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. Our results apply in particular to spherical Artin groups, where the associated arrangement is a reflection arrangement of a finite Coxeter group.

中文翻译：

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超平面排列的波状化及其曲线复合体

关于超平面排列的补充 ${\mathbb{C}}^{ñ}$对具有歧管的流形具有自然的修饰，该歧管具有通过去除（或“吹起”）超平面的管状邻域及其某些相交点而形成的角。当这种安排是一个实际的简单安排的复杂化时，该协调非常类似于Harvey的模空间协调。我们证明了通用化掩盖的面孔是由单纯复形的简化参数化的$\mathcal{C}$，其顶点是该安排补集的基本组的不可约的“抛物线子组”。所以，复杂$\mathcal{C}$对于排列补码起着类似的作用，就像曲线复数对模空间一样。同样，类似于曲线复合体和球形建筑物，我们证明了$\mathcal{C}$具有球面楔形的同伦类型。我们的结果尤其适用于球形Artin组，其中相关的排列是有限Coxeter组的反射排列。