Mathematical Sciences: Complex Manifold Theory and Kaehler Geometry

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8907582 Siu The object of this project is to consider several problems in complex differential geometry. They include analysis of functions and mappings defined on domains in the space of several complex variables as well as studies of the geometry of the domains and their boundaries. Work to be done will include an effort to determine whether the limit of a family of compact complex manifolds which are all biholomorphic images of the same irreducible Hermitian symmetric manifold is again a manifold of this type. New techniques for attacking this problem have been introduced recently. In addition, a complete proof for the case of complex projective space has been givng. This work - the global nondeformability of irreducible Hermitan syymetric manifolds of compact type is a weakened version of one of the unresolved parts of the theory of strong rigidity. The purpose of the theory is to show that certain compact complex manifolds can be characterized either by topological or curvature conditons. A second line of investigation concerns continuing research on the higher dimensional Nevanlinna theory of value distribution of holomorphic functions. The equidimensional case has been highly developed and is reasonably well understood. In this work, the nonequidimensional will be considered. Past methods do not work in this setting and newer methods of holomorphic connections will probably have to be used instead. Results in this area have direct connections with number theory. Additional work will be done on the construction of bounded holomorphic functions. In one sense, the classical uniformization theorem answers any such question in one dimension since any simply connected Riemann surface is either equivalent to the sphere, the plane or a disc. The same classification question concerning the universal covering space for compact negatively curved Kahler manifolds has never been resolved. Equivalently, one asks whether the covering space admits a large number of bounded holomorphic functions. Except for a few special cases, no one has produced any such functions (other than constants). The recent results of Kohn-Fefferman on Holder estimates of projections into spaces of holomorphic functions holds out the possibility that their method may be transplanted to manifolds and lead to a technique for construction of the desired functions. If that becomes a reality, the likelihood of an attack on the Corona problem in higher dimensions would finally be realized.