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The variations of Hodge structure of maximal dimension with associated Hodge numbers $ h\sp {2,0}>2$ and $ h\sp {1,1}=2q+1$ do not arise from geometry


Author: Azniv Kasparian
Journal: Trans. Amer. Math. Soc. 347 (1995), 4985-5007
MSC: Primary 32G20; Secondary 14C30, 14D07, 32J25
DOI: https://doi.org/10.1090/S0002-9947-1995-1290721-6
MathSciNet review: 1290721
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Abstract: The specified variations are proved to be covered by a bounded contractible domain $ \Omega $. After classifying the analytic boundary components of $ \Omega $ with respect to a fixed realization, the group of the biholomorphic automorphisms $ {\text{Aut}}\Omega $ and the $ {\text{Aut}}\Omega $-orbit structure of $ \Omega $ are found explicitly. Then $ \Omega $ is shown to admit no quasiprojective arithmetic quotients, whereas the lack of geometrically arising variations, covered by $ \Omega $.


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  • [1] W. Baily and A. Borel, Compactification of arithmetic quotients of bounded symmetric domains, Ann. of Math. (2) 84 (1966), 422-528. MR 0216035 (35:6870)
  • [2] R. Braun, W. Kaup, and H. Upmeier, On the automorphisms of circular and Reinhardt domains in complex Banach spaces, Manuscripta Math. 133 (1978), 25-97. MR 500878 (80g:32003)
  • [3] J. Carlson, Bounds on the dimension of variations of Hodge structure, Trans. Amer. Math. Soc. 294 (1986), 45-64. MR 819934 (87j:14010a)
  • [4] J. Carlson and L. Hernandez, Harmonic maps from compact Kahler manifolds to exceptional hyperbolic spaces, J. Geom. Analysis 1 (1991), 339-357. MR 1129347 (93a:58046)
  • [5] J. Carlson, A. Kasparian, and D. Toledo, Variations of Hodge structure of maximal dimension, Duke Math. J. 58 (1989), 669-694. MR 1016441 (90h:14015)
  • [6] J. Carlson and C. Simpson, Shimura varieties of weight two Hodge structures, Lecture Notes in Math., vol. 1246, Springer-Verlag, Berlin-Heidelberg-New York, 1987, pp. 1-15. MR 894038 (88j:14013)
  • [7] J. Carlson and D. Toledo, Integral manifolds, harmonic mappings, and the abelian subspace problem, Lecture Notes in Math., vol. 1352, Springer-Verlag, Berlin-Heidelberg-New York, 1988, pp. 60-74. MR 981818 (90a:32041)
  • [8] G. Fisher, Complex analytic geometry, Lecture Notes in Math., vol. 538, Berlin-Heidelberg-New York, 1976.
  • [9] P. Griffiths, Periods of integrals on algebraic manifolds. I, II, Amer. J. Math. 90 (1968), 568-626; 805-865.
  • [10] -, Periods of integrals on algebraic manifolds--summary of main results and discussion of open problems, Bull Amer. Math Soc. 75 (1970), 228-296.
  • [11] -, Periods of integrals on algebraic manifolds. III, Publ. Math. Inst. Hautes Ètudes Sci. 38 (1970), 125-180.
  • [12] -, Topics in transcendental algebraic geometry, Ann. of Math. Stud., no. 106, Princeton Univ. Press, Princeton, NJ, 1984. MR 756842 (86b:14004)
  • [13] P. Griffiths and J. Harris, Principles of algebraic geometry, Wiley-Interscience, New York, 1978. MR 507725 (80b:14001)
  • [14] S. Helgason, Differential geometry, Lie groups, and symmetric spaces, Academic Press, Orlando, FL, 1978. MR 514561 (80k:53081)
  • [15] A. Kasparian, Variations of Hodge structure of maximal dimension with associated Hodge numbers $ {h^{2,0}} > 2$ and $ {h^{1,1}} = 2q + 1$, Ph.D. Thesis, Univ. of Utah, 1993.
  • [16] S. Kobayashi, Hyperbolic manifolds and holomorphic mappings, Marcel Dekker, New York, 1970. MR 0277770 (43:3503)
  • [17] S. Krantz, Function theory of several complex variables, Wiley Interscience, New York, 1982. MR 635928 (84c:32001)
  • [18] A. Malčev, Commutative subalgebras of semisimple Lie algebras, Amer. Math. Soc. Transl. Ser. 2, vol. 40, Amer. Math. Soc., Providence, RI, 1951.
  • [19] R. Naramsinhan, Several complex variables, Chicago Lectures in Math, Chicago-London, 1971.
  • [20] I. Pyatetskii-Shapiro, Automorphic functions and the geometry of classical domains, Gordon and Breach, New York-London-Paris, 1969. MR 0252690 (40:5908)

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DOI: https://doi.org/10.1090/S0002-9947-1995-1290721-6
Article copyright: © Copyright 1995 American Mathematical Society

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