Tohoku Mathematical Journal

Polarized period map for generalized $K3$ surfaces and the moduli of Einstein metrics

Ryoichi Kobayashi and Andrey N. Todorov

Full-text: Open access

Article information

Source
Tohoku Math. J. (2) Volume 39, Number 3 (1987), 341-363.

Dates
First available in Project Euclid: 3 May 2007

Permanent link to this document
http://projecteuclid.org/euclid.tmj/1178228282

Mathematical Reviews number (MathSciNet)
MR0902574

Zentralblatt MATH identifier
0646.14029

Digital Object Identifier
doi:10.2748/tmj/1178228282

Subjects
Primary: 32G07: Deformations of special (e.g. CR) structures
Secondary: 14D22: Fine and coarse moduli spaces 32G13: Analytic moduli problems {For algebraic moduli problems, see 14D20, 14D22, 14H10, 14J10} [See also 14H15, 14J15] 32G20: Period matrices, variation of Hodge structure; degenerations [See also 14D05, 14D07, 14K30] 53C55: Hermitian and Kählerian manifolds [See also 32Cxx] 58E11: Critical metrics

Citation

Kobayashi, Ryoichi; Todorov, Andrey N. Polarized period map for generalized $K3$ surfaces and the moduli of Einstein metrics. Tohoku Math. J. (2) 39 (1987), no. 3, 341--363. doi:10.2748/tmj/1178228282. http://projecteuclid.org/euclid.tmj/1178228282.


Export citation

References

  • [Atl] M. ATIYAH, On analytic surfaces with double points, Proc. R. Soc. Lond. Ser. A 245 (1958), 237-244.
  • [At2] M. ATIYAH, Some examples of complex manifolds, Bonner Math. Schriften 6 (1958), 27
  • [AtHS] M. ATIYAH, N. HITCHIN AND I. SINGER, Self-duality in four-dimensional Riemannia geometry, Proc. R. Soc. Lond. Ser A 362 (1978), 425-461.
  • [Au] T. AUBIN, Nonlinear Analysis on Manifolds, Monge-Ampere Equations, Grundlehre der Math. 252, Springer-Verlag 1982.
  • [Be] A. BEAUVILLE, Application aux espaces de modules, in Geometrie des surfaces KS modules et periodes, Asterisque 126 (1985), 141-152.
  • [Bol] J. P. BOURGUIGNON, Deformations des metriques dinstein, Asterisque 80 (1980), 21-31
  • [Bo2] J. P. BOURGUIGNON ET AL., Premiere classe de Chern et courbure de Ricci: preuve d la conjecture de Calabi, Seminaire Palaiseau, Asterisque 58 (1958), Soc. Math. France.
  • [BuR] D. BURNS AND M. RAPOPORT, On the Torelli problem for Kahlerian 3 surfaces, Ann scient. Ec. Norm. Sup. 8 (1975), 235-274.
  • [CY] S. Y. CHENG AND S. T. YAU, Inequality between Chern numbers of singular Kahle surfaces and characterization of orbit spaces of discrete group of SU(2, 1), in "Com-plex Differential Geometry and Nonlinear Differential Equations" (edited by Y. T. Siu), Contemporary Mathematics 49 (1986), American Mathematical Society.
  • [F] A. FUJIKI, Closedness of the Douady space of compact Kahler spaces, Publ. RIMS, Kyoto Univ. 14 (1978), 1-52.
  • [H] N. HITCHIN, Compact four dimensional Einstein manifolds, J. Differential Geometr 9 (1974), 435-441.
  • [1] K. IVINSKIS, Normale Flachen und die Miyaoka-Kobayashi-Ungleichung, Diplomarbeit, Bonn 1985.
  • [Kb] R. KOBAYASHI, Einstein-Kahler F-metrics on open Satake F-surfaces with isolate quotient singularities, Math. Ann. 272 (1985), 385-398.
  • [Kd] K. KODAIRA, On the structure of compact analytic surfaces, I. Amer. J. Math. 8 (1964), 751-798.
  • [L] E. LOOIJENGA, A, Torelli theorem for Kahler-Einstein KS surfaces, Lecture Notes i Math. 894 (1981), 107-112.
  • [LP] E. LOOIJENGA AND C. PETERS, Torelli theorems for Kahler KS surfaces, Compositi math. 42 (1981), 145-186.
  • [Ma] A. MAYER, Families of KS surfaces, Nagoya Math. J. 48 (1972), 1-7
  • [Mi] Y. MIYAOKA, The maximal number of quotient singularities on surfaces with give numerical invariants, Math. Ann. 268 (1984), 159-171.
  • [Mo] B. MOISEZON, Singular Kahler spaces, Manifolds and related topics in topology, Toky (edited by A. Hatton), Univ. of Tokyo Press (1973), 343-351.
  • [Mrl] D. MORRISON, Some remarks on the moduli of KS surfaces, in Classification of Algebrai and analytic manifolds, Progress in Math. 39, Birkhauser, 1983, 303-332.
  • [Mr2] D. MORRISON, On uniformization of KS surfaces, preprint
  • [Na] Y. NAMIKAWA, On the surjectivity of the period mapof KS surfaces, in "Classificatio of Algebraic and Analytic Manifolds", Progress of Math. 39, Birkhauser, 1983.
  • [Ni] V. NIKULIN, On Kummer surfaces, Math. USSR Izv. 9 (1975), 261-275
  • [ShP] I. R. SHAFAREVICH AND A. SHAPIRO-PIATECKII, A Torelli theorem for algebraic surface of type KS, Math. USSR Izv. 5 (1971), 547-587.
  • [Si] Y. T. Siu, Every KS surface is Kahler, Inventiones math. 73 (1983), 139-150
  • [Tl] A. TODOROV, Applications of Kahler-Einstein-Calabi-Yau metric to moduli of K surfaces, Inventiones math. 61 (1980), 251-265.
  • [T2] A. TODOROV, HOW many Kahler metrics has a KS surface? Algebraic geometry an number theory: papers dedicated to I. R. Shafarevich, (1983).
  • [Va] J. VAROUCHAS, Stabilite des varietes kahleriennes par certains morphism propres, preprint, Universite de Nancy I.
  • [Vn] E. B. VINBERG, Discrete groups generated by reflections in Lobachevskii spaces, Math USSR Sbornik 1 (1967), 429-444.
  • [W] A. WEIL, Collected papers, vol. 2, pp. 393-395, Springer-Verlag 1979
  • [Yal] S. T. YAU, On the Ricci curvature of a compact Kahler manifold and the comple Monge-Ampere equation, I, Comm. Pure and Appl. Math. 31 (1978), 339-441.
  • [Ya2] S. T. YAU, Lectures given at the Institute for Advanced Study, Princeton (1980)
  • [YoKT] M. YOSHIDA, J. KANEKO AND S. TOKUNAGA, Complex crystallographic groups II, J. Math. Soc. Japan, vol. 34 (1982), 595-605.