Infinite staircases and reflexive polygons
Dan Cristofaro‐Gardiner, Tara S. Holm, Alessia Mandini, Ana Rita Pires · arXiv (Cornell University) · 2020
We explore the question of when an infinite staircase describes part of the ellipsoid embedding function of a convex toric domain. For rational convex toric domains in four dimensions, we conjecture a complete answer to this question, in terms of six families that are distinguished by the fact that their moment polygon is reflexive. To understand better when infinite staircases occur, we prove that any infinite staircase must have a unique accumulation point given as the solution to an explicit quadratic equation. We then provide a uniform proof of the existence of infinite staircases for our six families, using two tools. For the first, we use recursive families of almost toric fibrations to find symplectic embeddings into closed symplectic manifolds. In order to establish the embeddings for convex toric domains, we prove a result of potentially independent interest: a four-dimensional ellipsoid embeds into a closed symplectic toric four-manifold if and only if it can be embedded into a corresponding convex toric domain. For the second tool, we find recursive families of convex lattice paths that provide obstructions to embeddings. Our work contrasts the work of Usher, who finds infinite families of infinite staircases for irrationally shaped rectangles.