# The ghost stairs stabilize to sharp symplectic embedding obstructions

## TL;DR

This paper demonstrates that ghost staircase obstructions in symplectic embedding problems become sharp when stabilized, using novel methods from contact homology and intersection theory.

## Contribution

It introduces new techniques to analyze stabilized symplectic embeddings, showing ghost obstructions are sharp and providing explicit embeddings via symplectic folding.

## Key findings

- Ghost obstructions are sharp in stabilized embedding problems.
- New bounds for ECH grading of multiply covered orbits.
- Explicit bijections between lattice points and ECH generators.

## Abstract

In determining when a four-dimensional ellipsoid can be symplectically embedded into a ball, McDuff and Schlenk found an infinite sequence of "ghost" obstructions that generate an infinite "ghost staircase" determined by the even index Fibonacci numbers. The ghost obstructions are not visible for the four-dimensional embedding problem because strictly stronger obstructions also exist. We show that in contrast, the embedding constraints associated to the ghost obstructions are sharp for the stabilized problem; moreover, the corresponding optimal embeddings are given by symplectic folding. The proof introduces several ideas of independent interest, namely: (i) an improved version of the index inequality familiar from the theory of embedded contact homology (ECH), (ii) new applications of relative intersection theory in the context of neck stretching analysis, (iii) a new approach to estimating the ECH grading of multiply covered elliptic orbits in terms of areas and continued fractions, and (iv) a new technique for understanding the ECH of ellipsoids by constructing explicit bijections between certain sets of lattice points.