Hyperbolic hypersurfaces in $\mathbb P^n$ of Fermat-Waring type
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- by Bernard Shiffman and Mikhail Zaidenberg PDF
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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
- Robert Brody, Compact manifolds and hyperbolicity, Trans. Amer. Math. Soc. 235 (1978), 213–219. MR 470252, DOI 10.1090/S0002-9947-1978-0470252-3
- Jean-Pierre Demailly and Jawher El Goul, Hyperbolicity of generic surfaces of high degree in projective 3-space, Amer. J. Math. 122 (2000), no. 3, 515–546. MR 1759887
- Hirotaka Fujimoto, On meromorphic maps into the complex projective space, J. Math. Soc. Japan 26 (1974), 272–288. MR 346198, DOI 10.2969/jmsj/02620272
- Fujimoto H. A family of hyperbolic hypersurfaces in the complex projective space. Complex Variables Theory Appl. 43 (2001), 273–283.
- Mark Lee Green, Some Picard theorems for holomorphic maps to algebraic varieties, Amer. J. Math. 97 (1975), 43–75. MR 367302, DOI 10.2307/2373660
- Ha Huy Khoai, Hyperbolic surfaces in $\textrm {P}^3(\textbf {C})$, Proc. Amer. Math. Soc. 125 (1997), no. 12, 3527–3532. MR 1443160, DOI 10.1090/S0002-9939-97-04200-7
- Kazuo Masuda and Junjiro Noguchi, A construction of hyperbolic hypersurface of $\textbf {P}^n(\textbf {C})$, Math. Ann. 304 (1996), no. 2, 339–362. MR 1371771, DOI 10.1007/BF01446298
- M. McQuillan, Holomorphic curves on hyperplane sections of $3$-folds, Geom. Funct. Anal. 9 (1999), no. 2, 370–392. MR 1692470, DOI 10.1007/s000390050091
- Pacienza G. Rational curves on general projective hypersurfaces. E-print, math.AG/0010037, Oct. 2000, 21pp.
- Shiffman B., Zaidenberg M. Two classes of hyperbolic surfaces in ${\mathbf P}^3$. International J. Math. 11 (2000), 65–101.
- Manabu Shirosaki, Hyperbolic hypersurfaces in the complex projective spaces of low dimensions, Kodai Math. J. 23 (2000), no. 2, 224–233. MR 1768182, DOI 10.2996/kmj/1138044212
- Manabu Shirosaki, A hyperbolic hypersurface of degree 10, Kodai Math. J. 23 (2000), no. 3, 376–379. MR 1787671, DOI 10.2996/kmj/1138044265
- Yum-Tong Siu and Sai-Kee Yeung, Defects for ample divisors of abelian varieties, Schwarz lemma, and hyperbolic hypersurfaces of low degrees, Amer. J. Math. 119 (1997), no. 5, 1139–1172. MR 1473072
- Nobushige Toda, On the functional equation $\sum ^{p}_{i=0}\,a_{i}f_{i}^{n_{}i}=1$, Tohoku Math. J. (2) 23 (1971), 289–299. MR 291460, DOI 10.2748/tmj/1178242646
- M. G. Zaĭdenberg, The complement to a general hypersurface of degree $2n$ in $\textbf {CP}^n$ is not hyperbolic, Sibirsk. Mat. Zh. 28 (1987), no. 3, 91–100, 222 (Russian). MR 904640
- M. G. Zaĭdenberg, Stability of hyperbolic embeddedness and the construction of examples, Mat. Sb. (N.S.) 135(177) (1988), no. 3, 361–372, 415 (Russian); English transl., Math. USSR-Sb. 63 (1989), no. 2, 351–361. MR 937646, DOI 10.1070/SM1989v063n02ABEH003278
- Mikhail Zaidenberg, Hyperbolicity in projective spaces, Sūrikaisekikenkyūsho K\B{o}kyūroku 819 (1993), 136–156. International Symposium “Holomorphic Mappings, Diophantine Geometry and Related Topics” (Kyoto, 1992). MR 1247074
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
- MR Author ID: 196553
- Email: zaidenbe@ujf-grenoble.fr
- 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
- © Copyright 2001 American Mathematical Society
- 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
- MathSciNet review: 1896038