Moduli and Surfaces in Complex Geometry

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To understand properties of an object in mathematics or physics, sometimes it is more enlightening to put all objects of similar nature together and study properties on the resulted collection as a whole. This is termed as moduli in algebraic and complex geometry. The concept is used in quantum physics and string theory as well. The first part of the project is to understand properties of moduli of some naturally existing geometric objects. On the other hand, sometimes a geometric object with special properties such as existence of extra symmetries or special type of singularities provide good models for understanding of various subjects in science. The second part of the project is aimed at finding special complex surfaces or higher dimensional geometric objects that would capture some properties that mathematicians have been seeking after. Since a wide range of tools are to be used, including computer implementation, a significant portion of the proposed activities can be used to train students or collaborate with young scholars.In more concrete terms, the principal investigator proposed to conduct research in several problems related to complex and algebraic geometry. The first part of the proposal details possible applications and extensions of some analytic tool developed recently by Wing-Keung To and the investigator. The new tools allow them to approach problems related to moduli space of higher dimensional polarized complex manifolds in a new and efficient way. The project aims at exploring and understanding moduli of general polarized manifolds in comparison with analytic, geometric and arithmetic properties of moduli space of curves. The second part of the project aims at applying the classification of fake projective planes by Gopal Prasad and the principal investigator as building blocks to investigate various algebraic geometric or complex geometric open problems. It includes construction of some unknown examples with small canonical degree, construction of exotic symplectic manifolds of small invariants, more geometrical realization of fake projective planes and classification of arithmetic fake compact Hermitian symmetric spaces.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.