Calabi-Yau Foliations and Deformations

TL;DR

This paper studies the deformation theory of Calabi-Yau type foliations, establishing the structure of their deformation spaces and relating different types of deformations through geometric and product structures.

Contribution

It introduces a detailed analysis of deformation spaces for Calabi-Yau foliations, showing smoothness and product relations among different deformation types.

Findings

The space of unfoldings $K^f$ is smooth.

The holomorphic deformation space $K^h$ factors as a product $K^f \times K^{tr}$.

Deformations of the foliation can be understood via associated transversally holomorphic deformations.

Abstract

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 $\mathcal{G}$.