# Saturated and primitive smooth compactifications of ball quotients

 Title Saturated and primitive smooth compactifications of ball quotients Publication Type Journal Article Year of Publication 2019 Authors Beshkov P.G, Kasparyan A.K, Sankaran G.K Journal Annuaire de l’Université de Sofia “St. Kliment Ohridski”. Faculté de Mathématiques et Informatique Volume 106 Start Page 53 Pagination 53-77 ISSN 1313-9215 (Print) 2603-5529 (Online) Keywords Smooth toroidal compactifications of quotients of the complex 2-ball, unramified coverings 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 $Y$. 2010 MSC Primary: 14M27; Secondary: 14J25, 51H30
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