Finite complex reflection arrangements are $K(\pi,1)$

Abstract

Let $V$ be a finite dimensional complex vector space and $W\subseteq \mathrm{GL}(V)$ be a finite complex reflection group. Let $V^{\rm reg}$ be the complement in $V$ of the reflecting hyperplanes. We prove that $V^{\rm reg}$ is a $K(\pi,1)$ space. This was predicted by a classical conjecture, originally stated by Brieskorn for complexified real reflection groups. The complexified real case follows from a theorem of Deligne and, after contributions by Nakamura and Orlik-Solomon, only six exceptional cases remained open. In addition to solving these six cases, our approach is applicable to most previously known cases, including complexified real groups for which we obtain a new proof, based on new geometric objects. We also address a number of questions about $\pi_1(W\backslash V^{\rm reg})$, the braid group of $W$. This includes a description of periodic elements in terms of a braid analog of Springer’s theory of regular elements.