Volume 22, issue 5 (2018)

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Real line arrangements with the Hirzebruch property

Dmitri Panov

Geometry & Topology 22 (2018) 2697–2711
Abstract

A line arrangement of $3n$ lines in $ℂ{P}^{2}$ satisfies the Hirzebruch property if each line intersect others in $n+1$ points. Hirzebruch asked in 1985 if all such arrangements are related to finite complex reflection groups. We give a positive answer to this question in the case when the line arrangement in $ℂ{P}^{2}$ is real, confirming that there exist exactly four such arrangements.

Keywords
line arrangements, complex reflection groups, polyhedral manifolds, Kähler metrics
Mathematical Subject Classification 2010
Primary: 14N20, 32S22, 51F15, 52B70, 53C55
Secondary: 20F55, 32Q15