On subvarieties of singular quotients of bounded domains

Abstract

Let X $X$ be a quotient of a bounded domain in C n $\mathbb {C}^n$ . Under suitable assumptions, we prove that every subvariety of X $X$ not included in the branch locus of the quotient map is of log-general type in some orbifold sense. This generalizes a recent result by Boucksom and Diverio, which treated the case of compact, étale quotients. Finally, in the case where X $X$ is compact, we give a sufficient condition under which there exists a proper analytic subset of X $X$ containing all entire curves and all subvarieties not of general type (meant this time in in the usual sense as opposed to the orbifold sense).