Abstract

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.

Full Text
Published version (Free)

Talk to us

Join us for a 30 min session where you can share your feedback and ask us any queries you have

Schedule a call