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• W2076456931 abstract "Bloch, and later H. Cartan, showed that if H H, . I +2 are n + 2 hyperplanes in general position in complex projective space Pn, then Pn -Hi U ... U Hn+2 is (in current terminology) hyperbolic modulo A, where A is the union of the hyperplanes (H1 n... n Hk) D (Hk+ln n... Hn+2) for 2 2. Using the method of negative curvature, we give an explicit model for the Kobayashi metric when n = 2. Pour arriver a des resultats d'ordre quantatif, il faudrait construire... la fonction modulaire. Henri Cartan 1928 0. INTRODUCTION For a compact complex manifold M, Brody [5] has given a simple criterion for hyperbolicity: M is hyperbolic if and only if every holomorphic map f: C -* M is constant. For noncompact manifolds, hyperbolicity is much more difficult to establish, as Brody's Theorem is false (Green's example, see [14]). The situation is even more complicated for hyperbolicity modulo a subset, the most general case. For example, let D be a divisor on Pn, let X = Pn-D, and let Y be a proper subvariety of Pn . We say that X is hyperbolic modulo Y, if the Kobayashi distance dx satisfies dx(x, y) > 0 for all x, y distinct and not both in Y. If every nonconstant holomorphic map f: C X lies in the subvariety Y, then X is called Brody hyperbolic modulo Y [14]. It is unknown if Brody hyperbolicity modulo Y implies hyperbolicity modulo Y when X = Pn -D; a modification of Green's example shows this is false if Pn is replaced by an arbitrary variety. Furthermore, the results so far for hyperbolicity modulo Y are mostly qualitative, giving little insight into the quantitative behavior of the Kobayashi metric, which approaches 0 near Y in the tangent directions to Y. In this paper we give an explicit model for the Kobayashi metric on a particular X, namely P2 minus 4 lines. We prove that the model does indeed Received by the editors December 30, 1987 and, in revised form, September 1, 1988. 1980 Mathematics Subject Classification (1985 Revision). Primary 32H20. Supported in part by NSF grant DMS-8602197. i 1990 American Mathematical Society 0002-9947/90 $1.00+ $.25 perpage" @default.
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• W2076456931 date "1990-02-01" @default.
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• W2076456931 title "The method of negative curvature: the Kobayashi metric on ${bf P}sb 2$ minus $4$ lines" @default.
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• W2076456931 doi "https://doi.org/10.1090/s0002-9947-1990-0958888-x" @default.
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