Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

The Kobayashi pseudometric on algebraic manifolds of general type and in deformations of complex manifolds


Author: Marcus Wright
Journal: Trans. Amer. Math. Soc. 232 (1977), 357-370
MSC: Primary 32H20; Secondary 32G05
DOI: https://doi.org/10.1090/S0002-9947-1977-0466639-4
MathSciNet review: 0466639
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: This paper deals with regularity properties of the infinitesimal form of the Kobayashi pseudo-distance. This form is shown to be upper semicontinuous in the parameters of a deformation of a complex manifold. The method of proof involves the use of a parametrized version of the Newlander-Nirenberg Theorem together with a theorem of Royden on extending regular mappings from polydiscs into complex manifolds. Various consequences and improvements of this result are discussed; for example, if the manifold is compact hyperbolic the infinitesimal Kobayashi metric is continuous on the union of the holomorphic tangent bundles of the fibers of the deformation. This result leads to the fact that the coarse moduli space of a compact hyperbolic manifold is Hausdorff. Finally, the infinitesimal form is studied for a class of algebraic manifolds which contains algebraic manifolds of general type. It is shown that the form is continuous on the tangent bundle of a manifold in this class. Many members of this class are not hyperbolic.


References [Enhancements On Off] (What's this?)

  • [1] Robert Brody, Intrinsic metrics and measures on compact complex manifolds, Thesis, Harvard Univ., 1975.
  • [2] Robert Brody and Mark Green, A family of smooth hyperbolic hypersurfaces in $ {P_3}$ (unpublished).
  • [3] D. A. Eisenman, Intrinsic measures on complex manifolds and holomorphic mappings, Mem. Amer. Math. Soc. No. 96 (1970). MR 0259165 (41:3807)
  • [4] Richard S. Hamilton, Deformations of complex structures on pseudo-convex domains (to appear).
  • [5] S. Kobayashi, Hyperbolic manifolds and holomorphic mappings, Dekker, New York, 1970. MR 43 #3503. MR 0277770 (43:3503)
  • [6] S. Kobayashi and T. Ochai, Mappings into compact complex manifolds with negative first Chern class, J. Math Soc. Japan 23 (1971), 137-148. MR 44 #5514. MR 0288316 (44:5514)
  • [7] K. Kodaira, Holomorphic mappings of polydiscs into compact complex manifolds, J. Differential Geometry 6 (1971/72), 33-46. MR 46 #386. MR 0301228 (46:386)
  • [8] J. J. Kohn, Harmonic integrals of strongly pseudo-convex manifolds. I, Ann. of Math. (2) 78 (1963), 112-148. MR 27 #2999. MR 0153030 (27:2999)
  • [9] M. Kuranishi, Deformations of compact complex manifolds, Séminaire de Mathématiques Supérieures, No. 39, Univ. of Montreal Press, Montreal, Que., 1971. MR 50 #7588. MR 0355111 (50:7588)
  • [10] M. S. Narasimhan and R. R. Simha, Manifolds with ample canonical class, Invent. Math. 5 (1968), 120-128. MR 38 #5253. MR 0236960 (38:5253)
  • [11] A. Newlander and L. Nirenberg, Complex analytic coordinates in almost complex manifolds, Ann. of Math. (2) 65 (1957), 391-404. MR 19, 577. MR 0088770 (19:577a)
  • [12] D. A. Pelles (formerly D. A. Eisenman), Holomorphic maps which preserve intrinsic measure, Amer. J. Math. 97 (1975), 1-15. MR 0367300 (51:3542)
  • [13] H. L. Royden, Remarks on the Kobayashi metric, Several Complex Variables, II, Lecture Notes in Math., vol. 185, Springer, Berlin, 1971, pp. 125-137. MR 46 #3826. MR 0304694 (46:3826)
  • [14] -, The extension of regular holomorphic maps, Proc. Amer. Math. Soc. 43 (1974), 306-310. MR 49 #629. MR 0335851 (49:629)
  • [15] M. W. Wright, The behavior of the differential Kobayashi pseudo-metric in deformations of complex manifolds, Thesis, Standford Univ., 1974.
  • [16] S.-T. Yau, Intrinsic measures of compact complex manifolds, Math. Ann. 212 (1975), 317-329. MR 0367261 (51:3503)
  • [17] M. W. Wright, A note on the automorphism group of a compact algebraic manifold of general type, (to appear).

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 32H20, 32G05

Retrieve articles in all journals with MSC: 32H20, 32G05


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1977-0466639-4
Article copyright: © Copyright 1977 American Mathematical Society

American Mathematical Society