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 is an element of 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, hand Q implies a positive answer to the question above, and therefore implies stability of fixed points of the geometric structures described above. (c) 2025 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.