Mathematical Sciences: Complex Hyperbolicity and Manifolds of Negative Curvature

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9505067 Yeung The proposed research lies in complex differential geometry. The investigator seeks to prove that the complement of a generic hyperplane of sufficiently high degree in projective space is hyperbolic. This is a natural extension of the classical Picard theorem in function theory; was proved in dimension two recently by Yeung and Siu. Many problems in the theory of complex hyperbolic manifolds can be viewed as value distribution problems, and as such they relate also to diophantine approximation theory. Diophantine approximations are an important subarea of number theory dealing with integer equations; mathematicians have recently uncovered a deep and unexpected relation between diophantine equations and complex analytic geometry.