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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
where Z k {\displaystyle Z^{k}} 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 {\displaystyle \mathbf {P} ^{4}} , the group Griff 2 {\displaystyle \operatorname {Griff} ^{2}} is not a torsion group.