Griffiths group

In mathematics, more specifically in algebraic geometry, the Griffiths group of a projective complex manifold X measures the difference between homological equivalence and algebraic equivalence, which are two important equivalence relations of algebraic cycles.

More precisely, it is defined as

${\mathrm{Griff}}^k(X):=Z^k(X{)}_{\mathrm{h}\mathrm{o}\mathrm{m}}/Z^k(X{)}_{\mathrm{a}\mathrm{l}\mathrm{g}}$

where $Z^k(X)$ denotes the group of algebraic cycles of some fixed codimension k and the subscripts indicate the groups that are homologically trivial, respectively algebraically equivalent to zero.

This group was introduced by Phillip Griffiths who showed that for a general quintic in ${𝐏}^4$ (projective 4-space), the group ${\mathrm{Griff}}^2(X)$ is not a torsion group.

References

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