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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.

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