Special Complex Surfaces, Moduli Spaces, and Some Analytic Approach

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The Principle Investigator proposes to work on three directions in algebraic and complex geometry. The first is on investigation of properties related to hyperbolicity of moduli spaces of canonically polarized manifolds, such as a conjecture of Viehweg that the spaces are of log-general type. The idea is to study moduli of higher dimensional varieties guided by principles known for the moduli spaces of curves, though completely new approaches have to be taken. The main tools to be used are algebraic geometric and differential geometric in nature. The second is on the classification and understanding of special surfaces, such as fake projective planes, exotic quadrics, and complex ball quotients in general. Fake projective planes have been classified algebraically by Gopal Prasad and the PI in a project partially supported by a prior NSF grant. The goal here is to understand them geometrically. The project involves techniques from algebraic geometry, number theory, Bruhat-Tits theory and computer aided computations. The third is on development and application of analytic tools to problems which are algebraic or topological in nature. This includes a quest for classification of exotic quadrics and development of analytic tools for existence of rational curves on some varieties. The main techniques to be used are analytic in nature, such as estimates in partial differential equations and geometric analysis.An interesting aspect of mathematics is its intrinsic coherence and beauty. Time and again, ideas and tools from different areas of mathematics are brought together to solve seemingly unrelated problems spectacularly. A goal of the proposal is to study more in this direction by developing and applying techniques from various mathematical disciplines to understand various algebraic and complex geometric problems, some of which are long standing. It varies from investigation of general properties of moduli spaces which are collections of varieties of the same topological type, to detailed study and classification of special varieties such as fake projective planes and exotic quadrics which are algebraic surfaces with small topological invariants. The PI expects his graduate students to be actively involved in the mathematical projects proposed and to learn to appreciate the process of research. He also hopes that the interests in mathematics may be passed to other students through seminar talks and summer school presentations, both in US and in foreign countries.