On subvarieties of singular quotients of bounded domains

Abstract

Let $X$ be a quotient of a bounded domain in ${\mathbb{C}}^n$. Under suitable assumptions, we prove that every subvariety of $X$ 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 $X$ is compact, we give a sufficient condition under which there exists a proper analytic subset of $X$ 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).

26 pages, 3 figures, comments are very welcome! v2: the exposition has been (hopefully) improved and simplified, some references added. v3: several examples, remarks, and applications added in the introduction. v4: final version, to appear on J. Lond. Math. Soc. (2)