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Hyperbolic surfaces in $${\mathbb{P}}^ 3$$. (English) Zbl 0686.32015
The author constructs smooth hyperbolic surfaces (in the sense of S. Kobayashi) in the complex projective 3-space $$P^ 3({\mathbb{C}})$$, which is done using Main Theorem below.
Fix positive integers d,k $$(d>k)$$. Let $${\mathcal S}(d,k)$$ be the complex vector subspace of the vector space of all degree d homogeneous polynomials in $$Z^ 0,...,Z^ n$$ which is defined as follows.
$${\mathcal S}(d,k)$$ is spanned by monomials of the form $$(Z^ 0)^{d_ 0}(Z^ 1)^{d_ 1}...(Z^ n)^{d_ n}$$ where $$d_ i\geq d-k$$ for some $$i\in \{0,...,n\}$$. Here $$d_ 0,...,d_ n$$ are semipositive integers which satisfy $$d_ 0+...+d_ n=d.$$
The author defines the notion of $$S\in {\mathcal S}(d,k)$$ being nondegenerate. Then he proves the following.
Main theorem. Let $$M\subset P^ n({\mathbb{C}})$$ be a smooth hypersurface of degree d defined as the zero set of some nondegenerate $$S\in {\mathcal S}(d,k)$$. Assume $d-n-1 > [(n-1)(n-2)/2](k(n+1)+n+3).$ Then the image of any holomorphic curve f: $${\mathbb{C}}\to M$$ is not Zariski-dense in M.
Reviewer: T.Ochiai

MSC:
 32Q45 Hyperbolic and Kobayashi hyperbolic manifolds
Full Text:
References:
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