Moduli spaces of Lie algebras and foliations

Let $X$ be a smooth projective variety over the complex numbers and $S(d)$ the scheme parametrizing $d$-dimensional Lie subalgebras of $H^0(X,\mathcal{T} X)$. This article is dedicated to the study of the geometry of the moduli space $\text{Inv}$ of involutive distributions on $X$ around the points $\mathcal{F}\in \text{Inv}$ which are induced by Lie group actions. For every $\mathfrak{g} \in S(d)$ one can consider the corresponding element $\mathcal{F}(\mathfrak{g})\in \text{Inv}$, whose generic leaf coincides with an orbit of the action of $\exp(\mathfrak{g})$ on $X$. We show that under mild hypotheses, after taking a stratification $\coprod_i S(d)_i\to S(d)$ this assignment yields an isomorphism $\phi:\coprod_i S(d)_i\to \text{Inv}$ locally around $\mathfrak{g}$ and $\mathcal{F}(\mathfrak{g})$. This gives a common explanation for many results appearing independently in the literature. We also construct new stable families of foliations which are induced by Lie group actions.