Arboreal singularities from Lefschetz fibrations

TL;DR

This paper explores the geometric relationship between arboreal singularities and Lefschetz fibrations, extending the known categorical equivalences to a geometric correspondence involving tree-indexed spaces.

Contribution

It establishes a geometric matching between arboreal singularities and Lefschetz fibrations associated with tree plumbings, building on previous categorical results.

Findings

Categorical equivalence between microlocal sheaves and modules over tree quivers.

Geometric correspondence between arboreal singularities and Lefschetz fibrations.

Extension of categorical results to geometric matching.

Abstract

Nadler introduced certain Lagrangian singularities indexed by trees, and determined their microlocal sheaves to be the category of modules over the corresponding tree quiver. Another family of spaces indexed by trees: the tree plumbings of spheres. The Fukaya-Seidel category of the Lefschetz fibration with this plumbing as fiber and all spheres as vanishing cycles is well known to also be modules over the tree quiver. Here we upgrade this matching of categories to a matching of geometry.