Principal Investigator: David A. Gieseker and Huazhang Luo
In Geometric Invariant Theory, in order to study the moduli problems, the notion of stability for any polarized projective variety is introduced. However to check the stability is usually a difficult problem. It has been done by Mumford for curves with genus bigger than 1, by Gieseker for surfaces of general type, and by Viehweg for high dimensional varieties with semi-ample canonical line bundles. We will study the stability of a polarized smooth projective variety as used by Gieseker and Mumford from a differential geometric viewpoint. More specifically, we try to relate the Gieseker-Mumford stability to the existence of a special Kahler metric on the variety such that its scalar curvature is constant and its Ricci curvature satisfies some non-degenerate constraints. This metric is not Kahler-Einstein, but is closely related to Kahler-Einstein metric. One motivation for this project comes from the previous work done by Donaldson, Uhlenbeck and Yau on the relation between stability of vector bundle and the existence of Yang-Mills connection.
Our project is to relate two important but different fields in mathematics together. One is the study of special metrics, which involves the study of differential equations on manifold and usually has connection to physics. Another is the study of stability in moduli space theory which involves mainly algebra. Our approach is to establish a correspondence between these two fields, in particular to describe the meaning of stability in terms of the existence of special metrics. The advantage for doing is it will not only provide us with new insights about those special metrics, but also provide us with new technique to moduli space theory. For example, this kind of correspondence will enable us to solve some difficult mathematical questions such as checking the Gieseker-Mumford stability of some manifolds. This question is one of the fundamental questions in the moduli space theory but has not been solved in general.
