Rémi Danain-Bertoncini

Usually in deformation theory we try to find conditions ensuring that the Kuranishi space for a given structure is smooth. The most direct condition is to suppose that the space where obstructions can appear is equal to zero, thus no obstruction can appear during the resolution of the so called Maurer-Cartan equation. In the case of compact complex manifolds, there exists a particular class of very nice objects : Calabi-Yau manifolds. Without the precedent condition, we can however show that any deformation of such a manifold is unobstructed, thanks to Tian and Todorov works. The notion of Calabi-Yau manifold shows in particular the distinction between the space where obstructions can live, and the concrete space of obstructions.

For this talk I will present the precedent notions (Kuranishi space, obstructions ...) used in deformation theory and how it can be adapted for a theory of deformations of regular holomorphic foliations. I will finish by showing that a notion of Calabi-Yau foliation behaves well for certain deformations.