The complex projective plane as a ball quotient

Abstract

In 1986, Deligne and Mostow constructed a ball quotient $\mathbb{B}^2 / \Gamma$ 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 Poincar\'e.