Oka properties of some hypersurface complements

Hanysz, Alexander

doi:10.1090/S0002-9939-2013-11754-5

Oka properties of some hypersurface complements
HTML articles powered by AMS MathViewer

by Alexander Hanysz PDF

Proc. Amer. Math. Soc. 142 (2014), 483-496 Request permission

Abstract:

Oka manifolds can be viewed as the “opposite” of Kobayashi hyperbolic manifolds. Kobayashi asked whether the complement in projective space of a generic hypersurface of sufficiently high degree is hyperbolic. Therefore it is natural to investigate Oka properties of complements of low degree hypersurfaces. We determine which complements of hyperplane arrangements in projective space are Oka. A related question is which hypersurfaces in affine space have Oka complements. We give some results for graphs of meromorphic functions.

References

Gregery T. Buzzard and Steven S. Y. Lu, Algebraic surfaces holomorphically dominable by $\mathbf C^2$, Invent. Math. 139 (2000), no. 3, 617–659. MR 1738063, DOI 10.1007/s002220050021
Franc Forstnerič, Oka manifolds, C. R. Math. Acad. Sci. Paris 347 (2009), no. 17-18, 1017–1020 (English, with English and French summaries). MR 2554568, DOI 10.1016/j.crma.2009.07.005
Franc Forstnerič, Stein manifolds and holomorphic mappings, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 56, Springer, Heidelberg, 2011. The homotopy principle in complex analysis. MR 2975791, DOI 10.1007/978-3-642-22250-4
Franc Forstnerič and Finnur Lárusson, Survey of Oka theory, New York J. Math. 17A (2011), 11–38. MR 2782726
M. Gromov, Oka’s principle for holomorphic sections of elliptic bundles, J. Amer. Math. Soc. 2 (1989), no. 4, 851–897. MR 1001851, DOI 10.1090/S0894-0347-1989-1001851-9
Allen Hatcher, Algebraic topology, Cambridge University Press, Cambridge, 2002. MR 1867354
Björn Ivarsson and Frank Kutzschebauch, Holomorphic factorization of mappings into $\textrm {SL}_n(\Bbb C)$, Ann. of Math. (2) 175 (2012), no. 1, 45–69. MR 2874639, DOI 10.4007/annals.2012.175.1.3
Shoshichi Kobayashi, Hyperbolic manifolds and holomorphic mappings, Pure and Applied Mathematics, vol. 2, Marcel Dekker, Inc., New York, 1970. MR 0277770
Shoshichi Kobayashi, Hyperbolic complex spaces, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 318, Springer-Verlag, Berlin, 1998. MR 1635983, DOI 10.1007/978-3-662-03582-5
S. Lojasiewicz, Triangulation of semi-analytic sets, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 18 (1964), 449–474. MR 173265
Yum-Tong Siu and Sai-kee Yeung, Hyperbolicity of the complement of a generic smooth curve of high degree in the complex projective plane, Invent. Math. 124 (1996), no. 1-3, 573–618. MR 1369429, DOI 10.1007/s002220050064