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
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
Hellenica World - Scientific Library
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