Stability of fixed points in Poisson geometry and higher Lie theory

TL;DR

This work provides a unified deformation-theoretic framework for the stability of fixed points across a broad spectrum of Poisson- and Lie-theoretic structures. By recasting fixed-point stability as a question about gauge equivalence of Maurer–Cartan elements in a differential graded Lie algebra and its subalgebras, the authors derive a finite-dimensional cohomology vanishing criterion $H^1(\frak g/\frak h, \overline{\partial+[Q,-]})=0$ that guarantees local surjectivity of MC classes into the subalgebra. The results recover and extend Crainic–Fernandes stability for zero-dimensional leaves, and Dufour–Wade stability for higher-order fixed points, while providing new stability theorems for higher order fixed points of Lie $n$-algebroids, Lie bialgebroids, Courant algebroids, and Dirac structures. The approach yields concrete cohomological conditions, explicit gauge-construction maps, and a coherent pathway to analyze stability under additional structures such as Poisson, PN, and Dirac settings, with implications for singular foliations and related geometric structures.

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.

Theorems & Definitions (113)

• Definition 2.1
• Lemma 2.2
• Definition 2.3
• Lemma 2.4
• Definition 2.5
• Lemma 2.6
• Remark 2.7
• Theorem 2.8: CrFe,dufour2005stability
• proof
• Remark 2.9