The canonical wall structure and intrinsic mirror symmetry

TL;DR

This paper constructs and proves the consistency of a canonical wall structure derived from log Calabi-Yau pairs, providing a detailed description of the mirror family in the context of intrinsic mirror symmetry.

Contribution

It introduces a canonical wall structure based on punctured invariants, proving its consistency and its equivalence to the intrinsic mirror, advancing mirror symmetry understanding.

Findings

Proved the consistency of the canonical wall structure.

Established the equivalence of the constructed mirror family with the intrinsic mirror.

Provided a detailed algebro-geometric description of the mirror.

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.