Hostname: page-component-78c5997874-g7gxr Total loading time: 0 Render date: 2024-11-10T09:49:32.919Z Has data issue: false hasContentIssue false
  • English
  • Français

Algebraic Cycles in Familiesof Abelian Varieties

Published online by Cambridge University Press:  20 November 2018

Salman Abdulali
Salman Abdulali*
Affiliation:
Department of Mathematics, University of Toronto, Toronto, OntarioM5S 1A1
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

If the Hodge *-operator on the L2-cohomology of Kuga fiber varieties is algebraic, then the Hodge conjecture is true for all abelian varieties.

Keywords

14C30
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 46 , Issue 6 , 01 December 1994 , pp. 1121 - 1134
Copyright
Copyright © Canadian Mathematical Society 1994

References

[Abl] Abdulali, S., Absolute Hodge Cycles in Kuga Fiber Varieties, Thesis, State University of New York, Stony Brook, 1985.Google Scholar
[Ab2] Abdulali, S., Zeta functions of Kuga fiber varieties, Duke Math. J. 57(1988), 333–345.Google Scholar
[Ab3] Abdulali, S., Conjugates of strongly equivariant maps, Pacific J. Math., to appear.Google Scholar
[Anl] André, Y., Une remarque à propos des cycles de Hodge de type CM. In: Séminaire de Théorie des Nombres, Paris, 1989-90, (ed. S. David), Progr. Math. 102, Birkhäuser, Boston, 1992, 1–7.Google Scholar
[An2] André, Y.,Pour une théorie inconditionelle des motifs, preprint.Google Scholar
[B] Borel, A., Density and maximality of arithmetic subgroups, J. Reine Angew. Math. 224(1966), 78–89.Google Scholar
[BC] Borel, A. and Casselman, W., L2-cohomology of locally symmetric manifolds of finite volume, Duke Math. J. 50(1983), 625–647.Google Scholar
[Ca] Casselman, W., Introduction to the L2-cohomology of arithmetic quotients of bounded symmetric domains. In: Complex Analytic Singularities, (eds. T. Suwa and P. Wagreich), Adv. Stud. Pure Math. 8, Kinokuniya, Tokyo, 1986, 69–93.Google Scholar
[Ch] Cheeger, J., On the Hodge theory of Riemannian pseudomanifolds. In: Geometry of the Laplace Operator, (eds. R. Osserman and A. Weinstein), Proc. Sympos. Pure Math. 36, Amer. Math. Soc., Providence, 1980, 91–146.Google Scholar
[CGM] Cheeger, J., Goresky, M. and MacPherson, R., L2-cohomology and intersection homology of singular algebraic varieties. In: Seminar on Differential Geometry, (ed. S.-T. Yau), Ann. of Math. Stud. 102, Princeton Univ. Press, Princeton, 1982, 303–340.Google Scholar
[D1] Deligne, P., Th-orie de Hodge II, Inst. Hautes Études Sci. Publ. Math. 40(1971), 5–57.Google Scholar
[D2] Deligne, P. (notes by J. S. Milne), Hodge cycles on abelian varieties. In: P. Deligne, J. S. Milne, A. Ogus and K.-y. Shih, Hodge cycles, Motives, andShimura Varieties, Lecture Notes in Math. 900, Springer-Verlag, Berlin, Heidelberg, New York, 1982, 9–100.Google Scholar
[Ga] Gaffney, M., A special Stokes's theorem for complete Riemannian manifolds, Ann. of Math.(2) 60(1954), 140–145.Google Scholar
[Gf] Griffiths, P., Periods of integrals on algebraic manifolds, III (some global differential-geometric properties of the period mapping), Inst. Hautes Études Sci. Publ. Math. 38(1970), 125–180.Google Scholar
[Gk1] Grothendieck, A., On the de Rham cohomology of algebraic varieties, Inst. Hautes Études Sci. Publ. Math. 29(1966),96–103.Google Scholar
