Geodesic
On hyperbolic embedding of complements of divisors and the limiting behavior of the Kobayashi--Rroyden metric
Sbornik. Mathematics, Tome 55 (1986) no. 1, pp. 55-70

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In the following three cases criteria are found for complements of divisors in compact complex manifolds to be hyperbolically embedded in the sense of Kobayashi: for divisors with normal crossings, for arbitrary divisors in complex surfaces, and for unions of hyperplanes in projective space. A criterion is given for two-dimensional polynomial polyhedra to be hyperbolically embedded, and Iitaka's conjecture about conditions for hyperbolicity of the complement of a set of projective lines is confirmed. Upper semicontinuity is proved for the Kobayashi–Royden pseudometrics and Kobayashi–Eisenman pseudovolumes of a family of complex manifolds containing degenerate fibers, and conditions are given under which the hyperbolic length (volume) on the smooth part of a degenerate fiber is the limit of the hyperbolic length (volume) on the nonsingular fibers. Bibliography: 28 titles. 
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     author = {M. G. Zaidenberg},
     title = {On hyperbolic embedding of complements of divisors and the limiting behavior of the {Kobayashi--Rroyden} metric},
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M. G. Zaidenberg. On hyperbolic embedding of complements of divisors and the limiting behavior of the Kobayashi--Rroyden metric. Sbornik. Mathematics, Tome 55 (1986) no. 1, pp. 55-70. http://geodesic.mathdoc.fr/item/SM_1986_55_1_a3/