Calabi-Yau Foliations and Deformations

We propose in this article the study of the deformations of a Calabi-Yau type foliations $\mathcal{F}$. For three different types of deformations (unfoldings, holomorphic, transversally holomorphic) there exist Kuranishi spaces $K^f,K^h,K^{tr}$ parametrizing the corresponding families of deformations. We show that $K^f$ is smooth, and that we can obtain $K^h$ as the product $K^f\times K^{tr}$. At last, we show that we can see the $f$-deformations of $\mathcal{F}$ as the $tr$-deformations of a supplementary foliation $\mathcal{G}$.