Topological Aspects of Chow Quotients and Moduli Spaces

University Of Arizona, Tucson AZ

Investigators

Abstract

DMS-0104600 Yi Hu The investigator proposes to give topological constructions of the Chow quotient by introducing moduli spaces of stable orbits with prescribed momentum charges and moduli spaces of stable action-manifolds, respectively. The introductions of stable orbits and stable action-manifolds follow closely the idea of investigator's earlier work on stable degeneration of polygons. Note that just like a stable polygon is a collection of ordinary polygons, a stable orbit is a collection of ordinary orbits. The connections from stable orbits and stable action-manifolds to the corresponding Chow cycles are to be established by moment map. Next, he also attempts to interpret the Uhlenbeck compactification as a topological analogy of a Chow quotient. In the same vein, the investigator proposes a moduli problem for stable tuples of homogeneous polynomials of a fixed total degree in two variables. The investigator anticipates that the resulting moduli space is the smooth compactification of the variety of holomorphic maps from one-dimensional projective space to n-dimensional projective space as constructed in his earlier work on blowups along arrangements of subvarieties. This proposal deals with the fruitful interaction between topology and algebraic geometry. Algebraic Geometry as a field is both old and new. It is old because of its long history, it is new because it is still vigorously developing as one of the main stream mathematics. It has numerous beautiful connections and applications to other subjects. Roughly speaking, Algebraic Geometry studies spaces which are locally the set of solutions of some polynomial equations. Topology is another amazing mathematical achievement of this century and is the study of flexible geometrical properties of spaces. One of central problems in algebraic geometry is the classification of spaces in algebraic geometry. A key idea here is to treat every type of spaces as a point in a "master set" which is technically called a moduli space. This paper focus on the topological and geometrical aspects of moduli spaces. The proposal consists of some new ideas and methods which are substantially based upon investigator's earlier papers, among others.