1 results
On the
$K(\unicode[STIX]{x1D70B},1)$-problem for restrictions of complex reflection arrangements
- Part of
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- Journal:
- Compositio Mathematica / Volume 156 / Issue 3 / March 2020
- Published online by Cambridge University Press:
- 20 January 2020, pp. 526-532
- Print publication:
- March 2020
-
- Article
- Export citation
-
Let
$W\subset \operatorname{GL}(V)$ 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 
$K(\unicode[STIX]{x1D70B},1)$ 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 
$K(\unicode[STIX]{x1D70B},1)$. 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 
$K(\unicode[STIX]{x1D70B},1)$ property remains to be proved.
