# On subvarieties of singular quotients of bounded domains

## TL;DR

This paper investigates subvarieties of quotients of bounded domains, showing they are of log general type under certain conditions, and provides criteria for the existence of proper subsets containing all entire curves.

## Contribution

It generalizes previous results to non-compact quotients and introduces conditions for subvarieties and entire curves in such quotients.

## Key findings

- Subvarieties outside the branch locus are of log general type.
- Extension of results from compact to non-compact quotients.
- Criteria for proper subsets containing all entire curves.

## 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, \'etale 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).