Coupled contact systems and rigidity of maximal dimensional variations of Hodge structure
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Abstract:
In this article we prove that locally Griffiths’ horizontal distribution on the period domain is given by a generalized version of the familiar contact differential system. As a consequence of this description we obtain strong local rigidity properties of maximal dimensional variations of Hodge structure. For example, we prove that if the weight is odd (greater than one) then there is a unique germ of maximal dimensional variation of Hodge structure through every point of the period domain. Similar results hold if the weight is even with the exception of one case.References
- S. Bergmann and J. Marcinkiewicz, Sur les fonctions analytiques de deux variables complexes, Fund. Math. 33 (1939), 75–94 (French). MR 57, DOI 10.4064/fm-33-1-75-94
- R. L. Bryant, S. S. Chern, R. B. Gardner, H. L. Goldschmidt, and P. A. Griffiths, Exterior differential systems, Mathematical Sciences Research Institute Publications, vol. 18, Springer-Verlag, New York, 1991. MR 1083148, DOI 10.1007/978-1-4613-9714-4
- James A. Carlson, Bounds on the dimension of variations of Hodge structure, Trans. Amer. Math. Soc. 294 (1986), no. 1, 45–64. MR 819934, DOI 10.1090/S0002-9947-1986-0819934-6
- James A. Carlson, Hypersurface variations are maximal. II, Trans. Amer. Math. Soc. 323 (1991), no. 1, 177–196. MR 978385, DOI 10.1090/S0002-9947-1991-0978385-6
- James A. Carlson and Ron Donagi, Hypersurface variations are maximal. I, Invent. Math. 89 (1987), no. 2, 371–374. MR 894385, DOI 10.1007/BF01389084
- James A. Carlson and Carlos Simpson, Shimura varieties of weight two Hodge structures, Hodge theory (Sant Cugat, 1985) Lecture Notes in Math., vol. 1246, Springer, Berlin, 1987, pp. 1–15. MR 894038, DOI 10.1007/BFb0077525
- James A. Carlson, Aznif Kasparian, and Domingo Toledo, Variations of Hodge structure of maximal dimension, Duke Math. J. 58 (1989), no. 3, 669–694. MR 1016441, DOI 10.1215/S0012-7094-89-05832-8
- Phillip A. Griffiths, Periods of integrals on algebraic manifolds. I. Construction and properties of the modular varieties, Amer. J. Math. 90 (1968), 568–626. MR 229641, DOI 10.2307/2373545
- Phillip Griffiths (ed.), Topics in transcendental algebraic geometry, Annals of Mathematics Studies, vol. 106, Princeton University Press, Princeton, NJ, 1984. MR 756842, DOI 10.1515/9781400881659
- Wilfried Schmid, Variation of Hodge structure: the singularities of the period mapping, Invent. Math. 22 (1973), 211–319. MR 382272, DOI 10.1007/BF01389674
- Frank W. Warner, Foundations of differentiable manifolds and Lie groups, Graduate Texts in Mathematics, vol. 94, Springer-Verlag, New York-Berlin, 1983. Corrected reprint of the 1971 edition. MR 722297
Additional Information
- Richárd Mayer
- Affiliation: Department of Mathematics, University of Massachusetts, Amherst, Massachusetts 01003
- Email: mayer@math.umass.edu
- Received by editor(s): December 5, 1997
- Published electronically: July 26, 1999
- © Copyright 2000 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 352 (2000), 2121-2144
- MSC (1991): Primary 14C30
- DOI: https://doi.org/10.1090/S0002-9947-99-02395-8
- MathSciNet review: 1624194