Hyperbolic Hypersurfaces in P of Fermat-waring Type
نویسندگان
چکیده
In this note we show that there are algebraic families of hyperbolic, FermatWaring type hypersurfaces in P of degree 4(n − 1), for all dimensions n ≥ 2. Moreover, there are hyperbolic Fermat-Waring hypersurfaces in P of degree 4n − 2n + 1 possessing complete hyperbolic, hyperbolically embedded complements. Many examples have been given of hyperbolic hypersurfaces in P (e.g., see [ShZa] and the literature therein). Examples of degree 10 hyperbolic surfaces in P were recently found by Shirosaki [Shr2], who also gave examples of hyperbolic hypersurfaces with hyperbolic complements in P and P [Shr1]. Fujimoto [Fu2] then improved Shirosaki’s construction to give examples of degree 8. Answering a question posed in [Za3], Masuda and Noguchi [MaNo] constructed the first examples of hyperbolic projective hypersurfaces, including those with complete hyperbolic complements, in any dimension. Improving the degree estimates of [MaNo], Siu and Yeung [SiYe] gave examples of hyperbolic hypersurfaces in P of degree 16(n − 1). (Fujimoto’s recent construction [Fu2] provides examples of degree 2.) We remark that it was conjectured in 1970 by S. Kobayashi that generic hypersurfaces in P of (presumably) degree 2n− 1 are hyperbolic (for n = 3, see [DeEl] and [Mc]). The following result is an improvement of the example of Siu-Yeung [SiYe]: Theorem 1. Let d ≥ (m − 1), m ≥ 2n − 1. Then for generic linear functions h1, . . . , hm on C, the hypersurface Xn−1 = { z ∈ P : m ∑
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