Arboreal singularities

Abstract

Abstract. We introduce a class of combinatorial singularities of Lagrangian skeleta of symplectic manifolds. The link of each singularity is a finite regular cell complex homotopy equivalent to a bouquet of spheres. It is determined by its face poset which is naturally constructed starting from a tree (nonempty finite acyclic graph). The choice of a root vertex of the tree leads to a natural front projection of the singularity along with an orientation of the edges of the tree. Microlocal sheaves along the singularity, calculated via the front projection, are equivalent to modules over the quiver given by the directed tree.

Non-simplicity of Isocontact Embeddings in All Higher Dimensions

“…Remark 3.6. Recent advances in symplectic topology [38,54,63], in combination with the compactness theory of integral currents [36,68], strongly indicate that there should exist a Floer theory with singular boundary conditions. In particular, [21].…”

“…The theory of arboreal singularities [54,63] provides many interesting examples of singular Legendrians Λ ⊆ (R 2n+1 , ξ st ), arising as the Legendrian boundaries of singular Lagrangian arboreal skeleta [63,Section 2.4]. Arboreal singularities can more directly be described by using the following general class of singular Legendrians.…”

Arboreal singularities in Weinstein skeleta

“…Arboreal singularities. To each tree (acyclic connected graph), Nadler [Nad17] associates a topological stratified complex. If the tree has N vertices, then the highest dimension of the strata is N − 1 and the complex can be built from N top dimensional strata with boundary and corners, glued together in a manner determined by the edges of the tree.…”

“…The class of singularities we aim to restrict our skeleton to have was proposed by Nadler [Nad17], inspired by mirror symmetry. Kontsevich proposed a method to calculate certain microlocal sheaf invariants of a Weinstein manifold in terms of the singular topology of its skeleton, provided the singularities fall into a certain class which in this paper are referred to as A n singularities [Kon].…”

Weinstein manifolds revisited

“…A priori, a skeleton of a Weinstein domain can have very complicated singularities. However, David Nadler conjectured that up to Weinstein homotopy the singularities of the skeleton can be reduced to a finite list in any dimension, see [38]. For 2ndimensional symplectic Weinstein manifolds the list of Nadler's singularities, which he calls arboreal, are enumerated by decorated rooted trees with ≤ n + 1 vertices.…”