On general type surfaces with \(q=1\) and \(c_2=3p_g\) (Q1740379)

The article under review aims at the classification of complex surfaces of general type \(S\) with irregularity \(q(S) = 1\) satisfying \(c_2(S) = 3p_g(S)\). This is motivated by the well-known inequality \(3p_g(S)\leq c_2(S)\) for surfaces of general type with \(q(S) = 1\), which is a special case of the Bogomolov-Miyaoka-Yau inequality. The autor constructs a sequence of general type surfaces \(S_n\), \(n\geq 1\), such that \(q(S_n) = 1\) and \(c_2(S_n) = 3p_g(S_n) = 3n\). That is, the (finite) moduli spaces \(\mathcal M_n\) of surfaces of general type with these invariants are inhabited. Moreover, it is shown that the number of elements of \(\mathcal M_n\) is bounded by \(C_1 n < \#\mathcal M_n < e^{C_2 n}\) for universal constants \(C_1,C_2\) (independent of \(n\)). The first surface \(S_1\) is the so-called Cartwright-Steger surface (see, e.g., [\textit{D. I. Cartwright} et al., J. Algebr. Geom. 26, No. 4, 655--689 (2017; Zbl 1375.14120)]) and \(S_n\) arises as the abelian cover of \(S_1\) associated with any surjective group homomorphism \(\pi_1(S_1)\to G\) onto a finite abelian group \(G\) of order \(n\). In fact, it is shown that every finite étale abelian cover of the Cartwright--Steger surface has irregularity \(1\). Using the fact that the surfaces under investigation must be ball-quotients and applying Mostow-Siu rigidity, the statement about the bounds \(C_1 n < \#\mathcal M_n < e^{C_2 n}\) becomes a cautiously performed counting argument about the conjugacy classes of certain lattices in the Lie group of automorphisms of the ball \(\mathbb B^2\).