Stability of fixed points in Poisson geometry and higher Lie theory

TL;DR

This paper develops a unified criterion for the stability of fixed points in various geometric structures using differential graded Lie algebras and cohomology, extending known stability results to higher order singularities.

Contribution

It introduces a general cohomological criterion for stability of fixed points across multiple bracket structures in Poisson geometry and higher Lie theory.

Findings

Cohomological conditions imply stability of fixed points.

Unified approach applies to diverse structures like Lie algebroids and Courant algebroids.

Recovers and extends previous stability results.

Abstract

We provide a uniform approach to obtain sufficient criteria for a (higher order) fixed point of a given bracket structure on a manifold to be stable under deformations. Examples of bracket structures include Lie algebroids, Lie $n$-algebroids, singular foliations, Lie bialgebroids, Courant algebroids and Dirac structures in split Courant algebroids admitting a Dirac complement. We show that the stability problems are specific instances of the following problem: given a differential graded Lie algebra $\mathfrak g$, a differential graded Lie subalgebra $\mathfrak h$ of degreewise finite codimension in $\mathfrak g$ and a Maurer-Cartan element $Q\in \mathfrak h^1$, when are Maurer-Cartan elements near $Q$ in $\mathfrak g$ gauge equivalent to elements of $\mathfrak h^1$? We show that the vanishing of a finite-dimensional cohomology group associated to $\mathfrak g,\mathfrak h$ and $Q$ implies a positive answer to the question above, and therefore implies stability of fixed points of the geometric structures described above. In particular, we recover the stability results of Crainic-Fernandes for zero-dimensional leaves, as well as the stability results for higher order singularities of Dufour-Wade.