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 \(\overline{M}_{g, n}\) of stable genus g curves with n-marked points. Using the gluing theory for \(\overline{M}_{g, n} \) associated to horocycle structures, there is a natural orbifold gluing structure on \(\overline{M}_{g, n}\). We show this gluing atlas can be refined to provide a good orbifold gluing atlas and hence a smooth orbifold structure on \(\overline{M}_{g,n}\). 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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This work is supported by the Australian Research Council Grant and the National Natural Science Foundation of China Grant.
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Chen, B., Li, AM., Wang, BL. (2016). Gluing Principle for Orbifold Stratified Spaces. In: Futaki, A., Miyaoka, R., Tang, Z., Zhang, W. (eds) Geometry and Topology of Manifolds. Springer Proceedings in Mathematics & Statistics, vol 154. Springer, Tokyo. https://doi.org/10.1007/978-4-431-56021-0_2
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DOI: https://doi.org/10.1007/978-4-431-56021-0_2
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