The ghost stairs stabilize to sharp symplectic embedding obstructions

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 techniques of independent interest, namely: (i) new applications of ideas from embedded contact homology (ECH) in the context of nodal curves, (ii) an improved version of the index inequality familiar from the theory of embedded contact homology, (iii) new applications of relative intersection theory in the context of neck stretching analysis, (iv) a new approach to estimating the ECH grading of multiply covered elliptic orbits in terms of areas and continued fractions, and (v) a new technique for understanding the ECH of ellipsoids by constructing explicit bijections between certain sets of lattice points.