Moduli spaces of Lie algebras and foliations

TL;DR

This paper explores the geometry of the moduli space of involutive distributions on complex projective varieties, linking Lie algebra subalgebras to foliations induced by Lie group actions, and constructs new stable families of such foliations.

Contribution

It establishes a local isomorphism between moduli spaces of Lie subalgebras and involutive distributions, providing a unified framework for existing results and introducing new stable families of foliations.

Findings

The moduli space of involutive distributions can be locally described by Lie algebra subalgebras.

A stratification of the scheme parametrizing Lie subalgebras yields an isomorphism with the moduli space.

New stable families of foliations induced by Lie group actions are constructed.

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.