zbMATH — the first resource for mathematics

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

32Q45 Hyperbolic and Kobayashi hyperbolic manifolds
Full Text: DOI
[1] R. Brody, Intrinsic metrics and measures on compact complex manifolds , doctoral thesis, Harvard, 1975.
[2] R. Brody, Compact manifolds in hyperbolicity , Trans. Amer. Math. Soc. 235 (1978), 213-219. · Zbl 0416.32013
[3] 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
[4] 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
[5] H. Cartan, Sur les zéros des combinaisons linéaires de \(p\) fonctions holomorphes données , Mathematicaa (Cluj) 7 (1933), 5-31. · Zbl 0007.41503
[6] W. Fulton, Algebraic curves. An introduction to algebraic geometry , W. A. Benjamin, Inc., New York-Amsterdam, 1969. · Zbl 0181.23901
[7] M. L. Green, Some Picard theorems for holomorphic maps to algebraic varieties , Amer. J. Math. 97 (1975), 43-75. JSTOR: · Zbl 0301.32022
[8] 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
[9] S. Kobayashi, Hyperbolic manifolds and holomorphic mappings , Pure and Applied Mathematics, vol. 2, Marcel Dekker Inc., New York, 1970. · Zbl 0207.37902
[10] S. Lang, Hyperbolic and Diophantine analysis , Bull. Amer. Math. Soc. (N.S.) 14 (1986), no. 2, 159-205. · Zbl 0602.14019
[11] S. Lang, Higher dimensional diophantine problems , Bull. Amer. Math. Soc. 80 (1974), 779-787. · Zbl 0298.14014
[12] 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
[13] 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
[14] Y.-T. Siu, Nonequidimensional value distribution theory and meromorphic connections , · Zbl 0716.32016
[15] 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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.