A classification of infinite staircases for Hirzebruch surfaces
Year: 2025
Type: article
Abstract: The ellipsoid embedding function of a symplectic manifold gives the smallest amount by which the symplectic form must be scaled in order for a standard ellipsoid of the given eccentricity to embed symplectically into the manifold. It was first computed for the standard four‐ball (or equivalently, the complex projective plane) by McDuff and Schlenk, and found to contain the unexpected structure of an “infinite staircase,” that is, an infinite sequence of nonsmooth points arranged in a piecewise linear stair‐step pattern. Later work of Usher and Cristofaro‐Gardiner–Holm–Mandini–Pires suggested that while four‐dimensional symplectic toric manifolds with infinite staircases are plentiful, they are highly nongeneric. This paper concludes the systematic study of one‐point blowups of the complex projective plane, building on previous work of Bertozzi‐Holm‐Maw‐McDuff‐Mwakyoma‐Pires‐Weiler, Magill‐McDuff, Magill‐McDuff‐Weiler, and Magill on these Hirzebruch surfaces. We prove a conjecture of Cristofaro‐Gardiner–Holm–Mandini–Pires for this family: that if the blowup is of rational weight and the embedding function has an infinite staircase then that weight must be . We show also that the function for this manifold does not have a descending staircase. Furthermore, we give a sufficient and necessary condition for the existence of an infinite staircase in this family which boils down to solving a quadratic equation and computing the function at one specific value. Many of our intermediate results also apply to the case of the polydisk (or equivalently, the symplectic product of two spheres).
Source: Journal of Topology
DOI: 10.1112/topo.70017
Authors: Nicki Magill, Ana Rita Pires, Morgan Weiler
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,930
OpenAlex ID: W4408267529