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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!

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    $\begingroup$ Sure. There is even a book by Hirzebruch and his students on this. $\endgroup$
    – Moishe Kohan
    Commented Jun 8, 2022 at 1:05
  • $\begingroup$ Nice one, what's the name of the book? $\endgroup$
    – Selim G
    Commented Jun 8, 2022 at 9:27
  • $\begingroup$ Lev Borisov and collaborators have a series of papers on equations of fake projective planes arXiv:1710.04501, arXiv:1802.06333, arXiv:1804.00737, arXiv:2004.02637, arXiv:2008.09731, arXiv:2109.02070. $\endgroup$
    – Balazs
    Commented Jun 12, 2022 at 7:39

2 Answers 2

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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.

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.

Tretkoff, Paula, Complex ball quotients and line arrangements in the projective plane. With an appendix by Hans-Christoph Im Hof, 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.

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

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  • $\begingroup$ Thanks! Tretkoff's book is not open access, so I couldn't get much from these references unfortunately. It seems Hirzebruch in his original article builds only three such surfaces, do these methods yield more than that? Hopefully an infinity? $\endgroup$
    – Selim G
    Commented Jun 8, 2022 at 12:46
  • $\begingroup$ @SelimG: From what I know, there are only finitely many explicitly known commensurabilify classes. If you find one example, you get infinitely many by taking unramified finite coverings. $\endgroup$
    – Moishe Kohan
    Commented Jun 8, 2022 at 16:22
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Another relevant reference is:

Hirzebruch, F.(D-MPI) Chern numbers of algebraic surfaces: an example. Math. Ann.266(1984), no.3, 351–356.

The surface called W_3 there is an example.

For open surfaces (<=> nonuniform lattices) there are more examples, discussed in this paper of Hirzebruch and in the work of Holzapfel.

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