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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
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