5 Equivalence relations

We have introduced three equivalence relations in the previous two sections, which can be defined as follows for cycles in every codimension.

For simplicity, we restrict ourselves to the case k = $\Bbb$C. Now

$$\CH^{p}_{alg}$$(X) $$\subset$$ $$\CH^{p}_{hom}$$(X) $$\subset$$ $$\CH^{p}_{num}$$(X) $$\subset$$ $$\CH^{p}_{}$$(X).

One of Grothendieck's standard conjectures asserts that

$$\CH^{p}_{num}$$(X) = {x $$\in$$ $$\CH^{p}_{}$$(X) | nx $$\in$$ $$\CH^{p}_{hom}$$(X) for some positive integer n}.

Equivalently,

$$\CH^{p}_{hom}$$(X) $$\otimes$$ $$\Bbb$$Q = $$\CH^{p}_{num}$$(X) $$\otimes$$ $$\Bbb$$Q.

One way of proving this could be to attempt to show that $\CH^{p}_{alg}$(X) $\otimes$ $\Bbb$Q$\;\stackrel{\bf ?}{=}\;$$\CH^{p}_{num}$(X) $\otimes$ $\Bbb$Q. This holds for divisors and zero cycles in particular. So we introduce the quotient $\CH^{p}_{hom}$(X)/$\CH^{p}_{alg}$(X) = $\Griff^{p}_{}$(X) which is called the pth Griffiths group of X; note that $\Griff^{n}_{}$(X) = $\Griff^{1}_{}$(X) = 0 as remarked in the previous sections.

The terminology is because of the famous example of Griffiths [16] showing that $\Griff^{2}_{}$(X) $\neq$ 0 for a general hypersurface X of degree 5 in $\Bbb$P⁴_$\Bbb$C, and in fact $\Griff^{2}_{}$(X) has an element of infinite order. Later, Clemens [10] showed that $\Griff^{2}_{}$(X) has infinite rank in this case. Other examples of the non-triviality of the Griffiths group were given by Ceresa [9], who showed that if C is a generic curve of genus $\geq$ 3 embedded in its Jacobian variety X, and i(C) is the image under multiplication by -1 on X, then [C] - [i(C)] gives an element of infinite order in $\Griff^{g-1}_{}$(X); for g = 3, Nori [28] noted that using the action of Hecke correspondences, this in fact implies that $\Griff^{2}_{}$(X) has infinite rank in that case. Further examples of non-triviality or infinite dimensionality of $\Griff^{p}_{}$(X) $\otimes$ $\Bbb$Q were obtained by Bardelli [3], Voisin [42] and Paranjape [30].

B. Harris [18] showed that if C is the Fermat quartic curve U⁴ + V⁴ + W⁴ = 0 in $\Bbb$P²_$\Bbb$C, then Ceresa's cycle [C] - [i(C)] is non-trivial in $\Griff^{2}_{}$(X), where X is the Jacobian of C, by reducing this via iterated integrals to the observation that

$${\frac{2\displaystyle\int_0^1\left[\int_0^x\frac{dt}{(1-t^4)^{1/2... ...^4)^{1/2}}\right] \left[\displaystyle\int_0^1\frac{dt}{(1-t^4)^{3/4}}\right] }}$$

is not an integer! If this number is irrational, his method implies this element has infinite order in $\Griff^{2}_{}$(X), a fact which was proved by other methods by Bloch [4]. This gives an example of such a cycle defined over $\Bbb$Q. Schoen [35] showed that for a certain elliptic modular 3-fold X over $\overline{{\Bbb Q}}$ (the field of algebraic numbers), $\Griff^{2}_{}$(X) has infinite rank.

In all of these examples, the ambient variety has trivial canonical bundle (tangent bundle, in Ceresa's situation), and one uses image of the cycle under the Abel-Jacobi homomorphism to the intermediate Jacobian of Griffiths' (explained in Section 6). For example, in B. Harris' example, the number whose non-integrality is asserted is essentially an integral of holomorphic 3-form (an element of $\HH^{3,0}_{}$(X)), whose value is not a period of that 3-form.

There is a new class of examples of non-triviality of $\Griff^{p}_{}$(X) $\otimes$ $\Bbb$Q constructed by M. Nori [29], in which the canonical bundle of the variety is ample, and the intermediate Jacobian in question is 0. Nori has introduced a filtration of the Griffiths' group and one can show that every associated graded term in this filtration can be non-zero (Albano and Collino [1] have shown that it can even be of infinite rank). We discuss this further below in the context of conjectural Lefschetz theorems for Chow groups.

Bloch had asked if the Griffiths group is always divisible (for varieties over algebraically closed fields); very recently, Bloch and Esnault have found a counter-example [5]. Schoen [34] has an example (in positive characteristic) of a smooth variety X such that $\Griff^{p}_{}$(X) contains a (non-zero) divisible subgroup, for some p.

Other equivalences have been recently introduced and studied on Chow groups with the idea of settling the standard conjectures and also the Bloch conjecture.