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Mirror Symmetry for log Calabi-Yau Surfaces II

Jonathan Lai, Yan Zhou

Abstract

We show that the ring of regular functions of every smooth affine log Calabi-Yau surface with maximal boundary has a vector space basis parametrized by its set of integer tropical points and a $\mathbb{C}$-algebra structure with structure coefficients given by the geometric construction of Keel-Yu. To prove this result, we first give a canonical compactification of the mirror family associated with a pair $(Y,D)$ constructed by Gross-Hacking-Keel where $Y$ is a smooth projective rational surface, $D$ is an anti-canonical cycle of rational curves and $Y\setminus D$ is the minimal resolution of an affine surface with, at worst, du Val singularities. Then, we compute periods for the compactified family using techniques from work of Ruddat-Siebert and use this to give a modular interpretation of the compactified mirror family.

Mirror Symmetry for log Calabi-Yau Surfaces II

Abstract

We show that the ring of regular functions of every smooth affine log Calabi-Yau surface with maximal boundary has a vector space basis parametrized by its set of integer tropical points and a -algebra structure with structure coefficients given by the geometric construction of Keel-Yu. To prove this result, we first give a canonical compactification of the mirror family associated with a pair constructed by Gross-Hacking-Keel where is a smooth projective rational surface, is an anti-canonical cycle of rational curves and is the minimal resolution of an affine surface with, at worst, du Val singularities. Then, we compute periods for the compactified family using techniques from work of Ruddat-Siebert and use this to give a modular interpretation of the compactified mirror family.
Paper Structure (26 sections, 71 theorems, 167 equations, 14 figures)

This paper contains 26 sections, 71 theorems, 167 equations, 14 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. Connections to other works.
  4. Tropicalization of theta functions in dimension 2
  5. Theta functions in algebraic geometry
  6. The symplectic cohomology of smooth affine log Calabi-Yau surfaces.
  7. Outline of the paper.
  8. Acknowledgements
  9. Periods for Looijenga Pairs
  10. Generalized marked pairs
  11. Review of Mirror Symmetry for Log Calabi-Yau Surfaces
  12. Compactifications via Polygons
  13. Generalized Half Spaces and Polygons in $B$
  14. Passing to toric blowups and blowdowns.
  15. Compactified Families
  16. ...and 11 more sections

Key Result

Theorem 1.1

Let $U$ be a smooth affine log Calabi-Yau surface with maximal boundary. Then, there is a decomposition of the vector space of regular functions on $U$ into one-dimensional subspaces, canonical up to the choice of one of the two possible orientations of $U^{\operatorname{trop}}(\mathbb{Z})$.

Figures (14)

  • Figure 4.1: The developing map for the degree $1$ del Pezzo
  • Figure 4.2: The half space $Z(L)$ is a bounded polygon.
  • Figure 4.3: $P(W)$ for $(Y,D)$ where $n=1$ and $D^2 =3$.
  • Figure 4.4: $P(W)$ for the degree $1$ del Pezzo.
  • Figure 4.5: The toric fan for $\mathbb{P}(1,1,2)$.
  • ...and 9 more figures

Theorems & Definitions (178)

  • Theorem 1.1
  • Definition 1.2
  • Definition 1.3
  • Definition 1.4
  • Definition 1.5
  • Definition 1.6
  • Definition 1.7
  • Theorem 1.8
  • Theorem 1.9
  • Definition 2.1
  • ...and 168 more