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Augmentation varieties and disk potentials III

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

Abstract

This is the third in a series 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 prove that for connected Legendrian covers of monotone Lagrangian tori, the augmentation variety is equal to the image of the zero level set of the disk potential, as suggested by Dimitroglou-Rizell-Golovko. In particular, we show that Legendrian lifts of Vianna's exotic tori are not Legendrian isotopic. Using related ideas, we show that the Legendrian lift of the Clifford torus admits no exact fillings, extending results of Dimitroglou-Rizell and Treumann-Zaslow in dimension two. We consider certain disconnected Legendrians, and show, similar to another suggestion of Aganagic-Ekholm-Ng-Vafa that the components of the augmentation variety correspond to certain partitions and each component is defined by a (not necessarily exact) Lagrangian filling.

Augmentation varieties and disk potentials III

Abstract

This is the third in a series 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 prove that for connected Legendrian covers of monotone Lagrangian tori, the augmentation variety is equal to the image of the zero level set of the disk potential, as suggested by Dimitroglou-Rizell-Golovko. In particular, we show that Legendrian lifts of Vianna's exotic tori are not Legendrian isotopic. Using related ideas, we show that the Legendrian lift of the Clifford torus admits no exact fillings, extending results of Dimitroglou-Rizell and Treumann-Zaslow in dimension two. We consider certain disconnected Legendrians, and show, similar to another suggestion of Aganagic-Ekholm-Ng-Vafa that the components of the augmentation variety correspond to certain partitions and each component is defined by a (not necessarily exact) Lagrangian filling.
Paper Structure (12 sections, 38 theorems, 254 equations, 3 figures)

This paper contains 12 sections, 38 theorems, 254 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Augmentations and fillings
  3. Algebraic augmentation varieties
  4. Geometric augmentation varieties
  5. Toric examples
  6. Exact augmentations
  7. Fillings and partitions
  8. Ruling out exact fillings
  9. The multiple cover formula
  10. Localization for Euler numbers
  11. Localization for disk counts
  12. Comparison of perturbation schemes

Key Result

Theorem 1.1

Suppose that $Y$ is a monotone symplectic manifold with minimal Chern number at least two and $Z$ a contact manifold whose curvature is a negative multiple of the symplectic form. For a Legendrian lift $\Lambda \subset Z$ of a connected compact monotone Lagrangian $\Pi \subset Y$ with minimal Maslov

Figures (3)

  • Figure 1: Basis to ensure non-negative exponents
  • Figure 2: Newton polytope for two and three-dimensional tori corresponding to to $(a,b,c)$.
  • Figure 3: Homological relations between components of moduli spaces

Theorems & Definitions (91)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Remark 1.5
  • Example 2.1
  • Lemma 2.2
  • proof
  • Definition 2.3
  • Example 2.4
  • ...and 81 more