Rationality of integral cycles
type: Item
instance of: scholarly article
title: Rationality of integral cycles
zbMATH Open document ID: 1257.14003
author: Alexander Vishik
published in: Documenta Mathematica
publication date: 2010-12-16T00:00:00Z
full work available at URL: vishik.html
review text: 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 extensio…
reviewed by: Detlev Hoffmann
Mathematics Subject Classification ID: 14C15
Mathematics Subject Classification ID: 11E04
Mathematics Subject Classification ID: 14C25
Mathematics Subject Classification ID: 19D45
Mathematics Subject Classification ID: 55S25
zbMATH DE Number: 5827455
zbMATH Keywords: quadric
zbMATH Keywords: function field of a quadric
zbMATH Keywords: Chow group
zbMATH Keywords: algebraic cobordism
zbMATH Keywords: symmetric operation
zbMATH Keywords: Landweber-Novikov operation
zbMATH Keywords: rationality
MaRDI profile type: Publication
Recommended article: Characterization of minimal Pfister neighbors via Rost projectors
Recommended article: Q4466502
Recommended article: Q4711042
Recommended article: Witt groups of curves over local fields
Recommended article: On the problem of descent for quadratic forms
Recommended article: Q3594231
Recommended article: Fields of \(u\)-invariant \(9\)
Recommended article: Birational geometry of quadrics in characteristic 2
Recommended article: Birational geometry of quadrics
Recommended article: Essential dimension of quadrics
description: scientific article; zbMATH DE number 5827455