Hyperbolic hypersurfaces in 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

Published electronically:
December 27, 2001

MathSciNet review:
1896038

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Abstract | References | Similar Articles | Additional Information

Abstract: In this note we show that there are algebraic families of hyperbolic, Fermat-Waring type hypersurfaces in of degree , for all dimensions . Moreover, there are hyperbolic Fermat-Waring hypersurfaces in of degree possessing complete hyperbolic, hyperbolically embedded complements.

**[Br]**Robert Brody,*Compact manifolds and hyperbolicity*, Trans. Amer. Math. Soc.**235**(1978), 213–219. MR**0470252**, 10.1090/S0002-9947-1978-0470252-3**[DeEl]**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****[Fu1]**Hirotaka Fujimoto,*On meromorphic maps into the complex projecive space*, J. Math. Soc. Japan**26**(1974), 272–288. MR**0346198****[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]**Mark Lee Green,*Some Picard theorems for holomorphic maps to algebraic varieties*, Amer. J. Math.**97**(1975), 43–75. MR**0367302****[Kh]**Ha Huy Khoai,*Hyperbolic surfaces in 𝑃³(𝐶)*, Proc. Amer. Math. Soc.**125**(1997), no. 12, 3527–3532. MR**1443160**, 10.1090/S0002-9939-97-04200-7**[MaNo]**Kazuo Masuda and Junjiro Noguchi,*A construction of hyperbolic hypersurface of 𝑃ⁿ(𝐶)*, Math. Ann.**304**(1996), no. 2, 339–362. MR**1371771**, 10.1007/BF01446298**[Mc]**M. McQuillan,*Holomorphic curves on hyperplane sections of 3-folds*, Geom. Funct. Anal.**9**(1999), no. 2, 370–392. MR**1692470**, 10.1007/s000390050091**[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 . International J. Math. 11 (2000), 65-101. CMP**2000:12****[Shr1]**Manabu Shirosaki,*Hyperbolic hypersurfaces in the complex projective spaces of low dimensions*, Kodai Math. J.**23**(2000), no. 2, 224–233. MR**1768182**, 10.2996/kmj/1138044212**[Shr2]**Manabu Shirosaki,*A hyperbolic hypersurface of degree 10*, Kodai Math. J.**23**(2000), no. 3, 376–379. MR**1787671**, 10.2996/kmj/1138044265**[SiYe]**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****[To]**Nobushige Toda,*On the functional equation ∑^{𝑝}ᵢ₌₀𝑎ᵢ𝑓ᵢ^{𝑛𝑖}=1*, Tôhoku Math. J. (2)**23**(1971), 289–299. MR**0291460****[Za1]**M. G. Zaĭdenberg,*The complement to a general hypersurface of degree 2𝑛 in 𝐶𝑃ⁿ is not hyperbolic*, Sibirsk. Mat. Zh.**28**(1987), no. 3, 91–100, 222 (Russian). MR**904640****[Za2]**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****[Za3]**Mikhail Zaidenberg,*Hyperbolicity in projective spaces*, Sūrikaisekikenkyūsho Kōkyūroku**819**(1993), 136–156. International Symposium “Holomorphic Mappings, Diophantine Geometry and Related Topics” (Kyoto, 1992). MR**1247074**

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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