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By Michael Spivak

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Extra info for A Comprehensive Introduction to Differential Geometry, Vol. 5, Third Edition

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T u We now state and give a sketch of the proof of the Ahlfors-Schwarz lemma, in order to generalize the preceding proposition to higher order jets. 2 (Ahlfors-Schwarz). Let . / D i 0 . / d ^d be a hermtian metric on R , where log 0 is a subharmonic function such that i @@ log 0. / A . / in the sense of distribution, for some positive constant A. Then with the Poincaré metric of R as follows: . 1 j j2 =R2 /2 Proof. Assume first that 0 is smooth and defined on R . 1 j j2 =R2 /2 0 is maximum. Then its logarithmic i @@-derivative at 0 must be nonpositive, hence i @@ log 0.

C the normalization. D/. Then, we have the next. 3. C; D/ " deg! C/ X not contained in D. As in the compact case, analytic and algebraic hyperbolicity are closely related. 7 ([47]). X; D/ be a log-manifold such that X nD is hyperbolic and hyperbolically embedded in X. X; D/ is algebraically hyperbolic. The algebraic version of the Kobayashi conjecture is also verified. 8 ([47]). Let Xd Pn be a very generic hypersurface of degree n n d 2n C 1 in P . P ; Xd / is algebraically hyperbolic. 3 A brief history of the above results The chronicle of the above results about algebraic hyperbolicity is the following.

Xk over X, which identifies reg Jk V reg =Gk with Xk . • The direct image sheaf . Ek;m V / can be identified with the sheaf of holomorphic sections of Ek;m V . Let us say a few words about this result. 0/ is independent of the choice of the representative f . f ı '/Œk D fŒk ı ', we get a well-defined map B reg Jk V reg =Gk ! Xk : This map can be described explicitly in local coordinates. z1 ; : : : ; zn / near x0 2 X such that Vx0 D Vect @z1 ; : : : ; @zr . t/ D t. Xk is a k-stage tower of Pr 1 -bundles.

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