Towers of Looijenga pairs and asymptotics of ECH capacities

TL;DR

This paper explores the relationship between ECH capacities and algebraic geometry through towers of polarised toric surfaces, establishing new asymptotic results and obstructions for symplectic embeddings.

Contribution

It extends the connection between ECH capacities and algebraic surfaces to towers of Looijenga pairs, providing new criteria for asymptotic convergence and embedding obstructions.

Findings

Sub-leading asymptotics of ECH capacities are O(1) for all convex and concave toric domains.

Established criteria for convergence of sub-leading asymptotics in toric domains.

Proposed invariants to determine convergence in the toric case.

Abstract

ECH capacities are rich obstructions to symplectic embeddings in 4-dimensions that have also been seen to arise in the context of algebraic positivity for (possibly singular) projective surfaces. We extend this connection to relate general convex toric domains on the symplectic side with towers of polarised toric surfaces on the algebraic side, and then use this perspective to show that the sub-leading asymptotics of ECH capacities for all convex and concave toric domains are $O(1)$. We obtain sufficient criteria for when the sub-leading asymptotics converge in this context, generalising results of Hutchings and of the author, and derive new obstructions to embeddings between toric domains of the same volume. We also propose two invariants to more precisely describe when convergence occurs in the toric case. Our methods are largely non-toric in nature, and apply more widely to towers of polarised Looijenga pairs.