the author [Adv. Math. 213, No. 2, 489--552 (2007; Zbl 1129.14034)].
scientific article; zbMATH DE number 5827455