A classification of infinite staircases for Hirzebruch surfaces

TL;DR

This work classifies the presence and nature of infinite staircases for the ellipsoid embedding function associated to Hirzebruch surfaces $H_b$, showing that among rational $b$ only $b=\tfrac{1}{3}$ yields an infinite staircase and that the accumulation point curve $\mathrm{acc}(b)$ is irrational for non-principal staircases. It develops a rich fractal framework based on exceptional classes, blocked intervals, and mutation (including Cremona moves) to understand how staircases appear in the Block/Stair landscape and how symmetries extend the picture to all $b\in[0,1)$. For $b=\tfrac{1}{3}$ the paper proves there is no descending staircase and provides an explicit computation of $c_{1/3}$ on a key interval, complemented by a detailed ghost-stair analysis that explains regularity. The results connect to polydisks and finite toric blowups, highlighting a deep, fractal structure in four-dimensional symplectic embedding problems and offering a template for extending these ideas to broader targets.

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 non-generic. 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 $1/3$. 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).

Figures (7)

• Figure 1.1: This is ICERM, which depicts the parameterized accumulation point curve $(z,\lambda)=(\mathrm{acc}(b), \operatorname{vol}_b(\mathrm{acc}(b)))$ for $0\le b< 1$. The blue point indicates where $b=0$, at $(\tau^4, \tau^2)$. It is the accumulation point for the Fibonacci stairs of McDuffSchlenk12. The green point, where $b=1/3$, is the accumulation point for $c_{1/3}$, with $z=3+2\sqrt{2}$. The plot makes it clear that this is the minimum of the function $b\mapsto\mathrm{acc}(b)$. The black point has $z=6, b=b_0=1/5$ (see \ref{['eqn:bidef']}), and is the minimum of $\operatorname{vol}_b(\mathrm{acc}(b))$.
• Figure 1.2: The black dot represents the ball and the red dot the "cube" $P(1/2, 1/2)$. Points along the $b_1$ axis are Hirzebruch surfaces $H_{b_1}$, while points along the line $b_1+b_2=1$ are polydisks $P(b_1,b_2)$. The balanced blowups are represented by the line $b_1=b_2$.
• Figure 2.1: This figure depicts the moduli space of one-point blowups of $\mathbb{CP}^2(1)$ parametrized by the size of the blowup $b\in[0,1)$. The set $Block$ of $b$-values with $O(b)>0$ consists of the union of the interiors of the colored intervals. The complement of $Block$ consists of $Stair$ and the special rational $b$-values. The special rational $b$-values are the common limit points of the blue and orange intervals (e.g. $b=0.2$) and they do not have infinite staircases. The set $Stair$ consists of $b$-values with principal staircases, $b$-values with non-principal staircases, and the exceptional value $b=1/3$. The principal staircases occur at the endpoints of the colored intervals, where one endpoint has only an ascending staircase and the other only a descending one. In the figure, there are circles with Cantor sets; the non-principal staircases occur at the $b$-values in those Cantor sets that are not endpoints of colored intervals and have both an ascending and descending staircase. Finally, the exceptional value $b=1/3$, which has only an ascending staircase, appears as the limit of the nested circles in the figure. The structure of the region with red intervals is explained in Section \ref{['ss:fractal']} and the structure of the region with orange and blue intervals is explained in Section \ref{['ss:symmetries']}.
• Figure 2.2: MMW This schematic figure shows the relative locations of the centers of the classes obtained from a triple $\mathcal{T}=({\bf{E}}_{\lambda},{\bf{E}}_\mu,{\bf{E}}_\rho)$ by $x$ and $y$ mutations along with the obstruction functions near their centers. The curve in black is the graph of the function $z\mapsto V_b(z)$. If any of these obstructions are live for this value of $b$, then those will contribute a step towards the graph of $c_b$, possibly forming an infinite staircase (or two).
• Figure 2.3: This figure shows the relative position of the blocking $b$-intervals $J_{w{\bf{E}}}$ where $w$ is a word on $x$ and $y$, $w{\bf{E}}$ is the middle class of the generating triple $w\mathcal{T}$, and $\mathcal{T}=({\bf{E}}_{\lambda},{\bf{E}},{\bf{E}}_\rho)$.

Theorems & Definitions (99)

• Theorem 1.1.1
• Theorem 1.1.2
• Theorem 1.1.3
• Theorem 1.2.1
• Remark 1.2.2
• Theorem 1.2.3
• proof : Proof of Theorem \ref{['thm:13proved']}
• proof : Proof of Theorem \ref{['thm:main']}
• Remark 1.3.1
• Remark 2.2.1