Topological Aspects of Chow Quotients and Moduli Spaces

本文通过引入具有规定动量电荷的稳定轨道的模空间和稳定流形的模空间，分别给出了周商的拓扑结构。稳定轨道和稳定动作流形的引入与研究者早期关于多边形稳定退化的研究思想密切相关。注意，就像一个稳定多边形是普通多边形的集合一样，一个稳定轨道是普通轨道的集合。用矩图建立了稳定轨道和稳定作用流形与相应周氏环的联系。接下来，他还试图将乌伦贝克紧化解释为周氏商的拓扑类比。在同样的精神，研究者提出了一个模问题的稳定元组的齐次多项式的一个固定的总度在两个变量。研究者预期得到的模空间是从一维射影空间到n维射影空间的各种全纯映射的光滑紧化，正如他在早期关于沿子变体排列的膨胀的工作中构造的那样。这个建议处理拓扑和代数几何之间富有成效的相互作用。代数几何作为一个领域既古老又新颖。它是古老的，因为它的历史悠久，它是新的，因为它仍然蓬勃发展的主流数学之一。它与其他学科有许多美妙的联系和应用。粗略地说，代数几何研究的空间是局部多项式方程解的集合。拓扑学是本世纪另一个惊人的数学成就，是对空间的柔性几何性质的研究。代数几何的中心问题之一是代数几何空间的分类问题。这里的关键思想是将每种类型的空间视为“主集合”中的一个点，技术上称为模空间。本文主要讨论模空间的拓扑和几何问题。该提案包含了一些新的想法和方法，这些想法和方法在很大程度上是基于研究者早期的论文。

DMS-0104600Yi HuThe investigator proposes to give topological constructions ofthe Chow quotient by introducing moduli spaces of stable orbitswith prescribed momentum charges and moduli spaces of stableaction-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 solutionsof 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 consistsof some new ideas and methods which are substantially based upon investigator's earlier papers, among others.