Lectures on stabilized ellipsoid embeddings

TL;DR

The paper investigates stabilized four-dimensional ellipsoid embeddings, proving that for all stabilization levels $N\ge1$ the function $\mathcal{E}_{B^4}^N(1,a)$ exhibits a subcritical infinite Fibonacci staircase on $[1,\tau^4]$ and a supercritical regime for $a>\tau^4$ governed by the rational function $\tfrac{3a}{a+1}$ (in the stabilized setting). Central to the argument is a correspondence between symplectic embedding obstructions and index-zero $(p,q)$-sesquicuspidal curves in $\mathbb{CP}^2$, together with an enhanced inflation technique and a scattering-diagram/tropical-vertex framework that encodes curve counts as wall-crossing data. The authors develop a rich geometric- combinatorial toolkit—Looijenga pairs, toric models, elementary transformations, and the change-of-lattice trick—to translate cusp-counting into embedding bounds, culminating in a complete proof of the main stabilized-ellipsoid-theorem and a pathway to generalizations on monotone del Pezzo surfaces. This synthesis advances the understanding of how higher-dimensional stabilization alters embedding obstructions and links symplectic topology with cluster structures and tropical geometry, with potential implications for broader stabilized embedding problems.

Abstract

These notes are based on a five-part minicourse on stabilized symplectic embeddings given in Les Marécottes, Switzerland during a September 2025 workshop. Our main goal is to explain the recent resolution of the (restricted) stabilized ellipsoid embedding problem by D. McDuff and the author. Along the way we also introduce various other ideas which shed light on the context and hint at possible generalizations. Some of the concepts covered include sesquicuspidal curves, symplectic inflation, multidirectional tangency constraints, well-placed curves, cluster transformations, Looijenga pairs, toric models, scattering diagrams, and the tropical vertex theorem.

Figures (21)

• Figure 1: The image of $E(a_1,a_2)$ under the moment map $\mu: \mathbb{C}^2 \rightarrow \mathbb{R}_{\geq 0}^2$.
• Figure 2: The ellipsoid embedding function $\mathcal{E}_{B^4}(1,a)$ (not to scale).
• Figure 3: The box diagram giving the weight sequence for $a=17/5$.
• Figure 4: The stabilized ellipsoid embedding function $\mathcal{E}_{B^4}^N(1,a)$ (taken from sesqui).
• Figure 5: The local toric picture for the $(p,q)$-weighted symplectic blowup $M \rightsquigarrow \widetilde{M}$.

Theorems & Definitions (74)

• Remark 1.2
• Remark 1.4
• Definition 1.6
• Remark 1.7
• Example 1.12
• Remark 1.17
• Remark 1.18
• Corollary 1.20
• Lemma 1.21
• proof