Let $S$ be a compact complex surface. It is well-known that the following two facts are equivalent

1. $c_1^2(S) = 3 c_2(S)$

2. The universal cover of $S$ is biholomorphic to the unit ball in $\mathbb{C}^2$.

The unit ball in $\mathbb{C}^2$ is biholomorphic the complex hyperbolic plane $\mathbf{H}^2_{\mathbb{C}}$ and its group of biholomorphisms is $\mathrm{PU}(1,2)$. Consequently, a compact complex surface is the same thing as the datum of a torsion-free, co-compact lattice in $\mathrm{PU}(1,2)$.

My question is : are there constructions of such surfaces that do not transit through lattices in $\mathrm{PU}(1,2)$? For instance, are there constructions of surfaces satisfying $c_1^2(S) = 3 c_2(S)$ coming from algebraic geometry, or from ramified covering-type of construction?

Thanks!

Let $S$ be a compact complex surface. It is well-known that the following two facts are equivalent

1. $c_1^2(S) = 3 c_2(S)$ and $S \neq \mathbb{CP}^2$

2. The universal cover of $S$ is biholomorphic to the unit ball in $\mathbb{C}^2$.

The unit ball in $\mathbb{C}^2$ is biholomorphic the complex hyperbolic plane $\mathbf{H}^2_{\mathbb{C}}$ and its group of biholomorphisms is $\mathrm{PU}(1,2)$. Consequently, a compact complex surface is the same thing as the datum of a torsion-free, co-compact lattice in $\mathrm{PU}(1,2)$.

My question is : are there constructions of such surfaces that do not transit through lattices in $\mathrm{PU}(1,2)$? For instance, are there constructions of surfaces satisfying $c_1^2(S) = 3 c_2(S)$ coming from algebraic geometry, or from ramified covering-type of construction?

Thanks!