Curves in quadric and cubic surfaces whose complements are Kobayashi hyperbolically imbedded
Annales de l'Institut Fourier, Volume 65 (2015) no. 5, pp. 2057-2068.

We construct smooth irreducible curves of the lowest possible degrees in quadric and cubic surfaces whose complements are Kobayashi hyperbolically imbedded into those surfaces. Moreover we characterize line bundles on quadric and cubic surfaces such that the complete linear systems of the line bundles have a smooth irreducible curve whose complement is Kobayashi hyperbolically imbedded into those surfaces.

Nous construisons des courbes complexes lisses de degré minimal possible dans des surfaces quadriques et cubiques à complémentaire hyperboliquement plongé, au sens de Kobayashi. De plus, nous caractérisons les fibrés en droites sur de telles surfaces dont les systèmes linéaires associés possèdent des courbes lisses à complémentaire hyperboliquement plongé.

DOI: 10.5802/aif.2982
Classification: 32Q45, 14J26
Keywords: Kobayashi hyperbolic imbedding, holomorphic map
Mot clés : Plongement Kobayashi-hyperbolique, application holomorphes
Ito, Atsushi 1; Tiba, Yusaku 2

1 Department of Mathematics Kyoto University Kitashirakawa Oiwake-cho, Sakyo-ku, Kyoto (Japan)
2 Graduate School of Mathematical Sciences University of Tokyo Komaba, Meguro-ku, Tokyo (Japan)
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Ito, Atsushi; Tiba, Yusaku. Curves in quadric and cubic surfaces whose complements are Kobayashi hyperbolically imbedded. Annales de l'Institut Fourier, Volume 65 (2015) no. 5, pp. 2057-2068. doi : 10.5802/aif.2982. http://www.numdam.org/articles/10.5802/aif.2982/

[1] Beauville, Arnaud Complex algebraic surfaces, London Mathematical Society Student Texts, 34, Cambridge University Press, Cambridge, 1996, pp. x+132 (Translated from the 1978 French original by R. Barlow, with assistance from N. I. Shepherd-Barron and M. Reid) | DOI | Zbl

[2] Ciliberto, C.; Zaidenberg, M. Scrolls and hyperbolicity, Internat. J. Math., Volume 24 (2013) no. 4, pp. 1350026, 25 | DOI | Zbl

[3] Corvaja, Pietro; Noguchi, Junjiro A new unicity theorem and Erdös’ problem for polarized semi-abelian varieties, Math. Ann., Volume 353 (2012) no. 2, pp. 439-464 | DOI

[4] Demailly, Jean-Pierre; El Goul, Jawher Hyperbolicity of generic surfaces of high degree in projective 3-space, Amer. J. Math., Volume 122 (2000) no. 3, pp. 515-546 http://muse.jhu.edu/journals/american_journal_of_mathematics/v122/122.3demailly.pdf | Zbl

[5] Di Rocco, Sandra k-very ample line bundles on del Pezzo surfaces, Math. Nachr., Volume 179 (1996), pp. 47-56 | DOI | Zbl

[6] Dolgachev, Igor V. Classical algebraic geometry, Cambridge University Press, Cambridge, 2012, pp. xii+639 (A modern view) | DOI | Zbl

[7] Duval, Julien Une sextique hyperbolique dans P 3 (C), Math. Ann., Volume 330 (2004) no. 3, pp. 473-476 | DOI | Zbl

[8] El Goul, Jawher Logarithmic jets and hyperbolicity, Osaka J. Math., Volume 40 (2003) no. 2, pp. 469-491 http://projecteuclid.org/euclid.ojm/1153493095 | Zbl

[9] Fujimoto, Hirotaka Extensions of the big Picard’s theorem, Tôhoku Math. J. (2), Volume 24 (1972), pp. 415-422 | Zbl

[10] Fujimoto, Hirotaka On holomorphic maps into a taut complex space, Nagoya Math. J., Volume 46 (1972), pp. 49-61 | Zbl

[11] Green, Mark L. Holomorphic maps into complex projective space omitting hyperplanes, Trans. Amer. Math. Soc., Volume 169 (1972), pp. 89-103 | Zbl

[12] Hartshorne, Robin Algebraic geometry, Springer-Verlag, New York-Heidelberg, 1977, pp. xvi+496 (Graduate Texts in Mathematics, No. 52) | Zbl

[13] Kobayashi, Shoshichi Invariant distances on complex manifolds and holomorphic mappings, J. Math. Soc. Japan, Volume 19 (1967), pp. 460-480 | Zbl

[14] Kobayashi, Shoshichi Hyperbolic manifolds and holomorphic mappings, Pure and Applied Mathematics, 2, Marcel Dekker, Inc., New York, 1970, pp. ix+148 | Zbl

[15] Manin, Yu. I. Cubic forms: algebra, geometry, arithmetic, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., New York, 1974, pp. vii+292 (Translated from the Russian by M. Hazewinkel, North-Holland Mathematical Library, Vol. 4) | Zbl

[16] Masuda, Kazuo; Noguchi, Junjiro A construction of hyperbolic hypersurface of P n (C), Math. Ann., Volume 304 (1996) no. 2, pp. 339-362 | DOI | Zbl

[17] McQuillan, M. Holomorphic curves on hyperplane sections of 3-folds, Geom. Funct. Anal., Volume 9 (1999) no. 2, pp. 370-392 | DOI | Zbl

[18] Noguchi, Junjiro Holomorphic curves in algebraic varieties, Hiroshima Math. J., Volume 7 (1977) no. 3, pp. 833-853 | Zbl

[19] Noguchi, Junjiro Lemma on logarithmic derivatives and holomorphic curves in algebraic varieties, Nagoya Math. J., Volume 83 (1981), pp. 213-233 http://projecteuclid.org/euclid.nmj/1118786486 | Zbl

[20] Noguchi, Junjiro; Ochiai, Takushiro Geometric function theory in several complex variables, Translations of Mathematical Monographs, 80, American Mathematical Society, Providence, RI, 1990, pp. xii+283 (Translated from the Japanese by Noguchi) | Zbl

[21] Păun, Mihai Vector fields on the total space of hypersurfaces in the projective space and hyperbolicity, Math. Ann., Volume 340 (2008) no. 4, pp. 875-892 | DOI | Zbl

[22] Rousseau, Erwan Logarithmic vector fields and hyperbolicity, Nagoya Math. J., Volume 195 (2009), pp. 21-40 http://projecteuclid.org/euclid.nmj/1252934369 | Zbl

[23] Siu, Yum-Tong; Yeung, Sai-kee Hyperbolicity of the complement of a generic smooth curve of high degree in the complex projective plane, Invent. Math., Volume 124 (1996) no. 1-3, pp. 573-618 | DOI | Zbl

[24] Tiba, Yusaku Kobayashi hyperbolic imbeddings into toric varieties, Math. Ann., Volume 355 (2013) no. 3, pp. 879-892 | DOI | Zbl

[25] Zaĭdenberg, M.; Shiffman, B. New examples of Kobayashi hyperbolic surfaces in P 3 , Funktsional. Anal. i Prilozhen., Volume 39 (2005) no. 1, pp. 90-94 | DOI | Zbl

[26] Zaĭdenberg, M. G. Stability of hyperbolic embeddedness and the construction of examples, Mat. Sb. (N.S.), Volume 135(177) (1988) no. 3, p. 361-372, 415 | Zbl

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