(untitled) (Annual of Sofia University St. Kliment Ohridski. Faculty of Mathematics and Informatics, 2019)
Let X = ( B / Γ)’ be a smooth toroidal compactification of a quotient of the complex 2-ball B = PSU _2,1 / PS (U_2 × U_1) by a lattice Γ < PSU _2,1, D := X ( B /Γ) be the toroidal compactifying divisor of X, ρ : X → Y be a finite composition of blow downs to a minimal surface Y and E(ρ) be the exceptional divisor of ρ. The present article establishes a bijective correspondence between the finite unramified coverings of ordered triples (X, D, E) and the finite unramified coverings of ( ρ (X), ρ (D), ρ (E)). We say that (X, D,E(ρ)) is saturated if all the unramified coverings f: (X’, D’, E’ (ρ’)) → (X, D, E) are isomorphisms, while (X, D, E (ρ)) is primitive exactly when any unramified covering f: (X, D, E (ρ)) → ( f(X), f(D), f(E(ρ))) is an isomorphism. The covering relations among the smooth toroidal compactifications (B / Γ)’ 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 ρ = β : X = ( B / Γ )’ → Y of finitely many points of Y, we show that there is an isomorphism Φ : Aut (Y, β (D)) → Aut (X, D) of the relative automorphism groups and Aut (X, D) is a finite group. Moreover, when Y is an abelian surface then any finite unramified covering f: (X, D, E( β)) → ( f(X), f(D), f (E( β))) factors through an Aut (X, D)-Galois covering. We discuss the saturation and the primitiveness of X with Kodaira dimension κ (X)= - ∞, as well as of X with K3 or Enriques minimal model Y.