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*)/*F*^{n}(*X*,*C*) + *image* (*X*,*Z*),

where
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
*P*^{2n}_{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 (*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.