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

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

where ${\displaystyle Z^{k}(X)}$ 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 ${\displaystyle \mathbf {P} ^{4}}$ (projective 4-space), the group ${\displaystyle \operatorname {Griff} ^{2}(X)}$ is not a torsion group.