On infinite staircases in toric symplectic four-manifolds

TL;DR

The paper develops a general framework to study infinite staircases in ellipsoid embeddings within four-dimensional symplectic geometry, relating closed toric manifolds to finite-type convex toric domains via negative weight expansions. It proves that for finite-type targets, any infinite staircase must accumulate at a unique point $a_0$ determined by a quadratic equation $a^2-igl(rac{ ext{per}^2}{ ext{vol}}-2igr)a+1=0$, with the embedding value at $a_0$ matching the volume bound, and uses this to obstruct possible staircases. In the rational (finite-type) case, it identifies six reflexive-polygon families that yield twelve domains with uniform proofs of infinite staircases, using almost toric fibrations and convex lattice paths, and describes two distinct accumulation behaviors (depending on $J=2$ or $J=3$). The work further extends the framework to some non-toric closed targets and articulates a conjecture that these reflexive cases may be the only infinite staircases among rational convex toric domains, connecting the phenomenon to number-theoretic questions about Ehrhart-type functions and continued fractions. Overall, the paper provides a powerful obstruction criterion, a uniform constructive approach for a broad class of targets, and a compelling conjectural classification tied to reflexive geometry and arithmetic.

Abstract

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 recently led to considerable interest in understanding when the ellipsoid embedding function for other symplectic 4-manifolds 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 4-manifolds. 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 we show is powerful. 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. For the second tool, we find recursive families of convex lattice paths that provide obstructions to embeddings. We conclude by reducing 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.

Figures (26)

• Figure 1.1: This illustrates several regions in $\mathbb{R}^2_{\geq 0}$ that are Delzant polygons (with bottom left corner at the origin). These polygons correspond to the closed manifolds (a) $\mathbb{C} P^2$; (b) $\mathbb{C} P^2 \# \overline{\mathbb{C} P}^2$; (c) $\mathbb{C} P^1\times{\mathbb{C} P}^1$; (d) $\mathbb{C} P^2 \# 2\overline{\mathbb{C} P}^2$; and (e) $\mathbb{C} P^2 \# 3\overline{\mathbb{C} P}^2$. As we will see, the ellipsoid embedding functions for these examples admit infinite staircases.
• Figure 1.2: Regions corresponding to finite type rational convex toric domains, and their negative weight expansions $(b;b_1,\ldots,b_n)$. These negative weight expansions are precisely the ones that feature in Theorem \ref{['thm:main']}. The negative weight expansions $(3;1)$ and $(3;1,1)$ correspond to the $J=3$ case and all others are $J=2$; cf. Table \ref{['table:recurrence']} and Remark \ref{['rem: all cases']}. Note that all of these polygons are reflexive: each is a lattice polygon with precisely one interior lattice point.
• Figure 1.3: Plots of ellipsoid embedding functions for different domains, labeled by their negative weight expansion. The red curves are the volume curves and the vertical lines indicate where the accumulation points would necessarily be located, if a staircase existed, per Theorem \ref{['satisfies quadratic equation']}. The top two plots have infinite staircases: in (a) we have a $J=2$ case, where the inner corners touch the volume curve (a sharp infinite staircase); and in (b) we have a $J=3$ case, where the inner corners approach but do not touch the volume curve. The plots (c) and (d) do not have infinite staircases: (c) has non-zero staircase obstruction, while showing that (d) does not have a staircase is intricate cg ellipsoid.
• Figure 2.1: This figure illustrates how to compute $\ell_\Omega(\Lambda)$. The direction $\nu$ contributes one of the terms in the summation \ref{['eq:Omega length']}. In this case, $\nu = (2,-3)$ and $p_{\Omega,\nu}= (1,2)$, and so the contribution is $\det \left(21-32\right) = 4+3 = 7$.
• Figure 2.2: This figure illustrates how to compute the weight expansion for $a=\frac{8}{3}$. We must divide a $1\times \frac{8}{3}$ rectangle into squares. The process begins with $\ell_0=2$ squares of size $1\times 1$, then $\ell_1=1$ square of size $\frac{2}{3} \times \frac{2}{3}$, and finally $\ell_2=2$ squares of size $\frac{1}{3}\times\frac{1}{3}$.

Theorems & Definitions (67)

• Definition 1.2
• Theorem 1.3
• Remark 1.6
• Remark 1.7
• Definition 1.8
• Remark 1.9
• Remark 1.10
• Theorem 1.11
• Remark 1.13
• Remark 1.14