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Next: 7 Relation with cohomology Up: Algebraic Cycles Previous: 5 Equivalence relations

6 Chow groups and Abelian varieties

For any smooth projective variety we can form a countable collection of projective schemes Hilbn (the Hilbert or Chow schemes) that parametrise effective cycles of codimension p on X. Let C(X) be a subgroup of $ \CH^{p}_{}$(X). A homomorphism of groups from C(X) to the group of rational points of a group variety A is called regular if for any non-singular variety H parametrising cycles on X lying in C(X), the resulting set maps from the set of rational points on H to the set of rational points of A is induced by a morphism of varieties.

In analogy with the picture for divisors one may ask the following questions:

  1. Is there a finitely generated group N(X), a surjective map $ \xi$ : $ \CH^{p}_{}$(X)$ \to$$ \to$N(X), and a regular map from C(X) = ker$ \xi$ to the group of points on an Abelian variety A, so that any regular map from C(X) to an Abelian variety factors through A?
  2. Does this induce an isomorphism of $ \CH^{p}_{}$(X) with N(XA(k)?
    (Here A(k) denotes the group of k-rational points of A.)
  3. Is there a variety (or scheme) H parametrising effective cycles on X so that the morphism H$ \to$A is surjective? Or at least can we arrange $ \im$H = $ \im$$ \CH^{p}_{}$(X)?

There is no general geometric construction of the required Abelian variety. There is a complex torus associated to codimension p cycles, defined by Griffiths, which generalizes the Picard and Albanese varieties. This is called the pth intermediate Jacobian of X, and is defined by

Jp(X) = $\displaystyle {\frac{\HH^{2p-1}(X,{\Bbb C})}{F^p\HH^{2p-1}(X,{\Bbb C})+\im\HH^{2p-1}(X,{\Bbb Z})}}$.


Fp$\displaystyle \HH^{2p-1}_{}$(X,$\displaystyle \Bbb$C) = $\displaystyle \oplus_{p'\geq p}^{}$ $\displaystyle \HH^{p',2p-1-p'}_{}$(X)

is the pth piece of the Hodge filtration of $ \HH^{2p-1}_{}$(X,$ \Bbb$C). We can take the map $ \xi$ : $ \CH^{p}_{}$(X)$ \to$$ \HH^{2p}_{}$(X,$ \Bbb$C) and let C(X) = $ \CH^{p}_{hom}$(X) = ker$ \xi$ as before. There is a map, which Griffiths calls the Abel-Jacobi map,

$\displaystyle \CH^{p}_{hom}$(X)$\displaystyle \to$Jp(X).

However, this does not have as good properties as the Picard and Albanese maps, in general. The image of $ \CH^{p}_{alg}$(X) is an Abelian subvariety of J2(X) whose Lie algebra is contained in $ \HH^{p-1,p}_{}$(X) $ \subset$ $ \HH^{2p-1}_{}$(X, C)/Fp($ \HH^{2p-1}_{}$(X,$ \Bbb$C).

The Griffiths group $ \Griff^{p}_{}$(X) is always countable, since all effective algebraic cycles of a fixed degree are parametrized by the points of a (possibly reducible) Chow variety of X; taking the union over all degrees, all effective algebraic cycles lie in a countble collection of connected algebraic families, so that $ \CH^{p}_{}$(X)/$ \CH^{p}_{alg}$(X) is countable. Hence if $ \HH^{i,2p-1-i}_{}$(X) $ \neq$ 0 for some i > p, then the Abel-Jacobi map cannot be surjective. The restriction of the Abel-Jacobi map to $ \CH^{p}_{alg}$(X) is a regular homomorphism onto the Abelian variety which is its image; conjecturally, this is the universal regular homomorphism, as in the case of the Albanese map. One also expects the Abel-Jacobi map to be injective on torsion, in general. The injectivity of the Abel-Jacobi map on torsion is known for codimension 2 cycles, from work of Merkurjev and Suslin on the K-theory of division algebras, combined with results of Bloch and Ogus; we discuss this below. The universality of the Abel-Jacobi map on $ \CH^{2}_{alg}$(X) has been proved by Murre [27] using the injectivity on torsion.

next up previous
Next: 7 Relation with cohomology Up: Algebraic Cycles Previous: 5 Equivalence relations
Kapil Hari Paranjape 2002-11-21