Bounds on the dimension of variations of Hodge structure
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- by James A. Carlson
- Trans. Amer. Math. Soc. 294 (1986), 45-64
- DOI: https://doi.org/10.1090/S0002-9947-1986-0819934-6
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Erratum: Trans. Amer. Math. Soc. 299 (1987), 429.
Abstract:
We derive upper bounds on the dimension of a variation of Hodge structure of weight two and show that these bounds are sharp. Using them we exhibit maximal geometric variations of Hodge structure. Analogous results for higher weight are obtained in the presence of a nondegeneracy hypothesis, and variations coming from hypersurfaces are shown to be nondegenerate. Maximal geometric variations of higher weight are also constructed.References
- A. Ash, D. Mumford, M. Rapoport, and Y. Tai, Smooth compactification of locally symmetric varieties, Lie Groups: History, Frontiers and Applications, Vol. IV, Math Sci Press, Brookline, Mass., 1975. MR 0457437
- W. L. Baily Jr. and A. Borel, Compactification of arithmetic quotients of bounded symmetric domains, Ann. of Math. (2) 84 (1966), 442–528. MR 216035, DOI 10.2307/1970457
- Armand Borel, Compact Clifford-Klein forms of symmetric spaces, Topology 2 (1963), 111–122. MR 146301, DOI 10.1016/0040-9383(63)90026-0
- James A. Carlson and Phillip A. Griffiths, Infinitesimal variations of Hodge structure and the global Torelli problem, Journées de Géometrie Algébrique d’Angers, Juillet 1979/Algebraic Geometry, Angers, 1979, Sijthoff & Noordhoff, Alphen aan den Rijn—Germantown, Md., 1980, pp. 51–76. MR 605336
- James Carlson, Mark Green, Phillip Griffiths, and Joe Harris, Infinitesimal variations of Hodge structure. I, Compositio Math. 50 (1983), no. 2-3, 109–205. MR 720288
- Ron Donagi, Generic Torelli for projective hypersurfaces, Compositio Math. 50 (1983), no. 2-3, 325–353. MR 720291
- Mark L. Green, The period map for hypersurface sections of high degree of an arbitrary variety, Compositio Math. 55 (1985), no. 2, 135–156. MR 795711 —, Letter of September 1984. P. Griffiths, Periods of integrals on algebraic manifolds. II, Amer. J. Math. 90 (1968), 568-626, 805-865. —, Periods of rational integrals. I, II, Ann. of Math. (2) 90 (1969), 460-495, 498-591.
- A. Grothendieck, Hodge’s general conjecture is false for trivial reasons, Topology 8 (1969), 299–303. MR 252404, DOI 10.1016/0040-9383(69)90016-0 K. Kii, The local Torelli theorem for varieties with divisible canonical class, Math. USSR-Izv. 12 (1978), 53-67. F. S. Macaulay, Some properties of enumeration in the theory of modular systems, Proc. London Math. Soc. 26 (1927), 531-555.
- Chris Peters and Joseph Steenbrink, Infinitesimal variations of Hodge structure and the generic Torelli problem for projective hypersurfaces (after Carlson, Donagi, Green, Griffiths, Harris), Classification of algebraic and analytic manifolds (Katata, 1982) Progr. Math., vol. 39, Birkhäuser Boston, Boston, MA, 1983, pp. 399–463. MR 728615 H. V. Pittie, Letter of April 1984.
- Wilfried Schmid, Variation of Hodge structure: the singularities of the period mapping, Invent. Math. 22 (1973), 211–319. MR 382272, DOI 10.1007/BF01389674
Bibliographic Information
- © Copyright 1986 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 294 (1986), 45-64
- MSC: Primary 14D05; Secondary 14C30, 32G20
- DOI: https://doi.org/10.1090/S0002-9947-1986-0819934-6
- MathSciNet review: 819934