Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Hyperbolic hypersurfaces in $\mathbb P^n$ of Fermat-Waring type


Authors: Bernard Shiffman and Mikhail Zaidenberg
Journal: Proc. Amer. Math. Soc. 130 (2002), 2031-2035
MSC (2000): Primary 32Q45, 32H25; Secondary 14J70
DOI: https://doi.org/10.1090/S0002-9939-01-06417-6
Published electronically: December 27, 2001
MathSciNet review: 1896038
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this note we show that there are algebraic families of hyperbolic, Fermat-Waring type hypersurfaces in $\mathbb P^n$ of degree $4(n-1)^2$, for all dimensions $n\ge 2$. Moreover, there are hyperbolic Fermat-Waring hypersurfaces in $\mathbb P^n$ of degree $4n^2-2n+1$ possessing complete hyperbolic, hyperbolically embedded complements.


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

  • [Br] Brody R. Compact manifolds and hyperbolicity. Trans. Amer. Math. Soc. 235 (1978), 213-219. MR 57:10010
  • [DeEl] Demailly J.-P., El Goul J. Hyperbolicity of generic surfaces of high degree in projective 3-space. Amer. J. Math. 122 (2000), 515-546. MR 2001f:32045
  • [Fu1] Fujimoto H. On meromorphic maps into the complex projective space. J. Math. Soc. Japan 26 (1974), 272-288. MR 49:10924
  • [Fu2] Fujimoto H. A family of hyperbolic hypersurfaces in the complex projective space. Complex Variables Theory Appl. 43 (2001), 273-283. CMP 2001:10
  • [Gr] Green M. Some Picard theorems for holomorphic maps to algebraic varieties. Amer. J. Math. 97 (1975), 43-75. MR 51:3544
  • [Kh] Ha Huy Khoai. Hyperbolic surfaces in ${\mathbf P}^3({\mathbf C})$. Proc. Amer. Math. Soc. 125 (1997), 3527-3532. MR 98e:32046
  • [MaNo] Masuda K., Noguchi J. A construction of hyperbolic hypersurface of $\mathbb{P}^n(\mathbb C)$. Math. Ann. 304 (1996), 339-362. MR 97a:32025
  • [Mc] McQuillan M. Holomorphic curves on hyperplane sections of 3-folds. Geom. Funct. Anal. 9 (1999), 370-392. MR 2000f:32035
  • [Pa] Pacienza G. Rational curves on general projective hypersurfaces. E-print, math.AG/0010037, Oct. 2000, 21pp.
  • [ShZa] Shiffman B., Zaidenberg M. Two classes of hyperbolic surfaces in ${\mathbf P}^3$. International J. Math. 11 (2000), 65-101. CMP 2000:12
  • [Shr1] Shirosaki M. Hyperbolic hypersurfaces in the complex projective spaces of low dimensions. Kodai Math. J. 23 (2000), 224-233. MR 2001e:32038
  • [Shr2] Shirosaki M. A hyperbolic hypersurface of degree 10. Kodai Math. J. 23 (2000), 376-379. MR 2001i:32041
  • [SiYe] Siu Y.-T., Yeung S.-K. Defects for ample divisors of abelian varieties, Schwarz lemma, and hyperbolic hypersurfaces of low degrees. Amer. J. Math. 119 (1997), 1139-1172. MR 98h:32044
  • [To] Toda N. On the functional equation $\sum_{i=0}^p\,a_{i}f_{i}^{n_i}=1$. Tôhoku Math. J. (2) 23 (1971), 289-299. MR 45:551
  • [Za1] Zaidenberg M. The complement of a generic hypersurface of degree $2n$ in ${\mathbf CP}^n$ is not hyperbolic. Siberian Math. J. 28 (1987), 425-432. MR 88k:32063
  • [Za2] Zaidenberg M. Stability of hyperbolic imbeddedness and construction of examples. Math. USSR Sbornik 63 (1989), 351-361. MR 89f:32047
  • [Za3] Zaidenberg M. Hyperbolicity in projective spaces. Proc. Conf. ``Diophantine Problemes, Hyperbolic Spaces and related topics", RIMS, Kyoto, Japan, 26-31 October 1992; Tokyo, TIT, 1992, 136-156. MR 95a:32045

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 32Q45, 32H25, 14J70

Retrieve articles in all journals with MSC (2000): 32Q45, 32H25, 14J70


Additional Information

Bernard Shiffman
Affiliation: Department of Mathematics, Johns Hopkins University, Baltimore, Maryland 21218
Email: shiffman@math.jhu.edu

Mikhail Zaidenberg
Affiliation: Université Grenoble I, Institut Fourier, UMR 5582 CNRS-UJF, BP 74, 38402 St. Martin d’Hères cédex, France
Email: zaidenbe@ujf-grenoble.fr

DOI: https://doi.org/10.1090/S0002-9939-01-06417-6
Received by editor(s): January 26, 2001
Published electronically: December 27, 2001
Additional Notes: Research of the first author partially supported by NSF grant #DMS-9800479.
Communicated by: Steven R. Bell
Article copyright: © Copyright 2001 American Mathematical Society

American Mathematical Society