A number of compactifications familiar in complex-analytic geometry, in particular the Baily–Borel compactification and its toroidal variants, as well as the Deligne–Mumford compactifications, can be covered by open subsets whose nonempty intersections are classified by their fundamental groups.We exploit this fact to define a ‘stacky homotopy type’ for these spaces as the homotopy type of a small category. We thus generalize an old result of Charney–Lee on the Baily–Borel compactification of $\mathcal{A}_g$ and recover (and rephrase) a more recent one of Ebert–Giansiracusa on the Deligne–Mumford compactifications. We also describe an extension of the period map for Riemann surfaces (going from the Deligne–Mumford compactification to the Baily–Borel compactification of the moduli space of principally polarized varieties) in these terms.

中文翻译：

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Baily-Borel和同盟紧缩的同伦类型

复杂解析几何中熟悉的许多压缩，特别是Baily-Borel压缩及其环形变体，以及Deligne-Mumford压缩，都可以用开放子集覆盖，这些子集的非空交集按其基本组分类。这个事实为这些空间定义了一个“堆叠同伦类型”，作为一小类的同伦类型。因此，我们概括了Charney–Lee在$ \ mathcal {A} _g $的Baily–Borel压缩中的旧结果，并恢复（并重新措辞了）关于Deligne–Mumford压缩的最新的Ebert–Giansiracusa。我们还用这些术语描述了黎曼表面周期图的扩展（从Deligne-Mumford压实到主要极化变种模量的Baily-Borel压实）。