Saturated and primitive smooth compactifications of ball quotients

Abstract

Let $X=(\mathbb{B}/\mathrm{\Gamma }{)}^{\prime }$ be a smooth toroidal compactification of a quotient of the complex $2$-ball $\mathbb{B}={\mathrm{P}\mathrm{S}\mathrm{U}}_{2,1}/\mathrm{P}\mathrm{S}(U_2\times U_1)$ by a lattice $\mathrm{\Gamma }<{\mathrm{P}\mathrm{S}\mathrm{U}}_{2,1}$, $D:=X\setminus (\mathbb{B}/\mathrm{\Gamma })$ be the toroidal compactifying divisor of $X$, $\rho :X\to Y$ 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 $f:(X^{\prime },D^{\prime },E^{\prime }({\rho }^{\prime }))\to (X,D,E)$ are isomorphisms, while $(X,D,E(\rho ))$ is primitive exactly when any unramified covering $f:(X,D,E(\rho ))\to (f(X),f(D),f(E(\rho )))$ is an isomorphism. The covering relations among the smooth toroidal compactifications $(\mathbb{B}/\mathrm{\Gamma }{)}^{\prime }$ 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 $\rho =\beta :X=(\mathbb{B}/\mathrm{\Gamma }{)}^{\prime }\to Y$ of finitely many points of $Y$, we show that there is an isomorphism $\mathrm{\Phi }:\mathrm{A}\mathrm{u}\mathrm{t}(Y,\beta (D))\to \mathrm{A}\mathrm{u}\mathrm{t}(X,D)$ of the relative automorphism groups and $\mathrm{A}\mathrm{u}\mathrm{t}(X,D)$ is a finite group. Moreover, when $Y$ is an abelian surface then any finite unramified covering $f:(X,D,E(\beta ))\to (f(X),f(D),f(E(\beta )))$ factors through an $\mathrm{A}\mathrm{u}\mathrm{t}(X,D)$-Galois covering. We discuss the saturation and the primitiveness of $X$ with Kodaira dimension $\kappa (X)=-\mathrm{\infty }$, as well as of $X$ with $K3$ or Enriques minimal model $Y$.