Lefschetz theorems for torsion algebraic cycles in codimension 2
type: Item
instance of: scholarly article
title: Lefschetz theorems for torsion algebraic cycles in codimension 2
zbMATH Open document ID: 1388.14026
author: Deepam Patel
author: G. V. Ravindra
published in: Advances in Mathematics
publication date: 2017-08-01T00:00:00Z
review text: Consider \(\mathrm{CH}^2(X) \{ l \}\), the group of \(l\)-primary torsion in the Chow group of cycles codimension \(2\). Bloch defined a map \(\beta: \mathrm{CH}^2(X) \{ l \} \to N^1H^3 (X, \mathbb Q_{ l } / \mathbb Z_{ l })\), where \(N^1\) is the first piece of the coniveau filtration. Merkurjev-Suslin's theorems imply that \(\beta\) is an isomorphism, see [\textit{A. S. Merkur'ev} and \textit{A. A. Suslin}, Math. USSR, Izv. 21, 307--340 (1983; Zbl 0525.18008); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 46, No. 5, 1011--1046 (1982)]. Object of the present article is to exploit this identification, by proving first that \(N^1H^3\) satisfies Lefschetz and Noether properties. More precisely, given \(Y\) a smooth projective variety over \(\mathbb{C}\) and \(X\) a smooth ample hypersurface in \(Y\), the authors obtain that the natural restriction map \(\mathrm{CH}^2(Y) \{ l \} \to \mathrm{CH}^2(X) \{ l \}\) is an isomorphism when \(\dim Y \geq 5\). The Noether statement yields the vanishing of \(\mathrm{CH}^2(X) \{ l \}\) for the case of a general hypersurface \(X \subset \mathbb{P}^4\) of degree \(\geq 5\). The authors state that their demonstration of the Noether property follows closely the ideas of \textit{S. Bloch} and \textit{H. Esnault} [Math. Ann. 304, No. 2, 303--314 (1996; Zbl 0868.14004)]. This vanishing result gives some further evidence for a conjecture of Griffiths and Harris: On a general 3-fold \(X\) of degree \(\geq 6\), the Abel-Jacobi map \(\alpha\) from algebraic \(1\)-cycles on \(X\) homologically equivalent to zero to the intermediate Jacobian \(J^ 2(X)\) is zero. The image of \(\alpha\) is known to be of torsion, see [\textit{M. L. Green}, J. Differ. Geom. 29, No. 3, 545--555 (1989; Zbl 0692.14003)] and [\textit{C. Voisin}, C. R. Acad. Sci., Paris, Sér. I 307, No. 4, 157--160 (1988; Zbl 0673.14005)]. The last part deals with the Griffiths group. The authors have informed us that some of their statements here are only conjectural. The point …
reviewed by: Alberto Collino
Mathematics Subject Classification ID: 14C15
Mathematics Subject Classification ID: 14C25
zbMATH DE Number: 6755314
zbMATH Keywords: codimension two
zbMATH Keywords: torsion algebraic cycles
MaRDI profile type: Publication
full work available at URL: j.aim.2017.05.018
OpenAlex ID: W2730098486
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DOI: 10.1016/J.AIM.2017.05.018
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description: scientific article; zbMATH DE number 6755314