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# 4 Zero cycles on varieties of higher dimension

Another way of looking at divisors on curves is as zero dimensional cycles. For a higher dimensional X we now examine (X). Let dim X = n. We put (X)0 = ker(deg : (X)Z), the group of zero cycles of degree 0 (modulo rational equivalence). This conicides with cycles numerically equivalent to zero, and also with cycles algebraically equivalent to zero (as defined in the next section).

There is a surjective regular homomorphism (X)0(X)(k), where (X) is an Abelian variety, the Albanese variety of X. An algebraic construction of (X) is as follows. There is a universal line bundle P on X×(X) called the Poincaré bundle. By duality this induces a morphism X(X), where (X) denotes the dual Abelian variety to (X). Thus we obtain a morphism (X)(X) by additivity. As a complex torus,

(X) (X,C)/Fn(X,C) + image  (X,Z),

where F* is the Hodge filtration on cohomology.

Now it is not hard to show that for sufficiently large d, the morphism (X)(X) is surjective. However, the fibres of this map are not in general rational equivalence classes of effective zero cycles of degree d. It is true that (X)0(X)(k) is an isomorphism on torsion subgroups (Roitman's theorem; see [33]). However, if (X,) = (X) 0 for some i 2, then (X)0(X)(k) is not an isomorphism; in fact (X)0 is not the group of points of an Abelian variety in any natural way (this is a result of Mumford [26] for surfaces, generalised to arbitrary dimension by Roitman [32]).

In this situation, a well known conjecture of Bloch asserts that if X is a surface with (X) = 0, then in fact (X)0 (X). If this is the case, the natural map (C)(X) is surjective, for C any hyperplane section of X (or more generally, an ample divisor). This may be generalized as follows: Some examples are known in support of these conjectures; for example, Bloch, Kas and Lieberman [6] showed that Bloch's conjecture (for surfaces) is true for surfaces which are not of general type. Other (rather special) examples have been given by several authors; most recently Voisin [41] has shown that the conjecture holds for Godeaux surfaces. In higher dimensions, Roitman [33] shows that (X) = Z for smooth projective complete intersections with (X) = 0 (complete intersections always have (X) = 0 for i < n). In [8], it is shown that if X is the (desingularized) Kummer variety associated to an Abelian variety of odd dimension n, then there is a divisor D X such that (D)(X); here (X) = 0 but (X) 0.

The results of Mumford-Roitman on non-triviality of Chow groups of 0-cycles are over C, or rather, over universal domains; if X is defined over a field k, the above proofs (or variations of them) can be adapted to work over the algebraic closure of the function field k(X) of X over k. This raises the question as to whether the Chow group of 0-cycles is trivial in those cases over smaller algebraically closed fields. Schoen and Nori (see [36]) have constructed examples of surfaces over such that over , an algebraically closed field of transcendence degree 1, the Chow group of 0-cycles of degree 0 differs from the Albanese variety. Conjecturally, for any smooth projective surface over , the Chow group of 0-cycles of degree 0 is isomorphic to the Albanese; this is a particular instance of the Bloch-Beilinson conjectures. No non-trivial example of this conjecture has been verified, at present.

The above theory for zero-dimensional cycles admits generalizations to the case of singular projective varieties as well; see [38], [39], [37], [23].

Another area of application of the theory of zero cycles is when X is non-projective or even affine. The group (X) need not be 0 (unlike the top cohomology (X,Z)), in this case. For example, it is standard to use non-vanishing intersection numbers to provide obstructions to the existence of embeddings in P2nk of smooth projective varieties of dimension n; similar arguments can be given for affine varieties of dimension n if the analogous obstruction element in (X) is non-zero. Thus the theory of algebraic cycles has applications to the study of projective modules, and to affine algebraic geometry (see [7]). However, these results are usually much subtler than the analogous ones using intersection numbers.

Next: 5 Equivalence relations Up: Algebraic Cycles Previous: 3 Divisors on varieties
Kapil Hari Paranjape 2002-11-21