Topological Aspects of Chow Quotients

Abstract

This paper studies the canonical Chow quotient of a smooth projective variety by a reductive algebraic group. The main purpose is to give some topological interpretations and characterization of Chow quotient which have the advantage to be more intuitive and geometric. This is to be done over the field of complex numbers and in the languages that are familiar to topologists and differential geometers. More precisely, the main observation of this paper is that, over the field of complex numbers, the Chow quotient admits symplectic and other topological interpretations, namely, symplectically, the moduli spaces of stable orbits with prescribed momentum charges; and topologically, the moduli space of stable action-manifolds. In addition, we give a computable characterization of the Chow cycles of the Chow quotient, using the so-called perturbing-translating-specializing relation.

36 pages. Improved version. Introduction was completely rewritten with five newly added figures to help to illustrate the ideas. The introduction is written with nonspecialists in mind, in particular, a simple, yet quite informative example is used throughout to explain the ideas behind GIT quotients, the Chow quotient, and our topological approaches