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
- [math]\displaystyle{ \operatorname{Griff}^k(X) := Z^k(X)_\mathrm{hom} / Z^k(X)_\mathrm{alg} }[/math]
where [math]\displaystyle{ Z^k(X) }[/math] 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 [math]\displaystyle{ \mathbf P^4 }[/math] (projective 4-space), the group [math]\displaystyle{ \operatorname{Griff}^2(X) }[/math] is not a torsion group.
Notes
- ↑ (Voisin 2003)
References
- Carlson, James; Müller-Stach, Stefan; Peters, Chris (2017). Period Mappings and Period Domains. doi:10.1017/9781316995846. ISBN 9781107189867. https://books.google.com/books?id=XrYrDwAAQBAJ&q=%22griffiths+group%22.
- Griffiths, Philip A. (1969). "On the Periods of Certain Rational Integrals: I". Annals of Mathematics 90 (3): 460–495. doi:10.2307/1970746.
- Griffiths, Phillip A. (1969). "On the Periods of Certain Rational Integrals: II". Annals of Mathematics 90 (3): 496–541. doi:10.2307/1970747.
- Voisin, Claire (2000). "The Griffiths group of a general Calabi-Yau threefold is not finitely generated". Duke Mathematical Journal 102. doi:10.1215/S0012-7094-00-10216-5.
- 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. 26. pp. 3–71. doi:10.1007/978-3-030-18638-8_1. ISBN 978-3-030-18637-1. https://books.google.com/books?id=8vy0DwAAQBAJ&pg=PA26.
- Murre, Jacob (2014). "Lectures on Algebraic Cycles and Chow Groups". Hodge Theory (MN-49). Princeton University Press. pp. 410–448. ISBN 9780691161341. https://books.google.com/books?id=XrvzAgAAQBAJ&pg=PA434.
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