Appears in collection : Summer School 2019 - Foliations and algebraic geometry

Let X be a compact complex manifold. The Kuranishi space of X is an analytic space which encodes every small deformation of X. The Teichmüller space is a topological space formed by the classes of compact complex manifolds diffeomorphic to X up to biholomorphisms smoothly isotopic to the identity. F. Catanese asked when these two spaces are locally homeomorphic. Unfortunatly, this almost never occurs. I will reformulate this question replacing these two spaces with stacks. I will then show that, if X is Kähler, this new question has always a positive answer. Finally, I will discuss the non-Kähler case.