Generalization of Abel’s theorem and some finiteness property of zero- cycles on surfaces. (English) Zbl 0794.14002

Saito, Hiroshi

Compos. Math. 84, No. 3, 289-332 (1992).

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

Keywords:

polarized Hodge structures of weight 2; vanishing of cycles in the intermediate jacobian; filtration on the Chow group

References:

[1]	Angeniol, B. : Familles de cycles algébriques - schémas de Chow , Lecture Notes in Math. 896, Springer Verlag, Berlin, Heidelberg, New York, 1981. · Zbl 0496.14004 · doi:10.1007/BFb0091696
[2]	Angeniol, B. , and El Zein, F. : La classe fondamentale relative d’un cycle , Bull. Soc. Math. France, Memoire, 58 (1978), 67-93. · Zbl 0388.14003 · numdam:MSMF_1978__58__67_0 · eudml:94781
[3]	Artin, M. , Grothendieck, A. , Verdier, J.-P. : Théorie des Topos et Cohomologie étales de schémas, exposé XVIII par Deligne, P. , Lecture Notes in Math., 305, Springer Verlag, Berlin, Heidelberg, New York, 1973. · Zbl 0245.00002 · doi:10.1007/BFb0070714
[4]	Bloch, S. : Torsion algebraic cycles and a theorem of Roitman , Comp. Math., 39 (1979), 107-127. · Zbl 0463.14002 · numdam:CM_1979__39_1_107_0 · eudml:89412
[5]	Bloch, S. : Lectures on algebraic cycles , Duke University Mathematics Series IV, Durham, 1980. · Zbl 0436.14003
[6]	Deligne, P. : Theorie de Hodge III , Publ. math. IHES, 44 (1974) 5-77. · Zbl 0237.14003 · doi:10.1007/BF02685881 · numdam:PMIHES_1974__44__5_0 · eudml:103935
[7]	Kleiman, S. : Algebraic cycles and the Weil conjecture, in Dix exposés sur la cohomologie des schémas , North-Holland, Amsterdam, 1968. · Zbl 0198.25902
[8]	Merkur’Ev, A.S. , and Suslin, A.A. : K-cohomology of Brauer-Severi varieties and the norm residue homomorphism , Izv. Acad. Nauk, 46 (1982), 1011-1046(= Math. USSR. Izv. 21 (1983), 307-340.) · Zbl 0525.18008 · doi:10.1070/IM1983v021n02ABEH001793
[9]	Roitman, A.A. : The torsion of the group of 0-cycles modulo rational equivalence , Ann. of Math., 111 (1980), 553-569. · Zbl 0504.14006 · doi:10.2307/1971109
[10]	Saavedra, N. : Catégories Tannakiens , Lecture Notes in Math. 265, Springer Verlag, Berlin, Heidelberg, New York, 1972. · Zbl 0241.14008
[11]	Saito, H. : The Hodge cohomology and cubic equivalences , Nagoya Math. J., 94 (1984) 1-41. · Zbl 0574.14005 · doi:10.1017/S002776300002081X
[12]	Saito, H. : A note on cubic equivalences , Nagoya Math. J., 101 (1986) 1-26. · Zbl 0594.14008 · doi:10.1017/S0027763000000313
[13]	Samuel, P. : Relations d’équivalence en géométrie algébrique , in Proceeding of International Congress of Mathematicians, 1958, Cambridge University Press, Cambridge, 1960. · Zbl 0119.36901
[14]	Severi, F. : Ulterior sviluppi della teoria delle serie di equivalenza sulle suerfidie algebriche, appendice I a Geometria dei sistemi algebrici sopra una superficie e sopra una varietà algebrica , vol. terzo, Edizioni Cremonese, Roma, 1959.
[15]	Verdier, J.-P. : Classes d’homologie associée à un cycle , Astérisque, 36-37 (1976), 101-151 (exposé VI). · Zbl 0346.14005
[16]	Weil, A. : Oeuvres Scientifiques , Collected papers, Vol. II, Corrected second printing, Springer Verlag, New York, Heidelberg, Berlin, 1980. · Zbl 0428.01012