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 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 (projective 4-space), the group 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.
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 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 (projective 4-space), the group 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.