zbMATH — the first resource for mathematics

On the Griffiths group of the cubic sevenfold. (English) Zbl 0803.14022
In this paper we prove that the Griffiths group of a general cubic sevenfold is not finitely generated, even when tensored with \(\mathbb{Q}\). Using this result and a theorem of Nori, we provide examples of varieties which have some Griffiths group not finitely generated but whose corresponding intermediate Jacobian is trivial.
Reviewer: A.Albano (Torino)

14J40 \(n\)-folds (\(n>4\))
14K30 Picard schemes, higher Jacobians
14C15 (Equivariant) Chow groups and rings; motives
14C30 Transcendental methods, Hodge theory (algebro-geometric aspects)
Full Text: DOI EuDML
[1] Albano, A.: Infinite generation of the Griffiths group: a local proof. Thesis, University of Utah, 1986
[2] Bardelli, F. Curves of genus three on a general abelian threefold and the non-finite generation of the Griffiths group. In: Barth, W.-P., Lange H. (eds.) Arithmetic of complex manifolds (Erlangen, 1988). (Lecture Notes in Mathemathics, vol. 1399). Springer, New York Berlin Heidelberg, 1989, pp. 10-26
[3] Candelas, P., Derrick, E., Parkes, L.: Generalized Calabi-Yau Manifolds and the mirror of a rigid manifold. Preprint CERN-TH.6831/93, UTTG-24-92 · Zbl 0899.32011
[4] Ciliberto, C., Harris, J., Miranda, R.: General components of the Noether-Lefschetz locus and their density in the space of all surfaces. Math. Ann.282, (1988) 667-680 · Zbl 0671.14017 · doi:10.1007/BF01462891
[5] Clemens, H.: Homological equivalence, modulo algebraic equivalence, is not finitely generated. Inst. Hautes ?tudes Sci. Publ. Math.58 (1983) 19-38 · doi:10.1007/BF02953771
[6] Deligne, P.: Cohomologie des intersection compl?tes. In: Deligne, P., Katz, N. (eds.) Groupe de Monodromie en G?om?trie Alg?brique (S.G.A. 7 II). (Lecture Notes in Mathemathics, vol. 340). Springer, New York Berlin Heidelberg, 1973, pp. 39-61
[7] Green, M.: Griffiths’ infinitesimal invariant and the Abel-Jacobi map. Jour. Diff. Geom.29, (1989) 545-555 · Zbl 0692.14003
[8] Griffiths, P. A.: On the periods of certain rational integrals I, II. Ann. of Math.90 (1969) 460-541 · Zbl 0215.08103 · doi:10.2307/1970746
[9] Griffiths, P. A.: Infinitesimal invariant of normal functions. In: Griffiths, P. A. (ed.) Topics in transcendental algebraic geometry. (Annals of Mathematics Studies 106). Princeton University Press, Princeton, 1984, pp. 305-316
[10] Gross, B. H.: On the periods of abelian integrals and a formula of Chowla and Selberg (with an appendix by D. E. Rohrlich). Inventiones Math.45, (1978) 193-211 · Zbl 0418.14023 · doi:10.1007/BF01390273
[11] Kleiman, S.: Geometry on Grassmannians and applications to splitting bundles and smoothing cycles. Inst. Hautes ?tudes Sci. Publ. Math.36, (1969) 281-298 · Zbl 0208.48501 · doi:10.1007/BF02684605
[12] Morrison, D.: Mirror symmetry and rational curves on quintic threefolds: A guide for the mathematicians. Jour. A.M.S.6, (1993) 223-247 · Zbl 0843.14005
[13] Nori, M.: Cycles on the generic abelian threefold. Proc. Indian Acad. Sci.99, (1989) 191-196 · Zbl 0725.14006
[14] Nori, M.: Algebraic cycles and Hodge theoretic connectivity. Inventiones Math.111, (1993) 349-373 · Zbl 0822.14008 · doi:10.1007/BF01231292
[15] Ogus, A.: Griffiths transversality and crystalline cohomology. Ann. of Math.108, (1978) 395-419 · Zbl 0382.14005 · doi:10.2307/1971182
[16] Ran, Z.: Cycles on Fermat hypersurfaces. Compositio Math.42, (1980/81) 121-142 · Zbl 0463.14003
[17] Schoen, C.: Complex multiplication cycles on elliptic modular threefolds. Duke Math. Jour.53, (1986) 771-794 · Zbl 0623.14018 · doi:10.1215/S0012-7094-86-05343-3
[18] Shioda, T.: The Hodge conjecture for Fermat varieties. Math. Ann.245, (1979) 175-184 · Zbl 0408.14012 · doi:10.1007/BF01428804
[19] Steenbrink, J.: Some remarks about the Hodge conjecture. In: Cattani, E., Guill?n, F., Kaplan, A., Puerta, F. (eds.) Hodge theory. (Lecture Notes in Mathemathics, vol. 1246). Springer, New York Berlin Heidelberg, 1987, pp. 165-175 · Zbl 0629.14004
[20] Tretkoff, M. D.: The Fermat surface and its periods. In: Fornaess, J.E. (ed.) Recent developments in several complex variables. (Annals of Mathematics Studies 100). Princeton University Press, Princeton, 1981, pp. 413-428
[21] Voisin, C.: Une approche infinit?simale du th?or?me de H. Clemens sur les cycles d’une quintique g?n?rale de ?4. Jour. of Alg. Geom.1, (1992) 157-174 · Zbl 0787.14003
[22] Voisin, C.: Sur l’application d’Abel-Jacobi des vari?t?s de Calabi-Yau de dimension trois. Ann. Sci. E.N.S., (to appear) · Zbl 0808.14030
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.