Locally Hermitian symmetric spaces, non-positive curvature and complex hyperbolicity

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Abstract for DMS - 0104089The focus of the proposal is on the research of locally Hermitian symmetricspaces of non-compact type, considered as special complex manifolds of non-positive sectional curvature. Complex hyperbolicity can be considered as a weak notion of negative curvature. From analytic point of view, the principal investigator proposes to work on the rigidity of tangent bundles among the moduli space of vector bundles of locally Hermitian symmetric spaces.From the uniformization point of view, he proposes to work on characterization of arithmetic lattices in complex two balls. He proposes to investigate the analogueof Fijita's conjectures concerning very ampleness of pluricanonical line bundles on such manifolds, relating to algebraic geometry. He proposes to understand and estimate the growth of Betti numbers on a tower of coverings naturally associated with the manifolds, relating topology to geometry. In the direction of number theory, he proposes to study the question on finiteness of rational points on models of two ball quotients defined over an algebraic number field. He also proposes to work on several other problems in the area of complex hyperbolicityand Kaehler geometry, relating to his earlier research directions.In mathematics, people are interested in models which on one hand are elegant and relatively simple to describe and on the other hand display rich mathematicalstructures. Locally Hermitian symmetric spaces are such nice geometric models forwhich different disciplines of mathematics meet. The main purpose of this proposal is to understand several geometric and arithmetic aspects of such modelsand explain their interrelationships. Understanding of the various propertiesand problems proposed will enhance the knowledge of the subject and other disciplines of mathematics involved.