Moduli spaces of Lie algebras and foliations

Let X X be a smooth projective variety over the complex numbers and S ( d ) S(d) the scheme parametrizing d d -dimensional Lie subalgebras of H 0 ( X , T X ) H^0(X,\mathcal {T}X) . This article is dedicated to the study of the geometry of the moduli space Inv \text {Inv} of involutive distributions on X X around the points F ∈ Inv \mathcal {F}\in \text {Inv} which are induced by Lie group actions. For every g ∈ S ( d ) \mathfrak {g}\in S(d) one can consider the corresponding element F ( g ) ∈ Inv \mathcal {F}(\mathfrak {g})\in \text {Inv} , whose generic leaf coincides with an orbit of the action of exp ⁡ ( g ) \exp (\mathfrak {g}) on X X . We show that under mild hypotheses, after taking a stratification ∐ i S ( d ) i → S ( d ) \coprod _i S(d)_i\to S(d) this assignment yields an isomorphism ϕ : ∐ i S ( d ) i → Inv \phi :\coprod _i S(d)_i\to \text {Inv} locally around g \mathfrak {g} and F ( g ) \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.