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Journal Article

Bordifications of hyperplane arrangements and their curve complexes


Huang,  Jingyin
Max Planck Institute for Mathematics, Max Planck Society;

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Davis, M. W., & Huang, J. (2021). Bordifications of hyperplane arrangements and their curve complexes. Journal of Topology, 14(2), 419-459. doi:10.1112/topo.12184.

Cite as: http://hdl.handle.net/21.11116/0000-0008-FF83-5
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.