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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
##### Keywords:
hyperbolic surfaces; complex projective 3-space
Full Text:
##### References:
  R. Brody, Intrinsic metrics and measures on compact complex manifolds , doctoral thesis, Harvard, 1975.  R. Brody, Compact manifolds in hyperbolicity , Trans. Amer. Math. Soc. 235 (1978), 213-219. · Zbl 0416.32013  R. Brody and M. Green, A family of smooth hyperbolic hypersurfaces in $$P_3$$ , Duke Math. J. 44 (1977), no. 4, 873-874. · Zbl 0383.32009  H. Cartan, Sur les systèmes de fonctions holomorphes a variétés linéaires lacunaires et leurs applications , Ann. Sci École Norm. Sup. (3) 45 (1928), 255-346. · JFM 54.0357.06  H. Cartan, Sur les zéros des combinaisons linéaires de $$p$$ fonctions holomorphes données , Mathematicaa (Cluj) 7 (1933), 5-31. · Zbl 0007.41503  W. Fulton, Algebraic curves. An introduction to algebraic geometry , W. A. Benjamin, Inc., New York-Amsterdam, 1969. · Zbl 0181.23901  M. L. Green, Some Picard theorems for holomorphic maps to algebraic varieties , Amer. J. Math. 97 (1975), 43-75. JSTOR: · Zbl 0301.32022  M. Green and P. Griffiths, Two applications of algebraic geometry to entire holomorphic mappings , The Chern Symposium 1979 (Proc. Internat. Sympos., Berkeley, Calif., 1979), Springer, New York, 1980, pp. 41-74. · Zbl 0508.32010  S. Kobayashi, Hyperbolic manifolds and holomorphic mappings , Pure and Applied Mathematics, vol. 2, Marcel Dekker Inc., New York, 1970. · Zbl 0207.37902  S. Lang, Hyperbolic and Diophantine analysis , Bull. Amer. Math. Soc. (N.S.) 14 (1986), no. 2, 159-205. · Zbl 0602.14019  S. Lang, Higher dimensional diophantine problems , Bull. Amer. Math. Soc. 80 (1974), 779-787. · Zbl 0298.14014  M. Namba, Geometry of projective algebraic curves , Monographs and Textbooks in Pure and Applied Mathematics, vol. 88, Marcel Dekker Inc., New York, 1984. · Zbl 0556.14012  Y.-T. Siu, Defect relations for holomorphic maps between spaces of different dimensions , Duke Math. J. 55 (1987), no. 1, 213-251. · Zbl 0623.32018  Y.-T. Siu, Nonequidimensional value distribution theory and meromorphic connections , · Zbl 0716.32016  Yum Tong Siu, Nonequidimensional value distribution theory and subvariety extension , Complex analysis and algebraic geometry (Göttingen, 1985), Lecture Notes in Math., vol. 1194, Springer, Berlin, 1986, pp. 158-174. · Zbl 0591.32028
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