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 $\CH_{0}^{}$(X). Let dim X = n. We put $\CH_{0}^{}$(X)₀ = ker(deg : $\CH^{n}_{}$(X)$\to$$\Bbb$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 $\CH_{0}^{}$(X)₀$\to$$\Alb$(X)(k), where $\Alb$(X) is an Abelian variety, the Albanese variety of X. An algebraic construction of $\Alb$(X) is as follows. There is a universal line bundle P on X×$\Pic^{0}_{}$(X) called the Poincaré bundle. By duality this induces a morphism X$\to$$\Alb$(X), where $\Alb$(X) denotes the dual Abelian variety to $\Pic^{0}_{}$(X). Thus we obtain a morphism $\Sym^{d}_{}$(X)$\to$$\Alb$(X) by additivity. As a complex torus,

$$\Alb$$(X) $$\cong$$ $$\HH^{2n-1}_{}$$(X,$$\Bbb$$C)/$$\left(\vphantom{F^n\HH^{2n-1}(X,{\Bbb C})+{\rm image}\;\HH^{2n-1}(X,{\Bbb Z})}\right.$$Fⁿ$$\HH^{2n-1}_{}$$(X,$$\Bbb$$C) + image  $$\HH^{2n-1}_{}$$(X,$$\Bbb$$Z)$$\left.\vphantom{F^n\HH^{2n-1}(X,{\Bbb C})+{\rm image}\;\HH^{2n-1}(X,{\Bbb Z})}\right)$$,

where F^* is the Hodge filtration on cohomology.

Now it is not hard to show that for sufficiently large d, the morphism $\Sym^{d}_{}$(X)$\to$$\Alb$(X) is surjective. However, the fibres of this map are not in general rational equivalence classes of effective zero cycles of degree d. It is true that $\CH_{0}^{}$(X)₀$\to$$\Alb$(X)(k) is an isomorphism on torsion subgroups (Roitman's theorem; see [33]). However, if $\HH^{0}_{}$(X,$\Omega^{i}_{X/{\Bbb C}}$) = $\HH^{i,0}_{}$(X) $\neq$ 0 for some i $\geq$ 2, then $\CH_{0}^{}$(X)₀$\to$$\Alb$(X)(k) is not an isomorphism; in fact $\CH_{0}^{}$(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 $\HH^{2,0}_{}$(X) = 0, then in fact $\CH_{0}^{}$(X)₀ $\cong$ $\Alb$(X). If this is the case, the natural map $\CH^{1}_{}$(C)$\to$$\CH^{2}_{}$(X) is surjective, for C any hyperplane section of X (or more generally, an ample divisor). This may be generalized as follows: Some examples are known in support of these conjectures; for example, Bloch, Kas and Lieberman [6] showed that Bloch's conjecture (for surfaces) is true for surfaces which are not of general type. Other (rather special) examples have been given by several authors; most recently Voisin [41] has shown that the conjecture holds for Godeaux surfaces. In higher dimensions, Roitman [33] shows that $\CH^{n}_{}$(X) = $\Bbb$Z for smooth projective complete intersections with $\HH^{n,0}_{}$(X) = 0 (complete intersections always have $\HH^{i,0}_{}$(X) = 0 for i < n). In [8], it is shown that if X is the (desingularized) Kummer variety associated to an Abelian variety of odd dimension n, then there is a divisor D $\subset$ X such that $\CH^{n-1}_{}$(D)$\to$$\to$$\CH^{n}_{}$(X); here $\HH^{n,0}_{}$(X) = 0 but $\HH^{n-1,0}_{}$(X) $\neq$ 0.

The results of Mumford-Roitman on non-triviality of Chow groups of 0-cycles are over $\Bbb$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 $\overline{{\Bbb Q}}$ such that over $\overline{{\Bbb Q}(t)}$, 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 $\overline{{\Bbb Q}}$, 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 $\CH^{n}_{}$(X) need not be 0 (unlike the top cohomology $\HH^{2n}_{}$(X,$\Bbb$Z)), in this case. For example, it is standard to use non-vanishing intersection numbers to provide obstructions to the existence of embeddings in $\Bbb$P²ⁿ_k of smooth projective varieties of dimension n; similar arguments can be given for affine varieties of dimension n if the analogous obstruction element in $\CH^{n}_{}$(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.