Towers of Looijenga pairs and asymptotics of ECH capacities

Abstract

A. 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 p1q. 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.

Lattice Formulas For Rational SFT Capacities

“…the ECH capacities and the Gutt-Hutchings capacities, come in natural families indexed by the integers. The asymptotic behavior of these capacities as the index goes to 8 is of significant interest [9,10,13,34] and is key to some of the dynamical applications [17]. The RSFT capacities are naturally indexed by the codimension of the tangency constraint, and it is natural to ask how these capacities behave as the codimension diverges.…”

“…The symplectic capacities of toric domains have been studied extensively. See the work of Lu [21], Choi et al [3], Landry et al [20], Cristofaro-Gardiner [6], Gutt-Hutchings [13], Gutt at al [14] Hutchings [15,16], Siegel [28], Wormleighton [33,34] and Chaidez-Wormleighton [2].…”

“…This will be discussed in §3. 3. This paper is part of a program [2,[32][33][34] of the second author (and sometimes the first) to understand the relationship between the quantitative symplectic geometry of Ω and the algebraic geometry of Σ . 1.3.…”