Stability of hyperbolic imbeddedness and construction of examples

M. G. Zaidenberg

Sbornik. Mathematics, Tome 63 (1989) no. 2, pp. 351-361

Résumé

Methods are worked out for constructing smooth hyperbolic curves $\Gamma\subset\mathbf{CP}^2$ and surfaces $H\subset\mathbf{CP}^3$ with hyperbolically imbedded complements, and the methods are then used to construct examples of such curves with least possible degree 5. The existence of these curves agrees well with the 1970 conjecture of Kobayashi. It is proved that the sets of such curves and surfaces are open (in the classical topology). The proofs are based on tests obtained for stability of hyperbolicity and of hyperbolic imbeddedness of analytic subsets of complex manifolds under perturbations that can in general reconstruct the topology. Bibliography: 18 titles.

Bibliographie

[1] Green M. L., “Some examples and counterexamples in value distribution theory for several variables”, Compos. mathem., 30:3 (1975), 317–322 | MR | Zbl

[2] Azukawa K., Suzuki M., “Some examples of algebraic degeneracy and hyperbolic manifolds”, Rocky Mount. J. Math., 10:3 (1980), 655–659 | MR | Zbl

[3] Green M. L., “The complement of the dual of a plane curve and some new hyperbolic manifolds”, Value Distribution Theory, M. Dekker, N.Y., 1974, 119–131 | MR

[4] Carlson J., Green M. L., “A Picard theorem for holomorphic curves in the plane”, Duke Math. J., 43:1 (1976), 1–9 | DOI | MR | Zbl

[5] Grauert H., Peiernell U., “Hyperbolicity of the complement of plane curves”, Manuscr. Math., 50 (1985), 429–441 | DOI | MR | Zbl

[6] Kiernan P., “Hyperbolically imbedded spaces and the big Picard theorem”, Math. Ann., 204 (1973), 203–209 | DOI | MR | Zbl

[7] Kiernan P., Kobayashi E., “Holomorphic mappings into projective space with lacunary hyperplanes”, Nagoya Math. J., 50 (1973), 199–210 | MR

[8] Green M. L., “The hyperbolicity of complement of $2n+1$ hyperplanes in $P^n$ and related results”, Proc. Amer. Math. Soc., 66:1 (1977), 109–113 | DOI | MR | Zbl

[9] Kobayasi S., “Giperbolicheskie mnogoobraziya i golomorfnye otobrazheniya”, Matematika (sb. perevodov), 17:1 (1973), 47–96

[10] Zaidenberg M. G., “Dopolnenie k obschei giperpoverkhnosti stepeni $2n$ v $\mathbf{CP}^n$ ne giperbolichno”, Sib. matem. zhurn., 28:3 (1987), 91–100 | MR | Zbl

[11] Brody R., “Compact manifolds and hyperbolicity”, Trans. Amer. Math. Soc., 235 (1978), 213–219 | DOI | MR | Zbl

[12] Hahn K. T., Kim K. T., “Hyperbolicity of a complex manifold and other equivalent properties”, Proc. Amer. Math. Soc., 91:1 (1984), 49–53 | DOI | MR | Zbl

[13] Zaidenberg M. G., “Teoremy Pikara i giperbolichnost”, Sib. matem. zhurn, 24:6 (1983), 44–55 | MR | Zbl

[14] Zaidenberg M. G., “O giperbolicheskoi vlozhennosti dopolnenii k divizoram i predelnom povedenii metriki Kobayasi–Roidena”, Matem. sborn., 127(169) (1985), 55–71 | MR | Zbl

[15] Chirka E. M., Komleksnye analiticheskie mnozhestva, Nauka, M., 1985 | MR

[16] Green M. L., “Some Picard theorems for holomorphic maps to algebraic varieties”, Amer. J. Math., 97:1 (1975), 43–75 | DOI | MR | Zbl

[17] Babets V. A., “Teoremy pikarovskogo tipa dlya golomorfnykh otobrazhenii”, Sib. matem. zhurn., 25:2 (1984), 35–41 | MR | Zbl

[18] Brody R., Green M. L., “A family of smoth hyperbolic hypersurfaces in $P_3$”, Duke Math. J., 44:4 (1977), 873–874 | DOI | MR | Zbl