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.Contents Actually the construction in [8] only applies to compact subsets of the stabilized ellipsoid. To embed the whole product we are appealing to [23].q with gcd(p, q) = 1 and ε > 0 irrational and very small, and fix μ * > 0. Suppose that for all sufficiently small ε > 0 and generic admissible J there is a genus 0 curve C in X μ ,x with degree d, Fredholm index 0, and one negative end on {(β 1 , p)}, where gcd(d, p) = 1. Then, 3d = p + q, and for all k 0, we haveThe proof is given in § 3.6: see Proposition 3.6.1. The main point is that if C has genus 0, Fredholm index zero, and one negative end, then its Fredholm index remains zero under † In fact if the domain is an ellipsoid the two embedding problems are equivalent: see [19]. Here CP 2 (μ) denotes CP 2 with symplectic form ω scaled so that ω takes the value μ on the line L.‡ Here we assume that J is admissible, that is, adapted to the negative end of X: see § 2 for more details. † In fact, the ECH cobordism map detects buildings with ECH index 0 that may (and often do) consist of curves with both positive and negative ECH index. Further, it may not always be the case that curves with ECH index 2 and Fredholm index 0 must have double points, but as mentioned in Remark 2.2.3(ii) this is known in some situations.

A remark on the stabilized symplectic embedding problem for ellipsoids

“…The proofs of these results about c k , k > 0, rely on the following stabilization result, which was proved in [CGHM,Proposition 3.6.6] using arguments originating in [HK, CGH]. Before stating it, we describe the basic geometry.…”

“…Before stating it, we describe the basic geometry. (For more information, see for example [CGHM,§2.2…”

“…7 See [CGHM,§2] for more details of these calculations. Further, the degree of C is the number of times it intersects the line at infinity.…”

“…11 But it is easy to check that this condition is never satisfied if there also is a partition p = (p i ) that satisfies i p i y = s y and has more than one entry, since the corresponding piecewise linear path lies between the lines of slope 1/y and 1/s. (For relevant background see [Hu] or [CGHM,§2].) 3…”

“…In the limit the sphere breaks into at least two pieces, the top piece (called C U ) that lies in the completion X x of CP 2 (µ) Φ (∂E(1, x)), the bottom piece that lies in the positively completed blown up ellipsoid, and possibly some other components lying in the neck region ∂E(1, x) × R. It turns out that the top of the resulting curve has the right structure. (For details of this approach, see [CGHM,Remark 3.1.8 (ii)]. ) This argument does not work for m ≥ 3 because, as explained in [CGHM,Remark 3.5.4 (i)] although there is a suitable exceptional class, when the neck is stretched the resulting curve will have more than one negative end.…”