zbMATH — the first resource for mathematics

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)].

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