Orbifold分层空间的胶合原理
Gluing principle for orbifold stratified spaces
摘要：
本文研究了层化轨道空间的性质，建立了层化轨道空间是光滑轨道空间的一般判据。这个准则利用了轨道层空间的胶合丛上的线性层化的概念。我们引入了良好胶合结构的概念，以确保分层空间上的光滑结构。作为应用，我们给出了具有$n$标记点的稳定亏格$g$曲线的粗模空间$\bar{M}_{g, n}$上的轨道结构.根据$\bar{M}_{g, n} $与等环结构的胶合理论，在$\bar{M}_{g, n} $上存在一个自然的轨道胶合图谱。我们表明，这种胶合图谱可以细化，以提供一个良好的orbifold胶合结构，因此$\bar{M}_{g,n}$上的光滑orbifold结构。这个一般的胶合原理对于研究辛流形中稳定伪全纯曲线的紧化模空间的胶合理论是非常有用的。
英文摘要：
In this paper, we explore the theme of orbifold stratified spaces and establish a general criterion for them to be smooth orbifolds. This criterion utilizes the notion of linear stratification on the gluing bundles for the orbifold stratified spaces. We introduce a concept of good gluing structure to ensure a smooth structure on the stratified space. As an application, we provide an orbifold structure on the coarse moduli space $\bar{M}_{g, n}$ of stable genus $g$ curves with $n$-marked points. Using the gluing theory for $\bar{M}_{g, n} $ associated to horocycle structures, there is a natural orbifold gluing atlas on $\bar{M}_{g, n} $. We show this gluing atlas can be refined to provide a good orbifold gluing structure and hence a smooth orbifold structure on $\bar{M}_{g,n}$. This general gluing principle will be very useful in the study of the gluing theory for the compactified moduli spaces of stable pseudo-holomorphic curves in a symplectic manifold.
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