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.

Notes

^ (Voisin 2003, ch.8)

References

Carlson, James; Müller-Stach, Stefan; Peters, Chris (2017). Period Mappings and Period Domains. doi:10.1017/9781316995846. ISBN 9781107189867.
Griffiths, Philip A. (1969). "On the Periods of Certain Rational Integrals: I". Annals of Mathematics. 90 (3): 460–495. doi:10.2307/1970746. JSTOR 1970746.
Griffiths, Phillip A. (1969). "On the Periods of Certain Rational Integrals: II". Annals of Mathematics. 90 (3): 496–541. doi:10.2307/1970747. JSTOR 1970747.
Voisin, Claire (2000). "The Griffiths group of a general Calabi-Yau threefold is not finitely generated". Duke Mathematical Journal. 102. CiteSeerX 10.1.1.643.5313. doi:10.1215/S0012-7094-00-10216-5. S2CID 16342989.
Voisin, Claire (2003). "Nori's Work". Hodge Theory and Complex Algebraic Geometry II. pp. 215–242. doi:10.1017/CBO9780511615177.009. ISBN 9780521802833.
Voisin, Claire (2019). "Birational Invariants and Decomposition of the Diagonal". Birational Geometry of Hypersurfaces. Lecture Notes of the Unione Matematica Italiana. Vol. 26. pp. 3–71. doi:10.1007/978-3-030-18638-8_1. ISBN 978-3-030-18637-1. S2CID 164209404.
Murre, Jacob (2014). "Lectures on Algebraic Cycles and Chow Groups". Hodge Theory (MN-49). Princeton University Press. pp. 410–448. ISBN 9780691161341. JSTOR j.ctt6wpzdg.13.

[1] (Voisin 2003, ch.8)

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