Stability of fixed points in Poisson geometry and higher Lie theory

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 perturbations. 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 in particular recover stability results of Crainic-Fernandes for zero-dimensional leaves, as well as the stability results of higher order singularities of Dufour-Wade.

These stability problems can all be shown to be specific instances of the following problem: given a differential graded Lie algebra $g$, a differential graded Lie subalgebra $h$ of finite codimension in $g$ and a Maurer-Cartan element $Q\in h^1$, when are Maurer-Cartan elements near Q in $g$ gauge equivalent to elements of $h^1$?

We show that the vanishing of a finite-dimensional cohomology group associated to $g,h$ and Q implies a positive answer to the question above, and therefore implies stability of fixed points of the geometric structures described above.