Gluing principle for orbifold stratified spaces

TL;DR

This paper develops a general gluing principle for orbifold stratified spaces, providing criteria and structures to establish smooth orbifold structures, with applications to moduli spaces of stable curves.

Contribution

It introduces a new gluing principle and good gluing structures for orbifold stratified spaces, enabling the construction of smooth orbifold structures on complex moduli spaces.

Findings

Established a criterion for orbifold stratified spaces to be smooth orbifolds.

Constructed an orbifold structure on the moduli space ar;M_{g,n} using a refined gluing atlas.

Provided a framework applicable to the gluing theory of moduli spaces of pseudo-holomorphic curves.

Abstract

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.