Moduli spaces of Lie algebras and foliations

2022/09/05 by Velazquez, Sebastian Lucas
#Algebraic Geometry (math.AG) #Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.2209.01752

Abstract

Let X be a smooth projective variety over the complex numbers and S(d) the scheme parametrizing d-dimensional Lie subalgebras of H⁰(X,T X). This article is dedicated to the study of the geometry of the moduli space Inv of involutive distributions on X around the points F∈ Inv which are induced by Lie group actions. For every \mathfrakg ∈ S(d) one can consider the corresponding element F(\mathfrakg)∈ Inv, whose generic leaf coincides with an orbit of the action of exp(\mathfrakg) on X. We show that under mild hypotheses, after taking a stratification \coprod_i S(d)_i→ S(d) this assignment yields an isomorphism ϕ:\coprod_i S(d)_i→ Inv locally around \mathfrakg and F(\mathfrakg). 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.

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• Respectful decompositions of Lie algebras
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• Lie group foliations: Dynamical systems and integrators
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