Deformations of Lie algebras of vector fields arising from families of schemes

TL;DR

This paper develops a conceptual framework for understanding deformations of Lie algebras of vector fields derived from families of schemes, connecting geometric deformations with algebraic structures via stacks and moduli spaces.

Contribution

It formulates the deformation of Lie algebras of vector fields in a stack-theoretic setting and relates it to moduli of stable marked curves, extending prior explicit constructions.

Findings

Constructed a stack of deformations for Lie algebras of vector fields.

Established a morphism from the moduli stack of stable marked curves to the deformation stack.

Showed the morphism is nearly a monomorphism using Pursell-Shanks theory.

Abstract

Fialowski and Schlichenmaier constructed examples of global deformations of Lie algebras of vector fields from deforming the underlying variety. We formulate their approach in a conceptual way. Namely, we construct a stack of deformations and a morphism form the moduli stack of stable marked curves. The morphism associates to a family of marked curves the family of Lie algebras obtained by taking the Lie algebra of vertical vector fields on the family where one has extracted the marked points. We show that this morphism is almost a monomorphism by Pursell-Shanks theory.