## 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

### Citations:

Zbl 0691.14001; Zbl 0784.14003
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