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 Cet article a éte moissonné depuis la source Math-Net.Ru

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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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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/

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

[2] Kobayasi S., Okhiai T., “Kompaktifikatsiya Satake i bolshaya teorema Pikara”, Matematika (sb. perevodov), 17:2 (1973), 58–67

[3] Griffits F. A., “Doklad o variatsii struktury Khodzha (obzor osnovnykh rezultatov i obsuzhdenie otkrytykh problem i gipotez)”, UMN, 25:3 (1970), 175–234 | MR

[4] Bogomolov F. A., “Semeistva krivykh na poverkhnosti obschego tipa”, DAN SSSR, 236:5 (1977), 1041–1044 | MR | Zbl

[5] Shabat B. V., Vvedenie v kompleksnyi analiz, Ch. 2, Nauka, M., 1976 | MR

[6] Lvovskii M. N., “Integralnye predstavleniya giperbolicheskikh ob'emov”, Vestn. MGU. Ser. matem., mekh., 1976, no. 2, 21–27 | MR | Zbl

[7] Milnor Dzh., Osobye tochki kompleksnykh giperpoverkhnostei, Mir, M., 1971 | MR | Zbl

[8] Khu Sy-Tszyan, Teoriya gomotopii, Mir, M., 1964

[9] Zaidenberg M. G., “K teoremam Broudi i Grina ob usloviyakh polnoi giperbolichnosti kompleksnykh mnogoobrazii”, Funktsion. analiz, 14:4 (1980), 77–78 | MR | Zbl

[10] Zaidenberg M. G., “Teoremy Pikara i giperbolichnost”, Sib. matem. zhurn., 24:6 (1983), 44–45 | MR

[11] Kobayashi Sh., “Instrinsic distances, measures and geometric function theory”, Bull. Amer. Math., Soc., 82:3 (1976), 357–416 | DOI | MR | Zbl

[12] Ochiai T., “On holomorphic curves in algebraic varieties with ample irregularity”, Invent. Math., 43 (1977), 83–96 | DOI | MR | Zbl

[13] Kiernan P., Kobayashi Sh., “Holomorphic mappings into projective space with lacunary hyperplanes”, Nagoya Mat. J., 50 (1973), 199–216 | MR | Zbl

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

[15] Kiernan P., “Hyperbolic submanifolds of complex projective space”, Proc. Amer. Math. Soc., 22:3 (1969), 603–606 | DOI | MR | Zbl

[16] Royden H. L., “Remarks on the Kobayashi metric”, Lect. Notes in Math., 185 (1971), 125–137 | DOI | MR

[17] Royden H. L., “The extension of regular holomorphic maps”, Proc. Amer. Math. Soc., 43:2 (1974), 306–310 | DOI | MR | Zbl

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

[19] 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

[20] Green M. L., “Holomorphic maps to complex tori”, Amer. J. Math., 100:8 (1978), 615–620 | DOI | MR | Zbl

[21] Eastwood A., “A propos des varietes hyperboliques completes”, C. r. Acad. Sci., Ser. A, 280 (1975), 1071–1075 | MR

[22] Pelles D. A., “Holomorphic maps which preserve intrinsic measure”, Amer. J. Math., 97:1 (1975), 1–15 | DOI | MR | Zbl

[23] Wright M., “The Kobayashi pseudometric on algebraic manifold of general type and in deformations of complex manifolds”, Trans. Amer. Math. Soc., 232 (1977), 357–370 | DOI | MR | Zbl

[24] Kalka M., “A deformation theorem for the Kobayashi metric”, Proc. Amer. Mat. Soc., 59:2 (1976), 245–251 | DOI | MR | Zbl

[25] Siu Y.-T., “Every Stein subvariety admits a Stein neighborhoods”, Invent. Math., 38 (1976), 89–100 | DOI | MR | Zbl

[26] Hamm H., “Lokale topologische Eigenschaften komplexer Raume”, Math. Ann., 191 (1971), 235–252 | DOI | MR | Zbl

[27] Iitaka Sh., “Geometry of complements of lines in $P^2$”, Tokyo J. Math., 1:1 (1978), 1–19 | MR | Zbl

[28] Imayoshi Y., “Holomorphic families of Riemann surfaces and Teichmyller spaces”, Tôhoku Math. J., 31 (1979), 469–489 | DOI | MR | Zbl