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

\operatorname {Griff}^{k}(X):=Z^{k}(X)_{{\mathrm {hom}}}/Z^{k}(X)_{{\mathrm {alg}}}

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.^[1]

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

References

↑ Voisin, C., Hodge Theory and Complex Algebraic Geometry II, Cambridge University Press, 2003. See Chapter 8

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