PreprintCC BY 4.0arXiv
The complex projective plane as a ball quotient
Abstract
In 1986, Deligne and Mostow constructed a ball quotient 𝔹2 / Γ biholomorphic to the complex projective plane ℙ2 whose branch locus is a line arrangement. In this paper, we show that if ℙ2 is realized as a ball quotient whose branch divisor D is an arrangement of smooth pairwise normal-crossing curves, then the orbifold (ℙ2,D) is isomorphic to either the Deligne-Mostow example or a certain degree 9 cover of it. This classification of "ball quotient structures" on ℙ2 generalizes the ℙ1 case due to Poincaré.
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BibTeX
@misc{tan2026complex,
title = {{The complex projective plane as a ball quotient}},
author = {Cindy Tan},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2607.18710},
eprint = {2607.18710},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.18710},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The complex projective plane as a ball quotient
AU - Cindy Tan
PY - 2026
DA - 2026/09/28
PB - arXiv
DO - 10.48550/arxiv.2607.18710
UR - https://arxiv.org/abs/2607.18710
AB - In 1986, Deligne and Mostow constructed a ball quotient 𝔹^2 / Γ biholomorphic to the complex projective plane ℙ^2 whose branch locus is a line arrangement. In this paper, we show that if ℙ^2 is realized as a ball quotient whose branch divisor D is an arrangement of smooth pairwise normal-crossing curves, then the orbifold (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 ℙ^2 generalizes the ℙ^1 case due to Poincaré.
ER -