Geometry of the Moduli Spaces of Complex Manifolds

Lehigh University, Bethlehem PA

Investigators

Abstract

Moduli spaces of Calabi-Yau manifolds, Riemann surfaces and general type manifolds, stable bundles and stable maps have played fundamental roles in many subjects of mathematics from geometry, topology, algebraic geometry, to number theory and are also important in string theory. Mirror symmetry has motivated many deep mathematical results and theories. In this project the PI will study various local and global problems of the geometry of these moduli spaces. These problems include the deformation of canonical metrics such as the Kahler-Einstein metrics and Bergman metrics with respect to various gauges. This study lead to a geometric quantization of the Weil-Petersson type metrics. The PI will also study the global properties of the period map from the Teichmuller and moduli spaces to the classifying spaces of variation of Hodge structures and their quotients. The PI will also investigate the geometric properties on canonical and new Kahler metrics on these moduli spaces as well as the general existence of Kahler-Einstein and Hermitian-Einstein metrics on noncompact manifolds and bundles over them. Teichmuller and moduli spaces of Kahler-Einstein manifolds are important for both mathematics and theoretical physics. In string theory, many fundamental partition functions can be reduced to geometric or topological invariants of moduli spaces which also motivated many beautiful conjectures. The study of these important conjectures not only enhanced the development of various branches of mathematics but also help in verifying theory in high energy physics. The research of the PI also has potential applications in industry such as medical image processing and 3D cameras. For example, with the explicit construction of the period map the speed of compressing/decompressing pictures can be significantly increased.