Ramification conjecture and Hirzebruch's property of line arrangements. (25th October 2016)

• Record Type: Journal Article
• Authors: Panov, D.; Petrunin, A.
• Is Part Of: Compositio mathematica. Volume 152:Number 12(2016)
• Page Start: 2443
• Page End: 2460
• Publication Date: 2016-10-25
• Subjects: 51K10, -- 53C23, -- 57N65, -- 20F36; Alexandrov geometry, -- polyhedral spaces, -- CAT[0] spaces, -- hyperplane arrangements; Mathematics -- Periodicals; 510
• DOI: 10.1112/S0010437X16007648
• Languages: English
• ISSNs: 0010-437X

Abstract

Abstract : 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.