Ball quotients of nonpositive Kodaira dimension (Q2844626)

It is known that each complex projective surface is realizable as a finite cover of the projective plane \(\mathbb P ^ 2\), and it can be viewed as the quotient \(\mathbb B/\Gamma \) of the complex unit ball \(\mathbb B = \{(z _ 1, z _ 2) \in \mathbb C ^ 2\colon \;| z _ 1| ^ 2 + | z _ 2| ^ 2 < 1\}\) by a ball lattice \(\Gamma \). On this basis, R. Holzapfel has made a thorough investigation of Picard modular surfaces (see [Math. Nachr. 103, 117--153 (1981; Zbl 0495.14025)]. Later he initiated a programme for classifying these surfaces, up to birational equivalence [see his book ``Aspects of Mathematics'', E29, Wiesbaden: Vieweg, xiii, 414 p. (1998; Zbl 0980.14026)]. It incorporates a number of results providing presentations as ball quotients \(\mathbb B/\Gamma\) of surfaces of general type, as well as of rational, abelian, \(K3\), elliptic, hyperelliptic and Enriques surfaces, and of ruled surfaces with an elliptic base (an account of some of them, with references, is given in the reviewed paper). The starting point of the paper under review is Holzapfel's construction of a quotient \(\mathbb B/\Gamma _ {-1} ^ {(6,8)}\) of \(B\) by a fixed point free lattice \(\Gamma _ {-1} ^ {(6.8)}\), whose smooth toroidal compactification \(A _ {-1} ^ {\prime } = (B/\Gamma _ {-1} ^ {\prime }) ^ {\prime }\) by eight elliptic curves contains six rational \((-1)\)-curves and has an abelian minimal model \(A _ {-1} = (\mathbb C/\mathbb Z[i]) ^ 2\) cf. [Serdica Math. J. 30, 207--238 (2004; Zbl 1062.11035)]. For an abelian surface \(A\) and a finite subgroup \(H \leq \text{ Aut}(A)\), the authors determine birational invariants of \(A/H\) by properties of the \(H\)-action on \(A\). They classify all finite \(H\)-Galois quotients \(\overline {\mathbb B/\Gamma _ H}\) of \(A _ {-1} ^ {\prime }\). The present paper also finds the irregularity \(q(Y) = h ^ {1,0}(Y)\), the geometric genus \(p _ g(Y)\) and the Kodaira dimension \(\kappa (Y)\) of the smooth models \(Y\) of \(\overline {\mathbb B/\Gamma _ H}\). This is carried out by applying geometric invariant theory.