Explicit equations of a fake projective plane

Abstract

Fake projective planes are smooth, complex surfaces of general type with Betti numbers equal to those of the usual projective plane. They come in complex conjugate pairs and have been classified as quotients of the $2$-dimensional ball by explicitly written arithmetic subgroups. In the following, we find equations of a projective model of a conjugate pair of fake projective planes by studying the geometry of the quotient of such surface by an order $7$ automorphism.