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 Mat. Sb. (N.S.), 1988, Volume 135(177), Number 3, Pages 361–372 (Mi msb1706)

Stability of hyperbolic imbeddedness and construction of examples

M. G. Zaidenberg

Abstract: 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.

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English version:
Mathematics of the USSR-Sbornik, 1989, 63:2, 351–361

Bibliographic databases:

UDC: 515.171.7+517.5+512.7
MSC: Primary 32H20; Secondary 32H15, 32H25

Citation: M. G. Zaidenberg, “Stability of hyperbolic imbeddedness and construction of examples”, Mat. Sb. (N.S.), 135(177):3 (1988), 361–372; Math. USSR-Sb., 63:2 (1989), 351–361

Citation in format AMSBIB
\Bibitem{Zai88} \by M.~G.~Zaidenberg \paper Stability of hyperbolic imbeddedness and construction of examples \jour Mat. Sb. (N.S.) \yr 1988 \vol 135(177) \issue 3 \pages 361--372 \mathnet{http://mi.mathnet.ru/msb1706} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=937646} \zmath{https://zbmath.org/?q=an:0668.32023|0641.32016} \transl \jour Math. USSR-Sb. \yr 1989 \vol 63 \issue 2 \pages 351--361 \crossref{https://doi.org/10.1070/SM1989v063n02ABEH003278} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. Jawher El Goul, “Algebraic families of smooth hyperbolic surfaces of low degree in ℙ ℂ 3 ”, manuscripta math, 90:1 (1996), 521
2. BERNARD SHIFFMAN, MIKHAIL ZAIDENBERG, “TWO CLASSES OF HYPERBOLIC SURFACES IN ℙ3”, Int. J. Math, 11:01 (2000), 65
3. R. DEBALME, S. IVASHKOVICH, “COMPLETE HYPERBOLIC NEIGHBORHOODS IN ALMOST-COMPLEX SURFACES”, Int. J. Math, 12:02 (2001), 211
4. CIRO CILIBERTO, MIKHAIL ZAIDENBERG, “3-FOLD SYMMETRIC PRODUCTS OF CURVES AS HYPERBOLIC HYPERSURFACES IN ℙ4”, Int. J. Math, 14:04 (2003), 413
5. M. G. Zaidenberg, B. Shiffman, “New Examples of Kobayashi Hyperbolic Surfaces in $\mathbb{CP}^3$”, Funct. Anal. Appl., 39:1 (2005), 76–79
6. Junjiro Noguchi, Jörg Winkelmann, Katsutoshi Yamanoi, “Degeneracy of holomorphic curves into algebraic varieties”, Journal de Mathématiques Pures et Appliquées, 88:3 (2007), 293
7. Ta Thi Hoai An, Julie Tzu-Yueh Wang, Pit-Mann Wong, “Non-archimedean analytic curves in the complements of hypersurface divisors”, Journal of Number Theory, 128:8 (2008), 2275
8. M. G. Zaidenberg, “Hyperbolicity of Generic Deformations”, Funct. Anal. Appl., 43:2 (2009), 113–118
9. C. CILIBERTO, M. ZAIDENBERG, “SCROLLS AND HYPERBOLICITY”, Int. J. Math, 2013, 1350026
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