On the $K(\pi, 1)$-problem for restrictions of complex reflection arrangements

Abstract

Let $W\subset GL(V)$ be a complex reflection group, and ${\mathscr A}(W)$ the set of the mirrors of the complex reflections in $W$. It is known that the complement $X({\mathscr A}(W))$ of the reflection arrangement ${\mathscr A}(W)$ is a $K(\pi,1)$ space. For $Y$ an intersection of hyperplanes in $\mathscr A(W)$, let $X(\mathscr A(W)^Y)$ be the complement in $Y$ of the hyperplanes in $\mathscr A(W)$ not containing $Y$. We hope that $X(\mathscr A(W)^Y)$ is always a $K(\pi,1)$. We prove it in case of the monomial groups $W = G(r,p,\ell)$. Using known results, we then show that there remain only three irreducible complex reflection groups, leading to just eight such induced arrangements for which this $K(\pi,1)$ property remains to be proved.