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Algebraic families of smooth hyperbolic surfaces of low degree in $$\mathbb{P}_ \mathbb{C}^ 3$$. (English) Zbl 0887.32007
In [S. Kobayashi, Hyperbolic manifolds and holomorphic mappings, Marcel Dekker, New York (1970; Zbl 0207.37902)], Kobayashi conjectured that a generic hypersurface of $$\mathbb{P}_{\mathbb{C}}^n$$ of sufficiently high degree $$d$$ (expected bound being $$d \geq 2n-1$$) is hyperbolic. It is true for $$n=2$$, but for $$n>2$$ only some examples are known. For $$n=3$$, Brody and Green obtained the first example of a smooth hyperbolic surface in $$\mathbb{P}_{\mathbb{C}}^3$$ of any degree $$d=2p \geq 50$$ [R. Brody and M. Green, Duke Math. J. 44, 873-874 (1977; Zbl 0383.32009)], and Nadel obtained such examples of any degree $$d=6p+3\geq 21$$ [A. M. Nadel, Duke Math. J. 58, No. 3, 749-771 (1989; Zbl 0686.32015)].
In the present paper, the author improves Nadel’s technique and produces examples of smooth hyperbolic surfaces in $$\mathbb{P}_{\mathbb{C}}^3$$ of arbitrary degree $$d \geq 14$$, which is closer to the expected bound $$5$$. Furthermore, if $$H_{3,d}$$ is the subset of all hyperbolic surfaces in the projective space $$P_{3,d}$$ of all surfaces of degree $$d$$ in $$\mathbb{P}_{\mathbb{C}}^3$$, and if $$d>9+ \sum_{i=0}^3 k_i$$ where at least two of the integers $$k_i$$ are $$\geq 2$$, then the author proves that $$H_{3,d}$$ contains a Zariski open subset of $$P_{3,d}$$ of dimension $$\sum_{i=0}^3 \binom{k_i+3}{k_i} -1$$.

##### MSC:
 32Q45 Hyperbolic and Kobayashi hyperbolic manifolds 14H10 Families, moduli of curves (algebraic) 14H20 Singularities of curves, local rings 14N10 Enumerative problems (combinatorial problems) in algebraic geometry
##### Keywords:
hyperbolic surfaces; complex projective 3-space
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##### References:
 [1] [Be78]Beauville, A.–Surface algébrique complexes, Astérisque 54, 1978 [2] [Br78]Brody, R..–Compact manifolds and hyperbolicity, Trans. Amer. Math. Soc.,235 (1978), 213–219 [3] [BG77]Brody, R. &Green, M..–A family of smooth hyperbolic hypersufaces in $$\mathbb{P}$$ $$\mathbb{C}$$ 3 . Duke Math. J.,44 (1977), 873–874 · Zbl 0383.32009 [4] [De95]Demailly, J.-P.–Algebraic criteria for Kobayashi hyperbolic projective varieties and jet differentials, Proceedings of the AMS Summer Institute on Alg. Geom. held at Santa Cruz, ed. J. Kollar, July 1995, To appear [5] [GG79]Green, M. &Griffiths, P..–Two applications of algebraic geometry to entire holomorphic mappings, The Chern Symposium, 1979, (Proc. Internat. Sympos. Berkley, Calif., 1979), Springer-Verlag, New York, 1980, 41–74 [6] [Ko70]Kobayashi, S..–Hyperbolic manifols, and holomorphic mappings, Marcel Dekker, New York, 1970 [7] [Ko76]Kobayashi, S..–Intrinsic distances, mesures and geometric function theory, Bull. Amer. Math. Soc.,82 (1976), 357–416 · Zbl 0346.32031 [8] [La86]Lang, S..–Hyperbolic and Diophantine analysis, Bull. Amer. Math. Soc.,14 (1986), 159–205 · Zbl 0602.14019 [9] [MN93]Masuda, K. & Noguchi, N..–A construction of hyperbolic hypersurface of $$\mathbb{P}$$ $$\mathbb{C}$$ n , Preprint Tokyo Inst. Technology, Ohokayama, Tokyo, (1993) [10] [Mi68]Milnor, J..–Singular points of complex hypersurfaces, Princeton Univ. Press61 (1968) · Zbl 0184.48405 [11] [Na89]Nadel, A.M..–Hyperbolic surfaces in $$\mathbb{P}$$ $$\mathbb{C}$$ 3 , Duke Math. J.,58 (1989), 749–771 · Zbl 0686.32015 [12] [Si87]Siu, Y.T..–Defect relations for holomorphic maps between spaces of different dimensions, Duke Math. J.,55 (1987), 213–251 · Zbl 0623.32018 [13] [Za89]Zaidenberg, M.G..–Stability of hyperbolic imbeddedness and construction of examples, Math. USSR Sbornik,63 (1989), 351–361 · Zbl 0668.32023
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