Uniformization of Complex Surfaces

Abstract

This is an expository paper on the structure of complex surfaces which have the Hirzebruch proportionality $3c_2 = c_1^2$ or $2c_2 = c_1^2$ between their Chern numbers. We characterize surfaces with $3c_2 = c_1^2$ as ball quotients in the category of normal surfaces with branch loci. We discuss the uniformization problem for surfaces with $2c_2 = c_1^2$ from the point of view of Kähler–Einstein metrics and holomorphic conformal structures.