Applications of algebraic K-theory to the theory of algebraic cycles.

*(English)*Zbl 0561.14002
Algebraic geometry, Proc. Conf., Sitges (Barcelona)/Spain 1983, Lect. Notes Math. 1124, 216-261 (1985).

[For the entire collection see Zbl 0551.00002.]

This paper originated from lectures delivered during a conference on algebraic geometry in Sitges (Barcelona) in 1983. The main purpose was to discuss an application of a theorem of Merkur’ev and Suslin in algebraic K-theory to the theory of algebraic cycles. As such it is also an expanded version of my C. R. note on algebraic cycles [C. R. Acad. Sci., Paris, Sér. I 296, 981-984 (1983; Zbl 0532.14002)]. Let V be a smooth projective irreducible variety defined over an algebraically closed field k; \(d=\dim V\). Let \(CH^ i(V)\) be the i-th Chow group of V, i.e. the group of algebraic cycles on V of codimension i modulo rational equivalence. Let \(A^ i(V)\subset CH^ i(V)\) be the subgroup consisting of those classes which are themselves algebraically equivalent to zero. Consider algebraic families of cycles on V, i.e. couples (T,Z) with T a smooth projective irreducible variety and \(Z\in CH^ i(T\times V)\); fixing a point \(t_ 0\in T\) one gets a map \(Z: T\to A^ i(V).\) Finally consider couples (A,\(\phi)\) with A an abelian variety and \(\phi: A^ i(V)\to A\) a ”regular” homomorphism (”regular” means: for every algebraic family (T,Z) as above the composite map \(\phi \circ Z: T\to A\) is a morphism of algebraic varieties). Natural question: does there exist a universal regular homomorphism? If so, say \((A_ 0,\phi_ 0)\), then it is clearly unique up to a unique isomorphism and it is called the algebraic representative for \(A^ i(V)\). Except for \(i=1\) and \(i=d\) the answer is unknown [see for instance R. Hartshorne’s survey lecture on algebraic cycles in Algebraic Geom., Proc. Sympos. Pure Math. 29, Arcata 1974, 129-164 (1975; Zbl 0314.14001); p. 144]. The two main results of this paper answer the question for \(i=2\), i.e. for codimension two cycles.

Namely: Theorem I. (Any algebraically closed k) For \(i=2\) there exists such a universal \((A_ 0,\phi_ 0)\); moreover \(2\cdot \dim A_ 0\leq B_ 3(V)=\dim H^ 3_{et}(V,{\mathbb{Q}}_{\ell}),\) \(\ell \neq char(k)\). - Theorem II. Let \(k={\mathbb{C}}\), \(J^ 2(V)\) the second intermediate Jacobian, \(\psi: A^ 2(V)\to J^ 2(V)\) the Abel-Jacobi map; put \(J^ 2_ a(V)=Im(\psi).\) Then \((J^ 2_ a(V),\psi)\) is universal in the above sense. The proofs depend on results of H. Saito [Nagoya Math. J. 75, 95-119 (1979; Zbl 0433.14036)], of S. Bloch [Compos. Math. 39, 47-105 and 107-127 (1979; Zbl 0426.14018 and Zbl 0463.14002)], of S. Bloch and S. Ogus [Ann. Sci. Éc. Norm. Super., IV. Sér. 7(1974), 181-201 (1975; Zbl 0307.14008)] and on the above mentioned theorem of A. S. Merkur’ev and A. A. Suslin [Math. USSR, Izv. 21, 307-340 (1983); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 46, No.5, 1011-1046 (1982; Zbl 0525.18008)] which states that the norm residue map \(K_ 2(F)/nK_ 2(F)\to H^ 2_{et}(F,\mu_ n^{\otimes 2})\) is an isomorphism (where F is a field and \((n,char(F))=1).\) Before proving the two main results an effort is made in the paper to discuss the connection with étale cohomology and with the theory of algebraic cycles.

As a side result one obtains that there exists a higher Picard variety \(Pic^ 2(V)\) in the sense of D. I. Lieberman [Am. J. Math. 90, 366-374 (1968; Zbl 0159.505) provided one modifies slightly (and as it seems naturally) the requirements for such a higher Picard variety. The proof of this side result is entirely formal (however we take the opportunity to inform that in this proof one has to define the functor \({\mathcal F}\) in section 8.2 somewhat more carefully, namely \({\mathcal F}\) is ”generated” by the elements described in the footnote on page 261).

