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Un résultat en théorie des cycles algébriques de codimension deux. (French) Zbl 0532.14002
Let V be a smooth projective variety defined over an algebraically closed field of any characteristic. Let $$CH^ i(V)$$ be the Chow group of rational equivalence classes of algebraic cycles of codimension i on V and $$A^ i(V)\subset CH^ i(V)$$ the subgroup of those classes which are algebraically equivalent to zero. Let A be an abelian variety: a homomorphism $$\Phi:A^ i(V)\to A$$ is called ”regular” if for every triple $$(W,w_ 0,Z)$$ with W a smooth, projective variety, $$w_ 0\in W$$ and $$Z\in CH^ i(W\times V)$$, the composite map $$\Phi.Z:W\to A$$ defined by $$\Phi.Z(w)=\Phi \{Z(w)-Z(w_ 0)\}$$ is a morphism of algebraic varieties [see P. Samuel, Proc. Internat. Congr. Math., Edinburgh 1958, 470-487 (1960; Zbl 0119.369)]. The main result in this paper is as follows: For $$i=2$$ there exists a universal couple $$(A_ 0,\Phi_ 0)$$, consisting of an abelian variety $$A_ 0$$ and a regular homomorphism $$\Phi_ 0:A^ 2(V)\to A_ 0$$; i.e. for every other such couple $$(A,\Phi)$$ there is a unique homomorphism of abelian varieties $$f:A_ 0\to A$$ such that $$\Phi =f.\Phi_ 0$$. The proof is based upon results of H. Saito [Nagoya Math. J. 75, 95-119 (1979; Zbl 0433.14036)], S. Bloch and A. Ogus [Ann. Sci. Éc. Norm. Supér., IV. Sér. 7(1974), 181-201 (1975; Zbl 0307.14008)], S. Bloch [Groupe de Brauer, Sémin., Les Plans-sur-Bex 1980, Lect. Notes Math. 844, 76-102 (1981; Zbl 0467.12011)] and the theorem of A. S. Merkurjev and A. A. Suslin from algebraic K-theory [Izv. Akad. Nauk SSSR, Ser. Mat. 46, No.5, 1011-1046 (1982; Zbl 0525.18008)]. Next let $$k={\mathbb{C}}$$. Let $$J^ i(V)$$ be the (Weil or Griffiths) intermediate Jacobian, $$\Psi:A^ i(V)\to J^ i(V)$$ the Abel-Jacobi map and $$J^ i_ a(V)=Im(\Psi)$$ in $$J^ i(V)$$. It is proved that for $$i=2$$ the couple $$(J_ a^ 2(V),\Psi)$$ is universal in the sense described above. The proof uses besides the above mentioned theorem of Merkurjev-Suslin also the Bloch map for $$\ell$$-torsion cycles $$\lambda:CH^ 2(V)_{\ell -tors}\to H^ 3_{et}(V,{\mathbb{Q}}_{\ell}/{\mathbb{Z}}_{\ell}(2))$$, where $$\ell$$ is a prime number [cf. S. Bloch, Compos. Math. 39, 107-127 (1979; Zbl 0463.14002)]. Finally it is proved that the map on $$\ell$$-torsion points $$\Psi:A^ 2(V)_{\ell -tors}\to J^ 2_ a(V)_{\ell -tors}$$ is an isomorphism [compare with the theorem of A. A. Rojtman, Ann. Math., II. Ser. 111, 553-569 (1980; Zbl 0504.14006)].

##### MSC:
 14C15 (Equivariant) Chow groups and rings; motives 14C05 Parametrization (Chow and Hilbert schemes) 14K30 Picard schemes, higher Jacobians 14M07 Low codimension problems in algebraic geometry