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Tropical disk potential for almost toric manifolds

S. Venugopalan, C. T. Woodward

Abstract

Using our previous work we give a tropical formula for disk potentials for Lagrangian tori in almost toric four-manifolds, that is, fibrations by Lagrangian tori with only toric and focus-focus singularities, generalizing results of Mikhalkin for holomorphic spheres in the projective plane. As examples, we directly compute potentials for Lagrangian tori in del Pezzo surfaces equipped with monotone symplectic forms. These formulas were established in the monotone case by different methods in Pascaleff-Tonkonog, and investigated from the point of view of the Gross-Siebert program in Carl-Pumperla-Siebert, Bardwell-Evans--Cheung--Hong--Lin and also Lau-Lee-Lin.

Tropical disk potential for almost toric manifolds

Abstract

Using our previous work we give a tropical formula for disk potentials for Lagrangian tori in almost toric four-manifolds, that is, fibrations by Lagrangian tori with only toric and focus-focus singularities, generalizing results of Mikhalkin for holomorphic spheres in the projective plane. As examples, we directly compute potentials for Lagrangian tori in del Pezzo surfaces equipped with monotone symplectic forms. These formulas were established in the monotone case by different methods in Pascaleff-Tonkonog, and investigated from the point of view of the Gross-Siebert program in Carl-Pumperla-Siebert, Bardwell-Evans--Cheung--Hong--Lin and also Lau-Lee-Lin.

Paper Structure

This paper contains 42 sections, 48 theorems, 204 equations, 51 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Almost toric manifolds
  3. Focus-focus singularities
  4. Modifications of almost toric structures
  5. Tropical limit theorems
  6. Disk potentials
  7. Broken Maps
  8. Tropical limit theorems for moduli spaces
  9. Relative maps
  10. Intersecting polyhedral decompositions
  11. Curves in the elementary pieces
  12. Holomorphic spheres in toric varieties
  13. Disks in toric varieties
  14. Spheres meeting the focus-focus singularities
  15. Computing multiplicities via desingularization
  16. ...and 27 more sections

Key Result

Theorem 1.3

Let $X$ be a compact almost toric manifold of dimension four, $L$ a Lagrangian torus fiber of $\Phi$ and $\mathcal{P}$ an elementary polyhedral decomposition, for which any disk contributing to the potential $W_{\mathcal{X},L}$ has a tropical graph with all collisions in generic interior points. The where the multiplicities $m(v)$ are as in Definition dp:def:mvs below, and $\operatorname{Aut} (\Ga

Figures (51)

  • Figure 1: An almost toric diagram for the del Pezzo of degree four and the twelve tropical disks contributing to the potential of the monotone torus
  • Figure 2: An almost toric diagram for the Chekanov torus and the four tropical disks contributing to the potential
  • Figure 3: Cartoon diagrams for the 252 degree one curves in $\operatorname{Bl}^8 \mathbb{P}^2$
  • Figure 4: Perturbing incoming edges of $v$ with parameters $\ell_1>\ell_2>\ell_3$. The pairs $e_1$, $e_5$ and $e_2$, $e_3$ are coincident in $\Gamma$.
  • Figure 5: Perturbing $\Gamma$ in Figure \ref{['dp:fig:pert-edge']} with parameters $\ell_2>\ell_3>\ell_1$ produces two perturbed graphs.
  • ...and 46 more figures

Theorems & Definitions (162)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.3
  • Definition 1.4
  • Example 1.5
  • Example 1.6
  • Example 1.7
  • Definition 1.8
  • Definition 1.9
  • Definition 1.10
  • ...and 152 more