
## 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. #### Problem 2.05.

Can we find bordered generating function models for arboreal singularities?
• #### Problem 2.1.

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

#### 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."
• #### 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.

#### 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.

#### Problem 2.35.

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

#### 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).

#### 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.