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Griffiths group

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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

Griff k ⁡ ( X ) := Z k ( X ) h o m / Z k ( X ) a l g {\displaystyle \operatorname {Griff} ^{k}(X):=Z^{k}(X)_{\mathrm {hom} }/Z^{k}(X)_{\mathrm {alg} }}

where Z k ( X ) {\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.

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

Notes

  1. (Voisin 2003, ch.8)

References

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