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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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