Lefschetz theorems for torsion algebraic cycles in codimension 2
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 is that the proof they write depends on the identification of Murre's \(J^2_a\), the algebraic part of the intermediate Jacobian \(J\) , with \(J^2_0(Z)\), the largest abelian variety inside \(J\). This fact is at present unknown, but they note that it is a consequence of the generalized Hodge conjecture.
scientific article; zbMATH DE number 6755314
scientific article