Moduli spaces of lie algebras and foliations

Abstract:

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.