Real line arrangements with the Hirzebruch property

Abstract

A line arrangement of $3n$ lines in $ℂP^2$ 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 $ℂP^2$ is real, confirming that there exist exactly four such arrangements.