超平面构形的边化及其曲线复形
Bordifications of hyperplane arrangements and their curve complexes
摘要：
在$\mathbb C^n$中超平面排列的补集具有到流形的自然边界化，所述流形具有通过去除（或“吹胀”）超平面的管状邻域和它们的某些相交而形成的角。当排列是实单纯排列的复化时，边界化非常类似于模空间的哈维边界化。我们证明了边化的泛覆盖的面是由单纯复形$\mathcal{C}$的单形参数化的，其顶点是排列补的基本群的不可约“抛物子群“。因此，复形$\mathcal{C}$对于排列补的作用与曲线复形对于模空间的作用相似。此外，在类比与曲线复合体和球形建筑物，我们证明了$\mathcal{C}$具有同伦型的楔形球。
英文摘要：
The complement of an arrangement of hyperplanes in $\mathbb C^n$ has a natural bordification to a manifold with corners formed by removing (or "blowing up") tubular neighborhoods of the hyperplanes and certain of their intersections. When the arrangement is the complexification of a real simplicial arrangement, the bordification closely resembles Harvey's bordification of moduli space. We prove that the faces of the universal cover of the bordification are parameterized by the simplices of a simplicial complex $\mathcal{C}$, the vertices of which are the irreducible "parabolic subgroups" of the fundamental group of the arrangement complement. So, the complex $\mathcal{C}$ plays a similar role for an arrangement complement as the curve complex does for moduli space. Also, in analogy with curve complexes and with spherical buildings, we prove that $\mathcal{C}$ has the homotopy type of a wedge of spheres.
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