This paper originated from lectures delivered during a conference on algebraic geometry in Sitges (Barcelona) in 1983. The main purpose was to discuss an application of a theorem of Merkur’ev and Suslin in algebraic K-theory to the theory of algebraic cycles. As such it is also an expanded version of my C. R. note on algebraic cycles [C. R. Acad. Sci., Paris, Sér. I 296, 981-984 (1983; Zbl 0532.14002)]. Let V be a smooth projective irreducible variety defined over an algebraically closed field k; \(d=\dim V\). Let \(CH^ i(V)\) be the i-th Chow group of V, i.e. the group of algebraic cycles on V of codimension i modulo rational equivalence. Let \(A^ i(V)\subset CH^ i(V)\) be the subgroup consisting of those classes which are themselves algebraically equivalent to zero. Consider algebraic families of cycles on V, i.e. couples (T,Z) with T a smooth projective irreducible variety and \(Z\in CH^ i(T\times V)\); fixing a point \(t_ 0\in T\) one gets a map \(Z: T\to A^ i(V).\) Finally consider couples (A,\(\phi)\) with A an abelian variety and \(\phi: A^ i(V)\to A\) a ”regular” homomorphism (”regular” means: for every algebraic family (T,Z) as above the composite map \(\phi \circ Z: T\to A\) is a morphism of algebraic varieties). Natural question: does there exist a universal regular homomorphism? If so, say \((A_ 0,\phi_ 0)\), then it is clearly unique up to a unique isomorphism and it is called the algebraic representative for \(A^ i(V)\). Except for \(i=1\) and \(i=d\) the answer is unknown [see for instance R. Hartshorne’s survey lecture on algebraic cycles in Algebraic Geom., Proc. Sympos. Pure Math. 29, Arcata 1974, 129-164 (1975; Zbl 0314.14001); p. 144]. The two main results of this paper answer the question for \(i=2\), i.e. for codimension two cycles.

Namely: Theorem I. (Any algebraically closed k) For \(i=2\) there exists such a universal \((A_ 0,\phi_ 0)\); moreover \(2\cdot \dim A_ 0\leq B_ 3(V)=\dim H^ 3_{et}(V,{\mathbb{Q}}_{\ell}),\) \(\ell \neq char(k)\). - Theorem II. Let \(k={\mathbb{C}}\), \(J^ 2(V)\) the second intermediate Jacobian, \(\psi: A^ 2(V)\to J^ 2(V)\) the Abel-Jacobi map; put \(J^ 2_ a(V)=Im(\psi).\) Then \((J^ 2_ a(V),\psi)\) is universal in the above sense. The proofs depend on results of H. Saito [Nagoya Math. J. 75, 95-119 (1979; Zbl 0433.14036)], of S. Bloch [Compos. Math. 39, 47-105 and 107-127 (1979; Zbl 0426.14018 and Zbl 0463.14002)], of S. Bloch and S. Ogus [Ann. Sci. Éc. Norm. Super., IV. Sér. 7(1974), 181-201 (1975; Zbl 0307.14008)] and on the above mentioned theorem of A. S. Merkur’ev and A. A. Suslin [Math. USSR, Izv. 21, 307-340 (1983); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 46, No.5, 1011-1046 (1982; Zbl 0525.18008)] which states that the norm residue map \(K_ 2(F)/nK_ 2(F)\to H^ 2_{et}(F,\mu_ n^{\otimes 2})\) is an isomorphism (where F is a field and \((n,char(F))=1).\) Before proving the two main results an effort is made in the paper to discuss the connection with étale cohomology and with the theory of algebraic cycles.

As a side result one obtains that there exists a higher Picard variety \(Pic^ 2(V)\) in the sense of D. I. Lieberman [Am. J. Math. 90, 366-374 (1968; Zbl 0159.505) provided one modifies slightly (and as it seems naturally) the requirements for such a higher Picard variety. The proof of this side result is entirely formal (however we take the opportunity to inform that in this proof one has to define the functor \({\mathcal F}\) in section 8.2 somewhat more carefully, namely \({\mathcal F}\) is ”generated” by the elements described in the footnote on page 261).

##### MSC:

14C15 | (Equivariant) Chow groups and rings; motives |

14C05 | Parametrization (Chow and Hilbert schemes) |

14C35 | Applications of methods of algebraic \(K\)-theory in algebraic geometry |

14C22 | Picard groups |