The first chapter of the thesis contains some basic theory about stacks. Section 1.4 deals with the Deligne-Mumford local quotient characterization. The second chapter develops some homotopy theory for analytic stacks, and links it to the theory of covering spaces for them. We also mention a generalization of Van Kampen theorem for analytic stacks. The third chapter is about the uniformization result proven by Behrend and Noohi in [1]. Chapter 4 deals with a slightly different case: we consider stacks with proper diagonal, and try to prove that, locally around any point, they can be regarded as quotient stacks by come action of the inertia group at the point (which we prove to be a compact Lie group). The result fails if we consider stacks of groupoids over the category Comp of complex manifolds (see Example 4.1). We thus change our setting by looking at stacks over the category *Diff* of differentiable real manifolds. All the theory developed in Chapter 1 still holds true, and the counterexample above fails. I was not able to complete the proof, which is left as a conjecture. >>

### Uniformization results for some classes of analytic stacks

#### Abstract

The first chapter of the thesis contains some basic theory about stacks. Section 1.4 deals with the Deligne-Mumford local quotient characterization. The second chapter develops some homotopy theory for analytic stacks, and links it to the theory of covering spaces for them. We also mention a generalization of Van Kampen theorem for analytic stacks. The third chapter is about the uniformization result proven by Behrend and Noohi in [1]. Chapter 4 deals with a slightly different case: we consider stacks with proper diagonal, and try to prove that, locally around any point, they can be regarded as quotient stacks by come action of the inertia group at the point (which we prove to be a compact Lie group). The result fails if we consider stacks of groupoids over the category Comp of complex manifolds (see Example 4.1). We thus change our setting by looking at stacks over the category *Diff* of differentiable real manifolds. All the theory developed in Chapter 1 still holds true, and the counterexample above fails. I was not able to complete the proof, which is left as a conjecture. >>
##### Scheda Scheda DC
2017
stack
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Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/20.500.12608/28286`