The complex projective plane as a ball quotient

Abstract

In 1986, Deligne and Mostow constructed a ball quotient ${\mathbb{B}}^2/\hat{I}“$ biholomorphic to the complex projective plane ${\mathbb{P}}^2$ whose branch locus is a line arrangement. In this paper, we show that if ${\mathbb{P}}^2$ is realized as a ball quotient whose branch divisor $D$ is an arrangement of smooth pairwise normal-crossing curves, then the orbifold $({\mathbb{P}}^2,D)$ is isomorphic to either the Deligne-Mostow example or a certain degree 9 cover of it. This classification of "ball quotient structures" on ${\mathbb{P}}^2$ generalizes the ${\mathbb{P}}^1$ case due to Poincaré.