Arboreal Singularities in Weinstein Skeleta

TL;DR

This paper investigates the singularities of Weinstein skeleta, demonstrating how to deform and perturb them to classify most into arboreal types or reduce tangential singularities to arboreal forms, advancing the understanding of symplectic topology.

Contribution

It introduces a method to deform Weinstein skeleta into Morse-Bott* representatives and classifies their singularities as arboreal or tangential, providing a localization procedure to replace tangential singularities with arboreal ones.

Findings

Most singularities can be classified as arboreal after perturbation.

A localization procedure isolates non-arboreal singularities for modification.

Simplest tangential singularities can be replaced by arboreal singularities.

Abstract

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 type $\Sigma^{1,0}$) by arboreal singularities.