The Teichm\"uller Stack

This paper is a comprehensive introduction to the results of [7]. It grew as an expanded version of a talk given at INdAM Meeting Complex and Symplectic Geometry, held at Cortona in June 12-18, 2016. It deals with the construction of the Teichm\"uller space of a smooth compact manifold M (that is the space of isomorphism classes of complex structures on M) in arbitrary dimension. The main problem is that, whenever we leave the world of surfaces, the Teichm\"uller space is no more a complex manifold or an analytic space but an analytic Artin stack. We explain how to construct explicitly an atlas for this stack using ideas coming from foliation theory. Throughout the article, we use the case of $\mathbb{S}^3\times\mathbb{S}^1$ as a recurrent example.