| Register
\(\newcommand{\Cat}{{\rm Cat}} \) \(\newcommand{\A}{\mathcal A} \) \(\newcommand{\freestar}{ \framebox[7pt]{$\star$} }\)

2. Examples and basic questions

The theory of arboreal skeleta being very new, there is great demand for a good stock of examples, and many basic structures in the theory are not yet understood. In this section we catalog some fundamental questions about arboreal skeleta.


    1. Add remark

      Problem 2.05.

      Can we find bordered generating function models for arboreal singularities?


        • Add remark

          Problem 2.1.

          Can any arboreal singularity can be obtained as a symplectic reduction of a product of trivalent vertices?


            • Add remark

              Problem 2.15.

              For a Weinstein manifold $W$, what are the obstructions to finding a skeleton with a restricted class of singularities? What if we restrict to certain signed arboreal singularities?

                •     Call "strictly generalized arboreal singularities" those generalized arboreal singularities which do not appear on the original list of arboreal singularities. Such singularities play a necessary rôle in the arborealization program, but we might be interested in cases where they do not appear.

                  Add remark

                  Problem 2.2.

                  For a Weinstein manifold $W$, what are the obstructions to finding a skeleton without strictly generalized arboreal singularities?
                    1. Remark. We expect that flexible manifolds should not have such skeleta.
                        • Remark. We will refer to a compact arboreal space ("arborifold") which does not have any strictly generalized arboreal singularities as a "closed arboreal space."


                            • Add remark

                              Problem 2.25.

                              Is there a closed arboreal space whose sheaf category or symplectic cohomology is zero?

                                •     Many Weinstein manifolds which occur in practice appear as complements of Donaldson divisors. It would be interesting to better understand how to characterize such manifolds among all Weinstein manifolds.

                                  Add remark

                                  Problem 2.3.

                                  What restrictions are imposed on an arboreal singularity by the requirement that it be the skeleton of the complement of a Donaldon divisor? (For instance, can you say something about Reeb orbits on the boundary?)

                                    • Examples

                                          One difficulty with entering the theory of arboreal skeleta is the lack of a list of hands-on examples and toy models.

                                      Add remark

                                      Problem 2.35.

                                      What is an instructive, manageable playground for dealing with arboreal singularities?


                                        • Add remark

                                          Problem 2.4.

                                          What are naturally occurring examples of the $D_4$ singularity in skeleta of Weinstein manifolds?
                                            1. Remark. You can produce these artificially, but if would be nice to have, for instance, a description of some reasonable algebraic variety where the appearance of $D_4$ appears inevitable.


                                                • Add remark

                                                  Problem 2.45.

                                                  Give some notion of metric on the space of skeleta so that arboreal things are generic.

                                                    •     In the theory of Pham-Brieskorn singularities, we have a way of combining two Milnor fibers to get a higher-dimensional Milnor fiber via the join construction. It would be nice to generalize this to more general Weinstein manifolds.

                                                      Add remark

                                                      Problem 2.5.

                                                      Is there a notion of a "Weinstein join" of Weinstein manifolds which on the skeleton is really the join?

                                                        •     Given the skeleton of a nearby fiber of a Lefschetz fibration (or something worse) in $\mathbb{C}^n$, consider the cone on this skeleton (which is the skeleton associated to this Weinstein pair).

                                                          Add remark

                                                          Problem 2.55.

                                                          How can we understand the resulting very singular skeleton? For instance, how do we calculate its symplectic invariants?

                                                              Cite this as: AimPL: Arborealization of singularities of Lagrangian skeleta, available at http://aimpl.org/arborlagrange.