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Cohomological and cycle-theoretic connectivity. (English) Zbl 0828.14003

Cohomological theory of a smooth projective variety \(X\) and the group of cycles modulo rational equivalence, i.e. the Chow group, are related by a series of conjectures of A. A. Beilinson and S. Bloch [see U. Jannsen, “Mixed motives and algebraic \(K\)-theory”, Lect. Notes Math. 1400 (1990; Zbl 0691.14001)]. In this paper, the author examines some concrete cases.
For instance, the classical weak Lefschetz theorem, which asserts that if \(Y \subset X\) is an ample divisor then \(H^l (X) \to H^l (Y)\) is an isomorphism for \(l < \dim Y\), leads to the conjectural cycle-theoretic analogue that \(\text{CH}^p (X)_\mathbb{Q} \to \text{CH}^p (Y)_\mathbb{Q}\) is an isomorphism for \(p < \dim Y/2\) (the case \(p = 1\) is a classical theorem by S. Lefschetz and A. Grothendieck). The author proves the conjecture in some cases. More precisely, he shows that given a sequence of positive integers \(d_1, \ldots, d_r\) and a non-negative integer \(l\), if \(X \subset \mathbb{P}^n\) is a smooth subvariety of multidegree \((d_1, \ldots, d_r)\) and, \(n\) is sufficiently large, then \(\text{CH}^l(X)_\mathbb{Q} \cong \mathbb{Q}\).
Another example is given by the classical theorem of M. Noether and S. Lefschetz, which states that if \(X = \mathbb{P}^3\) and \(Y \subset X\) is a general surface with \(\deg Y \geq 4\), then \(\text{CH}^l(Y) = \mathbb{Z}\). In a recent paper, M. V. Nori shows that there is a cohomological analogue of this statement as well, from which one expects that if \(Y\) is a general complete intersection in \(X\) of sufficiently high multidegree then \(\text{CH}^p (X)_\mathbb{Q} \to \text{CH}^p (Y)_\mathbb{Q}\) is an isomorphism for \(p < \dim Y\). By improving Nori’s result, the author provides precise bounds for the degrees required.
Finally, continuing previous works of H. Esnault, M. V. Nori and V. Srinivas [Math. Ann. 293, No. 1, 1-6 (1992; Zbl 0784.14003)], the author studies cycles of small dimension \(l\) on intersections \(X\) of small multidegree \((d_1, \ldots, d_r)\) in \(\mathbb{P}^n\) and shows that, when \(n\) is large, then \(\text{CH}_l (X) = \mathbb{Z}\) and this group is generated by the class of a linear subspace \(\mathbb{P}^l \subset X\).

MSC:

14C25 Algebraic cycles
14C05 Parametrization (Chow and Hilbert schemes)
14M10 Complete intersections
14C15 (Equivariant) Chow groups and rings; motives
14F25 Classical real and complex (co)homology in algebraic geometry
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