Abstract
This paper deals with some families of projective hypersurfaces of degree 6 and dimension 3, left invariant under certain actions of the group of the cubic roots of unity.
The general Grothendieck-Hodge conjecture, for the families above and for their intersections, is verified. In particular certain families of elliptic curves on the general element of these families are closely investigated and their degeneration at the Fermat point is studied. These results are used to verify the general Infinitesimal Hodge Conjecture extending to the general type case a similar conjecture due to A. Albano and S. Katz.
Similar content being viewed by others
References
[A] M. Artin,On the solution of analytic equations, Invent. Math.5 (1968) 277–291
[AK] A. Albano, S. Katz,Lines on the Fermat quintic threefold and infinitesimal generalized Hodge conjecture, Trans. Amer. Math. Soc.324 (1991), 353–368
[B] F. Bardelli,On Grothendieck's generalized Hodge conjecture for a family of threefolds with trivial canonical bundle, Jour. für d. Reine und Angew. Math.422 (1991), 165–200
[CGGH] J. Carlson, M. Green, P. Griffiths, J. Harris,Infinitesimal variation of Hodge structure (I), Compositio Math.50 (1983), 109–205
[CKM] H. Clemens, J. Kollár, S. Mori,Higher dimensional complex geometry, Astérisque166 (1988)
[CM] A. Conte, J. Murre,The Hodge conjecture for fourfolds admitting a covering by rational curves, Math. Ann.238 (1978), 79–88
[CV] G. Ceresa, A. Verra,The Abel-Jacobi isomorphism for the sextic double solid, Pacific J. Math.124 (1986), 85–105
[G] P. A. Griffiths,On the periods of certain rational integrals, I, II, Ann. of Math.90 (1969), 460–541
[Gr] A. Grothendieck,Hodge general Conjecture is false for trivial reasons, Topology8 (1969), 299–303
[KL] S. L. Kleiman, D. Laksov,Schubert calculus, Amer. Math. Monthly79 (1972), 1061–1082
[KNS] K. Kodaira, L. Nirenberg, D. C. Spencer,On existence of deformation of complex analytic structures, Ann. of Math.68 (1958), 450–459
[M] J. P. Murre,Abel-Jacobi equivalence versus incidence equivalence for algebraic cycles of codimension two, Topology24 (1985), 361–367
[R] Z. Ran,Cycles on Fermat hypersurfaces, Compositio Math.42 (1980/81), 121–142
[S1] T. Shioda,The Hodge conjecture for Fermat varieties, Math. Ann.245 (1979), 175–184
[S2] T. Shioda,What is known about the Hodge conjecture?, Adv. Studies in Pure Math.1 (1983), 55–68
[SK] T. Shioda, T. KatsuraOn Fermat varieties, Tôhoku Math. Journ.31 (1979), 97–115
[T] H. Tango,On (n−1)-dimensional projective spaces contained in the Grassmann variety Gr(n−1). J. Math. Kyoto Univ. (JMKYAZ)14 (1974) 415–460
[W] G. E. Welters,Abel-Jacobi isogenies for certain types of Fano threefolds, Mathematical Center Tracts141, Mathematisch Centrum Amsterdam, 1981
Author information
Authors and Affiliations
Additional information
Research partially supported by italian M.U.R.S.T., by the Science Project “Geom. of Alg. Var.” and by G.N.S.A.G.A. (C.N.R.)
Rights and permissions
About this article
Cite this article
Rossi, M. The infinitesimal and generalized Hodge Conjecture for some families of sextic threefolds. Manuscripta Math 89, 511–544 (1996). https://doi.org/10.1007/BF02567532
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02567532