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The canonical wall structure and intrinsic mirror symmetry

Mark Gross, Bernd Siebert

Abstract

As announced "Intrinsic mirror symmetry and punctured invariants" in 2016, we construct and prove consistency of the canonical wall structure. This construction starts with a log Calabi-Yau pair (X,D) and produces a wall structure, as defined by Gross-Hacking-Siebert. Roughly put, the canonical wall structure is a data structure which encodes an algebro-geometric analogue of counts of Maslov index zero disks. These enumerative invariants are defined in terms of the punctured invariants of Abramovich-Chen-Gross-Siebert. There are then two main theorems of the paper. First, we prove consistency of the canonical wall structure, so that the canonical wall structure gives rise to a mirror family. Second, we prove that this mirror family coincides with the intrinsic mirror constructed in our paper "Intrinsic mirror symmetry". While the setup of this paper is narrower than that of the latter paper, it gives a more detailed description of the mirror.

The canonical wall structure and intrinsic mirror symmetry

Abstract

As announced "Intrinsic mirror symmetry and punctured invariants" in 2016, we construct and prove consistency of the canonical wall structure. This construction starts with a log Calabi-Yau pair (X,D) and produces a wall structure, as defined by Gross-Hacking-Siebert. Roughly put, the canonical wall structure is a data structure which encodes an algebro-geometric analogue of counts of Maslov index zero disks. These enumerative invariants are defined in terms of the punctured invariants of Abramovich-Chen-Gross-Siebert. There are then two main theorems of the paper. First, we prove consistency of the canonical wall structure, so that the canonical wall structure gives rise to a mirror family. Second, we prove that this mirror family coincides with the intrinsic mirror constructed in our paper "Intrinsic mirror symmetry". While the setup of this paper is narrower than that of the latter paper, it gives a more detailed description of the mirror.

Paper Structure

This paper contains 22 sections, 35 theorems, 237 equations, 3 figures.

Table of Contents

  1. The basic setup
  2. The tropicalization of a log Calabi-Yau pair
  3. Minimal log Calabi-Yau pairs
  4. The affine structure
  5. Punctured log maps and tropical geometry
  6. Review of notation
  7. Some tropical lemmas
  8. The canonical wall structure and logarithmic broken lines
  9. Recall of monomials on $B$
  10. The canonical wall structure
  11. Recall of wall structures
  12. The construction
  13. The relative case
  14. Logarithmic broken lines and theta functions
  15. A correspondence theorem for broken lines
  16. ...and 7 more sections

Key Result

Proposition 1.3

If $(X,D)$ satisfies Assumptions ass:absolute or ass:relative in the absolute or relative cases respectively, then $(B,\mathscr{P})$ is a pseudomanifold in the sense of satisfying conditions (1)--(5) of Theta. Further, for each one-dimensional good stratum $X_{\rho}$, either (a) $X_{\rho}\cong \math

Figures (3)

  • Figure 1.1: Sketch of $(X,D)$ in Example \ref{['ex:running example 1']}.
  • Figure 1.2: The two-dimensional stratum $X_\omega$.
  • Figure 4.1: The tropical type $\omega_0$ (codimension zero case).

Theorems & Definitions (95)

  • Proposition 1.3
  • proof
  • Example 1.4
  • Definition 1.5
  • Proposition 1.6
  • proof
  • Theorem 1.7
  • Proposition 1.8
  • proof
  • Remark 1.9
  • ...and 85 more