Arboreal Singularities in Weinstein Skeleta

We study the singularities of the isotropic skeleton of a Weinstein manifold in relation to Nadler’s program of arboreal singularities. By deforming the skeleton via homotopies of the Weinstein structure, we produce a Morse–Bott* representative of the Weinstein homotopy class whose stratified skeleton determines its symplectic neighborhood. We then study the singularities of the skeleta in this class and show that after a certain type of generic perturbation either (1) these singularities fall into the class of (signed Lagrangian versions of) Nadler’s arboreal singularities which are combinatorially classified into finitely many types in a given dimension or (2) there are singularities of tangency in associated front projections. We then turn to the singularities of tangency to try to reduce them also to collections of arboreal singularities. We give a general localization procedure to isolate the Liouville flow to a neighborhood of these non-arboreal singularities, and then show how to replace the simplest singularities of tangency (those of Thom-Boardman type \(\Sigma ^{1,0}\)) by arboreal singularities.