A geometric application of Nori’s connectivity theorem

Claire Voisin[1]

  • [1] Institut de mathématiques de Jussieu CNRS,UMR 7586

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze (2004)

  • Volume: 3, Issue: 3, page 637-656
  • ISSN: 0391-173X

Abstract

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We study (rational) sweeping out of general hypersurfaces by varieties having small moduli spaces. As a consequence, we show that general K -trivial hypersurfaces are not rationally swept out by abelian varieties of dimension at least two. As a corollary, we show that Clemens’ conjecture on the finiteness of rational curves of given degree in a general quintic threefold, and Lang’s conjecture saying that such varieties should be rationally swept-out by abelian varieties, contradict.

How to cite

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Voisin, Claire. "A geometric application of Nori’s connectivity theorem." Annali della Scuola Normale Superiore di Pisa - Classe di Scienze 3.3 (2004): 637-656. <http://eudml.org/doc/84544>.

@article{Voisin2004,
abstract = {We study (rational) sweeping out of general hypersurfaces by varieties having small moduli spaces. As a consequence, we show that general $K$-trivial hypersurfaces are not rationally swept out by abelian varieties of dimension at least two. As a corollary, we show that Clemens’ conjecture on the finiteness of rational curves of given degree in a general quintic threefold, and Lang’s conjecture saying that such varieties should be rationally swept-out by abelian varieties, contradict.},
affiliation = {Institut de mathématiques de Jussieu CNRS,UMR 7586},
author = {Voisin, Claire},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
language = {eng},
number = {3},
pages = {637-656},
publisher = {Scuola Normale Superiore, Pisa},
title = {A geometric application of Nori’s connectivity theorem},
url = {http://eudml.org/doc/84544},
volume = {3},
year = {2004},
}

TY - JOUR
AU - Voisin, Claire
TI - A geometric application of Nori’s connectivity theorem
JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
PY - 2004
PB - Scuola Normale Superiore, Pisa
VL - 3
IS - 3
SP - 637
EP - 656
AB - We study (rational) sweeping out of general hypersurfaces by varieties having small moduli spaces. As a consequence, we show that general $K$-trivial hypersurfaces are not rationally swept out by abelian varieties of dimension at least two. As a corollary, we show that Clemens’ conjecture on the finiteness of rational curves of given degree in a general quintic threefold, and Lang’s conjecture saying that such varieties should be rationally swept-out by abelian varieties, contradict.
LA - eng
UR - http://eudml.org/doc/84544
ER -

References

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  1. [1] M. Asakura – S. Saito, Noether-Lefschetz locus for Beilinson-Hodge cycles on open complete intersections, preprint 2003. 
  2. [2] J. Carlson – Ph. Griffiths, Infinitesimal variations of Hodge structures and the global Torelli problem, In: “Journées de géométrie algébrique", A. Beauville (eds.), Sijthoff-Nordhoff, 1980, pp. 51-76. Zbl0479.14007MR605336
  3. [3] E. Cattani – P. Deligne – A. Kaplan, On the locus of Hodge classes, J. Amer. Math. Soc. (2) 8 (1995), 483-506. Zbl0851.14004MR1273413
  4. [4] L. Chiantini – A.-F. Lopez – Z. Ran, Subvarieties of generic hypersurfaces in any variety, Math. Proc. Cambr. Philos. Soc. (2002). Zbl1068.14508MR1806776
  5. [5] H. Clemens, Curves in generic hypersurfaces, Ann. Sci. École Norm. Sup. 19 (1986), 629-636. Zbl0611.14024MR875091
  6. [6] H. Clemens, Curves on higher-dimensional complex projective manifolds, In: “Proceedings of the International Congress of Mathematicians", (2) 1 (Berkeley, Calif., 1986), 634-640. Zbl0682.14024MR934266
  7. [7] H. Clemens – J. Kollár – S. Mori, “Higher dimensional complex geometry”, Astérisque 166, SMF (1988). Zbl0689.14016MR1004926
  8. [8] H. Clemens – Z. Ran, Twisted genus bounds for subvarieties of generic hypersurfaces, Amer. J. Math. 126 (2004), 89-120. Zbl1050.14035MR2033565
  9. [9] P. Deligne, Théorèmes de Lefschetz et critères de dégénérescence de suites spectrales, Publ. Math. Inst. Hautes Études Sci. 35 (1968), 107-126. Zbl0159.22501MR244265
  10. [10] P. Deligne, Théorie de Hodge II, Publ. Math. Inst. Hautes Études Sci. 40 (1971), 5-57. Zbl0219.14007MR498551
  11. [11] P. Deligne, La conjecture de Weil pour les surfaces K 3 , Invent. Math. 15 (1972), 206-226. Zbl0219.14022MR296076
  12. [12] L. Ein, Subvarieties of generic complete intersections, Invent. Math. 94 (1988), 163-169. Zbl0701.14002MR958594
  13. [13] M. Green, Restrictions of linear series to hyperplanes, and some results of Macaulay and Gotzmann, In: “Algebraic curves and projective geometry", E. Ballico – C. Ciliberto (eds.), Lecture Notes in Mathematics 1389, Springer-Verlag 1989, pp. 76-86. Zbl0717.14002MR1023391
  14. [14] M. Green, A new proof of the explicit Noether-Lefschetz theorem, J. Differential Geom. 27 (1988), 155-159. Zbl0674.14005MR918461
  15. [15] Ph. Griffiths, Periods of certain rational integrals, I, II, Ann. of Math. 90 (1969), 460-541. Zbl0215.08103MR260733
  16. [16] S. Lang, Hyperbolic and diophantine Analysis, Bull. Amer. Math. Soc. (2) 14 (1986), 159-205. Zbl0602.14019MR828820
  17. [17] D. Mumford, Rational equivalence of zero-cycles on surfaces, J. Math. Kyoto Univ. 9 (1968), 195-204. Zbl0184.46603MR249428
  18. [18] M. Nori, Algebraic cycles and Hodge theoretic connectivity, Invent. Math. 111 (1993), 349-373. Zbl0822.14008MR1198814
  19. [19] A. Otwinowska, Asymptotic bounds for Nori’s connectivity theorem, preprint 2002. Zbl1086.14009MR2147354
  20. [20] C. Voisin, On a conjecture of Clemens on rational curves on hypersurfaces, J. Differential Geom. 44 (1996), 200-214, 49 (1998), 601-611. Zbl0883.14022MR1420353
  21. [21] C. Voisin, “Hodge Theory and Complex Algebraic Geometry II”, Cambridge University Press, 2003. Zbl1032.14002MR1997577
  22. [22] C. Voisin, Nori’s connectivity theorem and higher Chow groups, J. Inst. Math. Jussieu (2) 1 (2002), 307-329. Zbl1036.14004MR1954824

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