Covering relations between ball-quotient orbifolds (Q1434160)

The author discusses algebraic varieties with complex 2-ball \(B_2\) covering. Let \(M\) be a smooth algebraic surface of general type. If \(M\) is covered by the complex 2-ball \(B_2\), then its Chern number satisfies the equality \(c^2_1(M)=3e(M)\). Conversely if the Chern number of \(M\) satisfies this equality then the universal covering of \(M\) is \(B_2\). A branched Galois covering \(\varphi:M\to X\) endows \(X\) with a map \(\beta_\varphi: X\to\mathbb{N}\) sending \(p\in X\) to the order of the isotropy group above \(p\). The pair \((X,\beta_\varphi)\) is an orbifold, and \(M\) is a uniformization of \((X,\beta_\varphi)\). If the degree of \(\varphi\) is finite then \(M\) is a uniformization of \((X,\beta_\varphi)\), and if \(M\) is simply connected, it is called the universal uniformization of \((X, \beta_\varphi)\). In these cases, we say that \((X,\beta_\varphi)\) is uniformized by \(M\). The author proves that there exists infinite series of pairwise non-isomorphic orbifolds uniformized by the complex 2-ball \(B_2\). The corresponding lattices in \(SU(1,2)\) are all arithmetic. The author also gives an infinite series of irreducible curves along which \(\mathbb{P}^2\) is uniformized by the product of two complex 1-balls \(B_1 \times B_1\).