Geography of the Teichm\"uller stack

TL;DR

This paper explores the structure and local properties of the Teichmüller stack, identifying orbifold points and local models, with detailed analysis in the Kähler setting using cycle space compactness.

Contribution

It provides a detailed description of the geography of the Teichmüller stack and introduces a variant, addressing local models and orbifold points.

Findings

Identification of orbifold points in the Teichmüller stack

Description of local models for special points

Enhanced understanding in the Kähler setting using cycle spaces

Abstract

In this article, we describe the geography of the Teichm\"uller stack of \cite{LMStacks} and of one of its variants we introduce here, giving some answers to questions as: which points are orbifold points? What are the different local models of special points?... We give a rough description in the general case, and we use the compacity of the cycle spaces to get a much more detailed picture in the K\"ahler setting.