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Enumerative Geometry of Del Pezzo Surfaces

Yu-Shen Lin

Abstract

We prove an equivalence between the superpotential defined via tropical geometry and Lagrangian Floer theory for special Lagrangian torus fibres in del Pezzo surfaces constructed by Collins-Jacob-Lin. We also include some explicit calculations for the projective plane, which confirm some folklore conjecture in this case.

Enumerative Geometry of Del Pezzo Surfaces

Abstract

We prove an equivalence between the superpotential defined via tropical geometry and Lagrangian Floer theory for special Lagrangian torus fibres in del Pezzo surfaces constructed by Collins-Jacob-Lin. We also include some explicit calculations for the projective plane, which confirm some folklore conjecture in this case.

Paper Structure

This paper contains 20 sections, 41 theorems, 78 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Special Lagrangian Fibration on Del Pezzo Surfaces
  3. Affine Structures on the Base of SYZ Fibrations
  4. Some Useful Consequence of the Asymptotics
  5. Tropical Geometry
  6. Broken Lines and Tropical Superpotential
  7. Floer Theory of Special Lagrangian Fibration on HyperKähler Surfaces
  8. $A_{\infty}$ Structures from Lagrangian Floer Theory
  9. Fukaya's Trick and Open Gromov-Witten Invariants
  10. Correspondence Theorem
  11. Superpotential Functions and Tropical/Holomorphic Correspondence
  12. Definition of the Floer theoretic Superpotential
  13. On the Relative Gromov-Witten Invariants
  14. Tropical Geometry of $\mathbb{P}^2$
  15. Complex Affine Structure of $\mathbb{P}^2$
  16. ...and 5 more sections

Key Result

Theorem 1.1

(=Theorem 115) There exists an open neighborhood $U$ such that the special Lagrangian torus fibre contained in $U$ bounds a unique holomorphic disc of Maslov index two disc contained in $U$.

Figures (2)

  • Figure 1: The integral affine structure in Carl-Pumperla-Siebert
  • Figure 2: All the admissible tropical discs contribute to the calculation of Example \ref{['168']}. The bounded edges are contracted to a point and all the tropical discs have the same image.

Theorems & Definitions (99)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 2.1
  • Theorem 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • ...and 89 more