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Symplectic capacities, unperturbed curves, and convex toric domains

Dusa McDuff, Kyler Siegel

Abstract

We use explicit pseudoholomorphic curve techniques (without virtual perturbations) to define a sequence of symplectic capacities analogous to those defined recently by the second named author using symplectic field theory. We then compute these capacities for all four-dimensional convex toric domains. This gives various new obstructions to stabilized symplectic embedding problems which are sometimes sharp.

Symplectic capacities, unperturbed curves, and convex toric domains

Abstract

We use explicit pseudoholomorphic curve techniques (without virtual perturbations) to define a sequence of symplectic capacities analogous to those defined recently by the second named author using symplectic field theory. We then compute these capacities for all four-dimensional convex toric domains. This gives various new obstructions to stabilized symplectic embedding problems which are sometimes sharp.

Paper Structure

This paper contains 38 sections, 52 theorems, 128 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Overview
  3. Statement of main results
  4. Examples and applications
  5. Preliminaries on pseudoholomorphic curves
  6. Asymptotically cylindrical curves and their moduli
  7. Symplectic and contact manifolds
  8. Admissible almost complex structures
  9. Moduli spaces of pseudoholomorphic curves
  10. SFT compactifications
  11. Homology classes and energy
  12. Local tangency and skinny ellipsoidal constraints
  13. Formal curves
  14. Formal curve components
  15. Formal nodal curves and buildings
  16. ...and 23 more sections

Key Result

Theorem 1.2.2

For each $k \in \mathbb{Z}_{\geq 1}$, $\tilde{\mathfrak{g}}_k$ is independent of the choice of local divisor and is a symplectomorphism invariant. It satisfies the following properties:

Figures (1)

  • Figure 1: A configuration which could potentially arise in $\overline{\mathcal{M}}_X^J(\Gamma^+;\Gamma^-)\Langle \mathcal{T}^{(m)}p\Rangle$. Here the marked point $z_0$ mapping to $p$ lies on a ghost component, and $z_1,z_2,z_3$ are the special points near $z_0$ lying on nonconstant components. These satisfy respective constraints $\Langle \mathcal{T}^{(m_1)}p\Rangle, \Langle \mathcal{T}^{(m_2)}p\Rangle,\Langle \mathcal{T}^{(m_3)}p\Rangle$ such that $m_1+m_2+m_3 \geq m$. Such a configuration is also included in $\overline{\overline{\mathcal{M}}}_X^J(\Gamma^+;\Gamma^-)\Langle \mathcal{T}^{(m)}p\Rangle$ even if it does not arise as a limit of curves in $\mathcal{M}_X^J(\Gamma^+;\Gamma^-)\Langle \mathcal{T}^{(m)}p\Rangle$.

Theorems & Definitions (132)

  • Remark 1.2.1
  • Theorem 1.2.2
  • Remark 1.2.3: Stabilization hypotheses
  • Remark 1.2.4: Relationship with $\mathfrak{g}_k$
  • Remark 1.2.5: Relationship with Gutt--Hutchings capacities
  • Remark 1.2.6: Nondecreasing property
  • Remark 1.2.7: Generalizations
  • Theorem 1.2.8
  • Corollary 1.2.9
  • Remark 1.2.10
  • ...and 122 more