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Overview: Since the appearance of Kobayashi‘s book, there have been several re- sults at the basic level of hyperbolic spaces, for instance Brody‘s theorem, and results of Green, Kiernan, Kobayashi, Noguchi, etc. which make it worthwhile to have a systematic exposition. Although of necessity I re- produce some theorems from Kobayashi, I take a different direction, with different applications in mind, so the present book does not super- sede Kobayashi‘s. My interest in these matters stems from their relations with diophan- tine geometry. Indeed, if X is a projective variety over the complex num
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