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Ramification conjecture and Hirzebruch’s property of line arrangements
- Part of
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- Journal:
- Compositio Mathematica / Volume 152 / Issue 12 / December 2016
- Published online by Cambridge University Press:
- 25 October 2016, pp. 2443-2460
- Print publication:
- December 2016
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- Article
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The ramification of a polyhedral space is defined as the metric completion of the universal cover of its regular locus. We consider mainly polyhedral spaces of two origins: quotients of Euclidean space by a discrete group of isometries and polyhedral metrics on
$\mathbb{C}\text{P}^{2}$ with singularities at a collection of complex lines. In the former case we conjecture that quotient spaces always have a 
$\text{CAT}[0]$ ramification and prove this in several cases. In the latter case we prove that the ramification is 
$\text{CAT}[0]$ if the metric on 
$\mathbb{C}\text{P}^{2}$ is non-negatively curved. We deduce that complex line arrangements in 
$\mathbb{C}\text{P}^{2}$ studied by Hirzebruch have aspherical complement.
