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On the $K(\unicode[STIX]{x1D70B},1)$ -problem for restrictions of complex reflection arrangements

  • Nils Amend (a1), Pierre Deligne (a2) and Gerhard Röhrle (a3)

Abstract

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.

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On the $K(\unicode[STIX]{x1D70B},1)$ -problem for restrictions of complex reflection arrangements

  • Nils Amend (a1), Pierre Deligne (a2) and Gerhard Röhrle (a3)

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