Irregular ball-quotient surfaces with non-positive Kodaira dimension
Let \(\Gamma\subset \text{PU}(2,1)\) be an arithmetic lattice. In the non-cocompact case, one can study the Baily-Borel compactification \(X=\overline{\Gamma\backslash \mathbf{B}}\) which might be of special type, and if \(\Gamma\) is neat, the divisor \(D=X\setminus (\Gamma\backslash \mathbf{B})\) is a disjoint sum of elliptic curves. The author tackles the classification of such (irregular) surfaces with non-positive Kodaira dimension, by geometric methods. Let's denote \(\mathcal{T}\) the class of compact complex surfaces of the type \(X=U\cup D\) where \(U\) has the complex unit ball as universal holomorphic covering, and \(D\) is as above. It is proven that if \(Y\) is relatively minimal, irregular, with non-positive Kodaira dimension, then \(Y\) is dominated by a surface in \(\mathcal{T}\) if and only if it is abelian (and actually isogenous to a product of an elliptic curve). Results from \textit{R.-P. Holzapfel} [``Complex hyperbolic surfaces of abelian type'', Serdica Math. J. 30, No. 2--3, 207--238 (2004; Zbl 1062.11035)] which started this classification, and \textit{G. Tian} and \textit{S. T. Yau} [``Existence of Kähler-Einstein metrics on complete Kähler manifolds and their applications to algebraic geometry'', Mathematical aspects of string theory, Proc. Conf., San Diego/Calif. 1986, Adv. Ser. Math. Phys. 1, 574--629 (1987; Zbl 0682.53064)] are used throughout the paper. The first step is to prove that no irregular surface in \(\mathcal{T}\) has non-negative Kodaira dimension; then one has to treat the zero-dimensional case. A useful review of surface theory recalls that the latter happens only when its minimal model is abelian or hyperelliptic. One proceeds in the two above steps by reductio ad absurdum, and Tian-Yau's theorem provides the sought contradiction.
scientific article; zbMATH DE number 5505947
scientific article