Irregular ball-quotient surfaces with non-positive Kodaira dimension

Let $\Gamma \subset \mathbf{PU}(2,1)$ be a lattice. $\Gamma$ then acts on the complex open unit ball $\mathbf{B} \subset \mathbb{C}^2$. Suppose that $\Gamma$ is not cocompact and `sufficiently' neat. Then there exists a compactification $X = \overline{\Gamma \setminus \mathbf{B}}$ with the property that $X$ is a smooth projective surface and $D = X \setminus (\Gamma \setminus \mathbf{B})$ is a disjoint sum of elliptic curves. The main result is: $kod(X) \leq 0$, $h^1(\mathcal{O}_X) > 0$ $\Leftrightarrow X$ admits an abelian surface $Y$ as minimal model, and $Y$ carries a divisor $S = \sum S_i$ which consists of elliptic curves such that $4\cdot |sing\,S| = \sum |S_i \cap sing\,S|$.

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Published 1 January 2008