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Augmentation varieties and disk potentials II

Kenneth Blakey, Soham Chanda, Yuhan Sun, Chris T. Woodward

TL;DR

This work develops an algebraic framework for Legendrian contact geometry in circle-fibered contact manifolds by constructing the Chekanov-Eliashberg algebra CE(\Λ) and the Legendrian contact homology HE(\Λ) via a differential counting punctured holomorphic treed disks, with HE(\Λ)=\ker(\delta)/\mathrm{im}(\delta). It introduces cobordism-induced chain maps $\varphi(L,b)$ parameterized by bounding chains $b\in MC(L)$, establishes their independence from auxiliary choices, and proves a composition law that mirrors neck-stretching in symplectic cobordisms. The paper provides extensive classification results for disks in toric and Harvey–Lawson settings, computes leading abelianized differential terms, and connects disk potentials to augmentation data, enabling augmentation varieties to distinguish Legendrian isotopy classes. By developing special perturbations and an invariant abelianized theory, the framework yields practical tools for constructing augmentations from fillings and for comparing Legendrians via their cobordisms and potentials. Overall, the results extend Legendrian contact homology to circle-fibered geometries, offering new invariants and computational handles in higher dimensions with explicit toric and HL-filling examples.

Abstract

This is the second in a sequence of papers in which we construct Chekanov-Eliashberg algebras for Legendrians in circle-fibered contact manifolds and study the associated augmentation varieties. In this part, we first define the Chekanov-Eliashberg algebra and its Legendrian contact homology. For a tame Lagrangian cobordism between Legendrians, we define a chain map between their Chekanov-Eliashberg algebras.

Augmentation varieties and disk potentials II

TL;DR

This work develops an algebraic framework for Legendrian contact geometry in circle-fibered contact manifolds by constructing the Chekanov-Eliashberg algebra CE(\Λ) and the Legendrian contact homology HE(\Λ) via a differential counting punctured holomorphic treed disks, with HE(\Λ)=\ker(\delta)/\mathrm{im}(\delta). It introduces cobordism-induced chain maps parameterized by bounding chains , establishes their independence from auxiliary choices, and proves a composition law that mirrors neck-stretching in symplectic cobordisms. The paper provides extensive classification results for disks in toric and Harvey–Lawson settings, computes leading abelianized differential terms, and connects disk potentials to augmentation data, enabling augmentation varieties to distinguish Legendrian isotopy classes. By developing special perturbations and an invariant abelianized theory, the framework yields practical tools for constructing augmentations from fillings and for comparing Legendrians via their cobordisms and potentials. Overall, the results extend Legendrian contact homology to circle-fibered geometries, offering new invariants and computational handles in higher dimensions with explicit toric and HL-filling examples.

Abstract

This is the second in a sequence of papers in which we construct Chekanov-Eliashberg algebras for Legendrians in circle-fibered contact manifolds and study the associated augmentation varieties. In this part, we first define the Chekanov-Eliashberg algebra and its Legendrian contact homology. For a tame Lagrangian cobordism between Legendrians, we define a chain map between their Chekanov-Eliashberg algebras.
Paper Structure (18 sections, 45 theorems, 249 equations, 15 figures)

This paper contains 18 sections, 45 theorems, 249 equations, 15 figures.

Table of Contents

  1. Introduction
  2. Classification results
  3. Examples of punctured holomorphic disks
  4. Lifting disks from the base
  5. Disks bounding toric Lagrangians
  6. Disks in the Harvey-Lawson filling
  7. Circle-fibered Legendrian contact homology
  8. Coefficient rings
  9. The chain group
  10. The contact differential
  11. Special perturbations
  12. Chain maps from cobordisms
  13. Cobordism maps
  14. Bounding chains
  15. Independence of the cobordism maps from choices
  16. ...and 3 more sections

Key Result

Theorem 1.1

Suppose that $\Lambda$ is a tame Legendrian in a contact manifold $(Z, \alpha)$. Counting punctured holomorphic disks $u: C \to \mathbb{R} \times Z$ bounding $\mathbb{R} \times \Lambda$ defines Legendrian contact homology groups independent of the choice of almost complex structures and perturbations, and is an invariant of the Legendrian isotopy class of $\Lambda$.

Figures (15)

  • Figure 1: A zero-area map with three punctures
  • Figure 2: Adding multiple trajectories at an intersection point
  • Figure 3: Examples of the forgetful map
  • Figure 4: Treed disks defining the chain map
  • Figure 5: Holomorphic curves defining the curvature
  • ...and 10 more figures

Theorems & Definitions (139)

  • Theorem 1.1
  • Theorem 1.2
  • Example 2.1
  • Example 2.2
  • Example 2.3
  • Example 2.4
  • Definition 2.5
  • Lemma 2.6
  • proof
  • Definition 2.7
  • ...and 129 more