HYPERBOLIC HYPERSURFACES IN P n OF FERMAT - WARING TYPE Bernard SHIFFMAN

In this note we show that there are algebraic families of hyperbolic, Fermat-Waring type hypersurfaces in P of degree 4(n− 1)2, for all dimensions n 2. Moreover, there are hyperbolic Fermat-Waring hypersurfaces in P of degree 4n2−2n +1 possessing complete hyperbolic, hyperbolically embedded complements. Many examples have been given of hyperbolic hypersurfaces in P3 (e.g., see [ShZa] and the literature therein). Examples of degree 10 hyperbolic surfaces in P3 were recently found by Shirosaki [Shr2], who also gave examples of hyperbolic hypersurfaces with hyperbolic complements in P3 and P4 [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)2. (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]: T 1. — Let d . (m − 1)2, m . 2n − 1. Then for generic linear functions h1, . . . ,hm on Cn+1, the hypersurface Xn−1 =   ∈ P n : m