Hyperbolic Properties of Families of Polarized Manifolds and Problems Related to Fake Compact Hermitian Symmetric Spaces

Purdue University, West Lafayette IN

Investigators

Abstract

To understand the mathematical properties of a geometric object, often one needs to group families of objects of similar nature together and study a family as a whole. This concept arises naturally in quantum physics and string theory as well. A large part of this research project is devoted to the understanding the geometric properties of the parameter space inherited indirectly from the objects that it parameterizes. The Principal Investigator plans to investigate universal properties of such families in terms of various notions of negativity in curvature. Some of the problems to be studied are longstanding ones in algebraic and complex geometry. Another line of research is on the classification of some special geometric objects known as fake compact Hermitian symmetric spaces. These objects provide interesting geometric models for studying a wide range of problems from different areas of mathematics, including algebraic geometry, differential geometry, number theory, and representation theory. More specifically, the Principal Investigator will pursue research in several directions in algebraic and complex geometry. In the first part of the project, he intends to apply recently-developed tools involving generalized Weil-Petersson metrics to understand various hyperbolicity properties of specific families of polarized manifolds, including the hyperbolicity for families of Kahler Ricci flat manifolds, families of special log-general type manifolds, and families of general-type manifolds. In particular, he will investigate a conjecture of Viehweg about log-general type properties of moduli spaces of canonically polarized manifolds. In the second part, he plans to complete a collaborative project on the classification of arithmetic fake compact Hermitian symmetric spaces, a natural continuation of earlier work on fake projective planes. In addition, he will study problems related to these fake structures, such as the investigation of exceptional collections on fake projective planes from the point of view of derived categories, the characterization of the universal covering of a complex exotic quadric or a fake quadric, and the understanding of geometric properties of Cartwright-Steger surfaces.