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!



Constructions of complex surfaces covered by the ball of $\mathbb{C}^2$
Below are two book references, both originating in the 1983 paper by Hirzebruch:


*Hirzebruch, Friedrich*, Arrangements of lines and algebraic surfaces, Arithmetic and geometry, Pap. dedic. I. R. Shafarevich, Vol. II: Geometry, Prog. Math. 36, 113-140 (1983). [ZBL0527.14033](https://zbmath.org/?q=an:0527.14033).


*Barthel, Gottfried; Hirzebruch, Friedrich; Höfer, Thomas*, Geradenkonfigurationen und algebraische Flächen. (Configurations of lines and algebraic surfaces). Eine Veröffentlichung des Max-Planck-Instituts für Mathematik, Bonn, Aspects of Mathematics, D4. Braunschweig/Wiesbaden: Friedr. Vieweg & Sohn. XII, 308 S.; (1987). [ZBL0645.14016](https://zbmath.org/?q=an:0645.14016).


*Tretkoff, Paula*, [**Complex ball quotients and line arrangements in the projective plane. With an appendix by Hans-Christoph Im Hof**](http://dx.doi.org/10.1515/9781400881253), Mathematical Notes (Princeton) 51. Princeton, NJ: Princeton University Press (ISBN 978-0-691-14477-1/pbk; 978-1-400-88125-3/ebook). ix, 215 p. (2016). [ZBL1342.14001](https://zbmath.org/?q=an:1342.14001).


In this approach, ball quotients are constructed via a combination of blow-ups and (finite) ramified coverings of $P^2$.


