Saturated and primitive smooth compactifications of ball quotients

Abstract

Let $X = ( {\mathbb B} / \Gamma)'$ be a smooth toroidal compactification of a quotient of the complex $2$-ball ${\mathbb B} = {\rm PSU} _{2,1} / {\rm PS} (U_2 \times U_1)$ by a lattice $\Gamma < {\rm PSU} _{2,1}$, $D := X \setminus ( \mathbb{B} /\Gamma)$ be the toroidal compactifying divisor of $X$, $\rho : X \rightarrow Y$ be a finite composition of blow downs to a minimal surface $Y$ and $E(\rho)$ be the exceptional divisor of $\rho$. The present article establishes a bijective correspondence between the finite unramified coverings of ordered triples $(X, D, E)$ and the finite unramified coverings of $( \rho (X), \rho (D), \rho (E)).$ We say that $(X, D,E(\rho))$ is saturated if all the unramified coverings $f: (X', D', E' (\rho')) \rightarrow (X, D, E)$ are isomorphisms, while $(X, D, E (\rho))$ is primitive exactly when any unramified covering $f: (X, D, E (\rho)) \rightarrow ( f(X), f(D), f(E(\rho)))$ is an isomorphism. The covering relations among the smooth toroidal compactifications $(\mathbb{B} / \Gamma)'$ are studied by Uludag's [7], Stover's [6], Di Cerbo and Stover's [2] and other articles.In the case of a single blow up $\rho = \beta : X = ( {\mathbb B} / \Gamma )' \rightarrow Y$ of finitely many points of $Y$, we show that there is an isomorphism $\Phi : {\rm Aut} (Y, \beta (D)) \rightarrow {\rm Aut} (X, D)$ of the relative automorphism groups and ${\rm Aut} (X, D)$ is a finite group. Moreover, when $Y$ is an abelian surface then any finite unramified covering $f: (X, D, E( \beta)) \rightarrow ( f(X), f(D), f (E( \beta)))$ factors through an ${\rm Aut} (X, D)$-Galois covering. We discuss the saturation and the primitiveness of $X$ with Kodaira dimension $\kappa (X)= - \infty$, as well as of $X$ with $K3$ or Enriques minimal model $Y$.