Mathematical Sciences: Complex Manifolds and Meromorphic Mappings

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This project continues mathematical research into problems associated with the geometry of several complex variables and mappings defined on regions within such spaces. The work focuses on complex varieties, meromorphic mappings and holomorphic vector bundles. Work on meromorphic mappings seeks generalizations of Hartogs' theorems on separate analyticity and on holomorphic extendibility. Also to be investigated are Hartogs-type results for analytic subvarieties with possible application to the characterization of holomorphic chains of codimension greater than unity. Studies of holomorphic vector bundles concerns singular hermitian metrics on numerically effective line bundles on projective-algebraic manifolds. A continuation of work on cohomology vanishing theorems and sectional curvatures of indecomposable Hermitian-Einstein vector bundles on complex projective space will be carried out. Finally, the geometry of affine algebraic varieties and degree bounds for the division problem for special polynomial ideals will be studied. Complex function theory encompasses the study of differentiable functions of a several complex variables and related classes of functions such as boundary values of holomorphic functions. The subject is highly geometric; many of the problems concern the properties of various sets and how they transform by mappings composed of holomorphic functions. Applications of the theory to potential theory and fluid dynamics is now standard in engineering circles.