[Paper Review] Equivariant symplectic homology, linearized contact homology and the Lagrangian capacity

Miguel Pereira|arXiv (Cornell University)|May 26, 2022

Homotopy and Cohomology in Algebraic Topology4 citations

TL;DR

This paper computes the Lagrangian capacity of 4-dimensional convex toric domains, proving it equals the diagonal capacity via comparisons with McDuff–Siegel and Gutt–Hutchings symplectic capacities. Using equivariant symplectic homology and linearized contact homology under a virtual perturbation assumption, the author establishes transversality for holomorphic curves in ellipsoids, enabling the computation in arbitrary dimensions for convex or concave toric domains.

ABSTRACT

We establish computational results concerning the Lagrangian capacity from "Cieliebak and Mohnke - Punctured holomorphic curves and Lagrangian embeddings". More precisely, we show that the Lagrangian capacity of a 4-dimensional convex toric domain is equal to its diagonal. The proof involves comparisons between the Lagrangian capacity, the McDuff-Siegel capacities from "McDuff and Siegel - Symplectic capacities, unperturbed curves, and convex toric domains", and the Gutt-Hutchings capacities from "Gutt and Hutchings - Symplectic capacities from positive S1-equivariant symplectic homology". Working under the assumption that there is a suitable virtual perturbation scheme which defines the curve counts of linearized contact homology, we extend the previous result to toric domains which are convex or concave and of any dimension. For this, we use the higher symplectic capacities from "Siegel - Higher symplectic capacities". The key step is showing that moduli spaces of asymptotically cylindrical holomorphic curves in ellipsoids are transversely cut out.

Motivation & Objective

• To compute the Lagrangian capacity of 4-dimensional convex toric domains.
• To extend the computation to higher-dimensional convex or concave toric domains using advanced symplectic capacities.
• To establish transversality for moduli spaces of asymptotically cylindrical holomorphic curves in ellipsoids.
• To relate the Lagrangian capacity to other symplectic capacities (McDuff–Siegel, Gutt–Hutchings) via inequality chains.
• To provide a computational framework using equivariant symplectic homology and linearized contact homology under a virtual perturbation assumption.

Proposed method

• Uses the Gutt–Hutchings capacities $ c^{ ext{GH}}_k $ and higher symplectic capacities $ ilde{rak{g}}^{ ext{≤1}}_k $, $ rak{g}^{ ext{≤1}}_k $ defined via action and word-length filtrations.
• Applies action filtration $ rak{A}^{ ext{≤a}} $ and word-length filtration $ rak{B}^{ ext{≤ℓ}} $ to define capacities from linearized contact homology.
• Employs equivariant symplectic homology to analyze holomorphic curves in symplectic manifolds with $ S^1 $-actions.
• Relies on a virtual perturbation scheme to define curve counts in linearized contact homology, assuming its existence.
• Proves transverse transversality for moduli spaces of asymptotically cylindrical holomorphic curves in ellipsoids.
• Composes inequalities between capacities to derive the Lagrangian capacity as the infimum of action levels where certain homology maps are non-zero.

Experimental results

Research questions

• RQ1 What is the Lagrangian capacity of a 4-dimensional convex toric domain?
• RQ2 How do the Lagrangian capacity, McDuff–Siegel capacities, and Gutt–Hutchings capacities relate in symplectic geometry?
• RQ3 Can the Lagrangian capacity be computed for higher-dimensional convex or concave toric domains using holomorphic curve invariants?
• RQ4 Under what conditions are moduli spaces of asymptotically cylindrical holomorphic curves in ellipsoids transversely cut out?
• RQ5 To what extent can the Lagrangian capacity be computed via spectral invariants from linearized contact homology?

Key findings

• The Lagrangian capacity of a 4-dimensional convex toric domain equals its diagonal capacity.
• The computation extends to convex or concave toric domains of any dimension under the assumption of a virtual perturbation scheme for linearized contact homology.
• Transversality is established for moduli spaces of asymptotically cylindrical holomorphic curves in ellipsoids, a key technical step.
• The Lagrangian capacity is shown to be bounded above and below by the same value via chains of symplectic capacity inequalities.
• The higher symplectic capacities $ rak{g}^{ ext{≤1}}_k $ and $ ilde{rak{g}}^{ ext{≤1}}_k $, defined via filtrations on linearized contact homology, are instrumental in the computation.
• The result demonstrates that Lagrangian capacity, defined via embedded Lagrangian tori, can be computed via holomorphic curve invariants in symplectic topology.