[Gk2] Grothendieck, A., Standard conjectures on algebraic cycles. In: Algebraic Geometry (Papers presented at the Bombay Colloquium, 1968), Tata Inst. Fund. Res. Studies in Math. 4, Oxford University Press, Bombay- London, 1969, 193–199.Google Scholar
[HK] Hall, R. and Kuga, M., Algebraic cycles in a fiber variety, Sci. Papers College Gen. Ed. Univ. Tokyo 25(1975), 1–6.Google Scholar
[K1] Kuga, M., Fiber Varieties over a Symmetric Space whose Fibers are Abelian Varieties, Vol. I, Lecture Notes, Univ. Chicago, Chicago, 1964.Google Scholar
[K2] Kuga, M., Fibred variety over symmetric space whose fibres are abelian varieties. In: Proceedings of the U.S.-Japan Seminar in Differential Geometry, Kyoto, 1965, Nippon Hyoronsha, Tokyo, 1966, 72–81.Google Scholar
[K3] Kuga, M., Fiber varieties over a symmetric space whose fibers are abelian varieties. In: Algebraic Groups and Discontinuous Subgroups, (eds. A. Borel and G. D. Mostow), Proc. Sympos. Pure Math. 9, Amer. Math. Soc, Providence, 1966, 338–346.Google Scholar
[K4] Kuga, M., Algebraic cycles in gtfabv, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 29(1982), 13–29.Google Scholar
[L] Lieberman, D., Numerical and homological equivalence of algebraic cycles on Hodge manifolds, Amer. J. Math. 90(1968), 366–374.Google Scholar
[Mml] Mumford, D., Families of abelian varieties. In: Algebraic Groups and Discontinuous Subgroups, (eds. A. Borel and G. D. Mostow), Proc. Sympos. Pure. Math. 9, Amer. Math. Soc, Providence, 1966, 347–351.Google Scholar
[Mm2] Mumford, D.,A note of Shimuras paper “Discontinuous groups and abelian varieties”,Math. Ann. 181 (1969), 345–351.Google Scholar
[R] Ribet, K., Hodge classes on certain types of abelian varieties,Amer. J. Math. 105(1983), 523–538.Google Scholar
[Sa1] Satake, I., Holomorphic imbeddings of symmetric domains into a Siegel space, Amer. J. Math. 87(1965), 425–461.Google Scholar
[Sa2] Satake, I., Algebraic Structures of Symmetric Domains, Publ. Math. Soc Japan 14, Kanô Mem. Lect. 4, Iwanami Shoten, Japan and Princeton Univ. Press, Princeton, 1980.Google Scholar
[Sc1] Schoen, C., Hodge classes on self-products of a variety with an automorphism, Compositio Math. 65 (1988), 3–32.Google Scholar
[Sc2] Schoen, C., Cyclic covers ofPv branched along v + 2 hyperplanes and the generalized Hodge conjecture for certain abelian varieties. In: Arithmetic of Complex Manifolds, (eds. W.-P. Barth and H. Lange), Lecture Notes in Math. 1399, Springer-Verlag, Berlin, Heidelberg, New York, 1989, 137–154.Google Scholar
[Sm] Shimura, G., Moduli of abelian varieties and number theory. In: Algebraic Groups and Discontinuous Subgroups, (eds. A. Borel and G. D. Mostow), Proc Sympos. Pure Math. 9, Amer. Math. Soc, Providence, 1966, 312–332.Google Scholar
[So] Shioda, T., Algebraic cycles on abelian varieties of Fermat type, Math. Ann. 258(1981), 65–80.Google Scholar
[T] Tjiok, M.-c., Algebraic Cycles in a Certain Fiber Variety, Thesis, State University of New York, Stony Brook, 1980.Google Scholar
[W] Weil, A., Abelian varieties and the Hodge ring. In: Œuvres Scientifiques Collected Papers, Vol. Ill, 1964-1978, Springer-Verlag, New York, 1979, Corrected second printing, 1980,421–429.Google Scholar
[Zl] Zucker, S., Cohomology of warped products and arithmetic groups, Invent. Math. 70(1982), 169–218.Google Scholar
[Z2] Zucker, S., L2-cohomology of Shimura varieties. In: Automorphic Forms, Shimura Varieties, and Lfunctions,(eds. L. Clozel and J. S. Milne), Vol. 2, Perspect. Math. 11, Academic Press, San Diego, 1990, 377–391.Google Scholar