Analytic stacks and hyperbolicity

TL;DR

This paper extends Brody's theorem, which links two notions of hyperbolicity in complex spaces, to Deligne-Mumford analytic stacks, establishing their equivalence under compactness assumptions.

Contribution

It introduces definitions of Kobayashi and Brody hyperbolicity for stacks and proves their equivalence, generalizing classical results to a broader geometric context.

Findings

Established equivalence of hyperbolicity notions for stacks

Extended classical hyperbolicity results to Deligne-Mumford stacks

Provided foundational definitions for hyperbolicity in stack theory

Abstract

The classical Brody's theorem asserts the equivalence between two notions of hyperbolicity for compact complex spaces, one named after Kobayashi and one expressed in terms of lack of non constant holomorphic entire functions (compactness is only used to prove the harder implication). We extend this theorem to Deligne-Mumford analytic stacks, by first providing definitions of what we think of Kobayashi and Brody hyperbolicity for such objects and then proving the equivalence of these concepts under an assumption of compactness.