Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 5 (2009), 087, 40 pages      arXiv:0909.2201      http://dx.doi.org/10.3842/SIGMA.2009.087
Contribution to the Special Issue “Élie Cartan and Differential Geometry”

Variations of Hodge Structure Considered as an Exterior Differential System: Old and New Results

James Carlson a, Mark Green b and Phillip Griffiths c
a) Clay Mathematics Institute, United States
b) University of California, Los Angeles, CA, United States
c) The Institute for Advanced Study, Princeton, NJ, United States

Received April 20, 2009, in final form August 31, 2009; Published online September 11, 2009

Abstract
This paper is a survey of the subject of variations of Hodge structure (VHS) considered as exterior differential systems (EDS). We review developments over the last twenty-six years, with an emphasis on some key examples. In the penultimate section we present some new results on the characteristic cohomology of a homogeneous Pfaffian system. In the last section we discuss how the integrability conditions of an EDS affect the expected dimension of an integral submanifold. The paper ends with some speculation on EDS and Hodge conjecture for Calabi-Yau manifolds.

Key words: exterior differential systems; variation of Hodge structure, Noether-Lefschetz locus; period domain; integral manifold; Hodge conjecture; Pfaffian system; Chern classes; characteristic cohomology; Cartan-Kähler theorem.

pdf (448 kb)   ps (275 kb)   tex (39 kb)

References

  1. Allaud E., Nongenericity of variations of Hodge structure for hypersurfaces of high degree, Duke Math. J. 129 (2005), 201-217, math.AG/0503346.
  2. Bryant R.L., Chern S.S., Gardner R.B., Goldschmidt H.L., Griffiths P.A., Exterior differential systems, Mathematical Sciences Research Institute Publications, Vol. 18, Springer-Verlag, New York, 1991.
  3. Bryant R.L., Griffiths P.A., Some observations on the infinitesimal period relations for regular threefolds with trivial canonical bundle, in Arithmetic and Geometry, Vol. II, Progr. Math., Vol. 36, Birkhäuser Boston, Boston, MA, 1983, 77-102.
  4. Bryant R.L, Griffiths P.A., Characteristic cohomology of differential systems. I. General theory, J. Amer. Math. Soc. 8 (1995), 507-596.
  5. Bryant R.L, Griffiths P.A., Characteristic cohomology of differential systems. II. Conservation law for a class of parabolic equations, Duke Math. J. 78 (1995), 531-676.
  6. Carlson J.A., Bounds on the dimension of variations of Hodge structure, Trans. Amer. Math. Soc. 294 (1986), 45-64, Erratum, Trans. Amer. Math. Soc. 299 (1987), 429.
  7. Carlson J.A., Donagi R., Hypersurface variations are maximal. I, Invent. Math. 89 (1987), 371-374.
  8. Carlson J.A., Kasparian A., Toledo D., Variations of Hodge structure of maximal dimension, Duke Math. J. 58 (1989), 669-694.
  9. Carlson J.A., Simpson C., Shimura varieties of weight two Hodge structures, in Hodge Theory (Sant Cugat, 1985), Lecture Notes in Math., Vol. 1246, Springer, Berlin, 1987, 1-15.
  10. Carlson J.A., Toledo D., Generic integral manifolds for weight-two period domains, Trans. Amer. Math. Soc. 356 (2004), 2241-2249, math.AG/0501078.
  11. Carlson J.A., Toledo D., Variations of Hodge structure, Legendre submanifolds and accessibility, Trans. Amer. Math. Soc. 311 (1989), 391-411.
  12. Carlson J.A., Müller-Stach S., Peters C., Period mappings and period domains, Cambridge Studies in Advanced Mathematics, Vol. 85, Cambridge University Press, Cambridge, 2003.
  13. Donagi R., Generic Torelli for projective hypersurfaces, Compositio Math. 50 (1983), 325-353.
  14. Green M., Koszul cohomology and the geometry of projective varieties. II, J. Differential Geom. 20 (1984), 279-289.
  15. Green M., Griffiths P., Algebraic cycles and singularities of normal functions. II, in Inspired by S.S. Chern, Nankai Tracts Math., Vol. 11, World Sci. Publ., Hackensack, NJ, 2006, 2006, 179-268.
  16. Griffiths P., Hermitian differential geometry and the theory of positive and ample holomorphic vector bundles, J. Math. Mech. 14 (1965), 117-140.
  17. Ivey T.A., Landsberg J.M., Cartan for beginners: differential geometry via moving frames and exterior differential systems, Graduate Studies in Mathematics, Vol. 61, American Mathematical Society, Providence, RI, 2003.
  18. Mayer R., Coupled contact systems and rigidity of maximal dimensional variations of Hodge structure, Trans. Amer. Math. Soc. 352 (2000), 2121-2144, alg-geom/9712001.
  19. Otwinowska A., Composantes de petite codimension du lieu de Noether-Lefschetz: un argument asymptotique en faveur de la conjecture de Hodge pour les hypersurfaces, J. Algebraic Geom. 12 (2003), 307-320.
  20. Otwinowska A., Composantes de dimension maximale d'un analogue du lieu de Noether-Lefschetz, Compositio Math. 131 (2002), 31-50.
  21. Voisin C., Hodge loci and absolute Hodge classes, Compositio Math. 143 (2007), 945-958, math.AG/0605766.

Previous article   Next article   Contents of Volume 5 (2009)