Duke Mathematical Journal

Nongenericity of variations of hodge structure for hypersurfaces of high degree

Emmanuel Allaud

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

In this paper, we are interested in proving that the infinitesimal variations of Hodge structure for hypersurfaces for hypersurfaces of high-enough degree lie in a proper subvariety of the variety of all integral elements of the Griffiths transversality system. That is, this proves that in this case, the geometric infinitesimal variations of Hodge structure satisfy further conditions rather than just being integral elements of the Griffiths system. This is proved using the Jacobian ring representation of the (primitive) cohomology of the hypersurfaces and a space of symmetrizers as defined by Donagi, but here, we use the Jacobian ring representation to identify a geometric structure carried by the variety of all integral elements.

Article information

Source
Duke Math. J., Volume 129, Number 2 (2005), 201-217.

Dates
First available in Project Euclid: 27 September 2005

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1127831437

Digital Object Identifier
doi:10.1215/S0012-7094-05-12921-0

Mathematical Reviews number (MathSciNet)
MR2165541

Zentralblatt MATH identifier
1082.14014

Subjects
Primary: 14D07: Variation of Hodge structures [See also 32G20]

Citation

Allaud, Emmanuel. Nongenericity of variations of hodge structure for hypersurfaces of high degree. Duke Math. J. 129 (2005), no. 2, 201--217. doi:10.1215/S0012-7094-05-12921-0. https://projecteuclid.org/euclid.dmj/1127831437


Export citation

References

  • E. Allaud, Variations de structures de Hodge et systèmes différentiels extérieurs, Ph.D. dissertation, Université Paul Sabatier, Toulouse, France, 2002.
  • J. Carlson, M. Green, P. A. Griffiths, and J. Harris, Infinitesimal variations of Hodge structure, I, Compositio. Math. 50 (1983), 109--205. MR 0720288
  • R. Donagi, Generic Torelli for projective hypersurfaces, Compositio Math. 50 (1983), 325--353. MR 0720291
  • P. A. Griffiths, Periods of integrals on algebraic manifolds, I: Construction and properties of the modular varieties, Amer. J. Math. 90 (1968), 568--626.; II: Local study of the period mapping, 805--865. MR 0229641; MR 0233825
  • —, On the periods of certain rational integrals, I, Ann. of Math. (2) 90 (1969), 460--495.; II, 496--541. MR 0260733
  • —, Infinitesimal variations of Hodge structure, III: Determinantal varieties and the infinitesimal invariant of normal functions, Compositio Math. 50 (1983), 267--324. MR 0720290
  • P. A. Griffiths and J. Harris, Infinitesimal variations of Hodge structure, II: An infinitesimal invariant of Hodge classes, Compositio Math. 50 (1983), 207--265. MR 0720289
  • —, Principles of Algebraic Geometry, Wiley Classics Lib., Wiley, New York, 1994. MR 1288523
  • R. Mayer, Coupled contact systems and rigidity of maximal dimensional variations of Hodge structure, Trans. Amer. Math. Soc. 352 (2000), 2121--2144. MR 1624194