On infinite staircases in toric symplectic four-manifolds

An influential result of McDuff and Schlenk asserts that the function that encodes when a four-dimensional symplectic ellipsoid can be embedded into a four-dimensional ball has a remarkable structure: the function has infinitely many corners, determined by the odd-index Fibonacci numbers, that fit together to form an infinite staircase. This work has led to considerable interest in understanding when the ellipsoid embedding function for other symplectic fourmanifolds is partly described by an infinite staircase. We provide a general framework for analyzing this question for a large family of targets, called finite type convex toric domains, which we prove generalizes the class of closed toric symplectic fourmanifolds. When the target is of finite type, we prove that any infinite staircase must have a unique accumulation point $a_0$ , given as the solution to an explicit quadratic equation. Moreover, we prove that the embedding function at $a_0$ must be equal to the classical volume lower bound. In particular, our result gives an obstruction to the existence of infinite staircases that experimentally seems strong. Our methods apply equally well to any closed manifold which is a blowup of the complex projective plane. In the special case of rational convex toric domains, we can say more. We conjecture a complete answer to the question of existence of infinite staircases, in terms of six families that are distinguished by the fact that their moment polygon is reflexive. 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 toric symplectic 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. We conclude by connecting our conjecture that these are the only infinite staircases among rational convex toric domains to a question in number theory related to a classic work of Hardy and Littlewood.