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Generalization of Abel’s theorem and some finiteness property of zero- cycles on surfaces. (English) Zbl 0794.14002
Let \(C\) be a nonsingular projective curve, and \(J(C)\) be its jacobian. To a divisor \(D\) of degree zero on \(C\) is associated a point \(\gamma (D)\) of the jacobian by integration, and so-called Abel’s theorem states that the image of \(D\) in the jacobian vanishes if and only if \(D\) is rationally equivalent to zero, i.e., \(D\) is a divisor of a rational function of \(C\), in other words, it gives an algebraic condition for \(\gamma (D)=0\), whereas the jacobian is defined complex-analytically. The following conjecture of S. Bloch [“Lectures on algebraic cycles”, Duke Univ. Math. Ser. 4 (1980; Zbl 0436.14003)] can be regarded as the weight 2 counterpart of the above equivalence:
For any smooth projective variety \(V\) over \(\mathbb{C}\), there exists a filtration on the Chow group of 0-cycles \(\text{CH}_ 0(V)\). Let \(S\) be a surface and let \(z\) be a cycle on \(V \times S\) with \(\dim z=m=\dim V\). Then \(z\) induces a map \(z:\text{CH}_ 0 (V) \to \text{CH}_ 0 (S)\). - - We get also \([z]:gr^ \bullet \text{CH}_ 0(V) \to gr^ \bullet \text{CH}_ 0(S)\). Conjecture (S. Bloch). The map \([z]\) depends only upon the cohomology class \(\{c\} \in H^ 4(V \times S)\).
Metaconjecture. There is an equivalence of category between a suitable category of polarized Hodge structures of weight 2 and a category built up from \(gr^ 2\text{CH}_ 0 (S)\).
The aim of this article is two-fold: to give a condition for the vanishing of cycles in the intermediate jacobian, and to construct filtrations on the Chow groups which satisfy the above conjectures.

14C05 Parametrization (Chow and Hilbert schemes)
14C25 Algebraic cycles
14K30 Picard schemes, higher Jacobians
14H40 Jacobians, Prym varieties
14J99 Surfaces and higher-dimensional varieties
Full Text: Numdam EuDML
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