Riemannian metrics on complemements of subvarieties

Let C be a smooth complex projective curve and D a finite, possibly empty, collection of distinct points on C. Classical uniformization says that M = C-D admits a complete metric of finite volume and constant curvature -1 precisely when the Euler number e(M) is negative. In other words, there is a purely topological necessary and sufficient condition.

I will discuss topological methods for proving existence of a metric of constant holomorphic sectional curvature -1 on the complement of curves in a smooth complex projective surface, and some applications. I will mainly focus on an interesting example due to Hirzebruch. This is mostly joint with Luca Di Cerbo.