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.

arXiv comment77 pages, 3 figures, accepted version, to appear in Inventiones

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Cited by in corpus (8)

• Equations of mirrors to log Calabi--Yau pairs via the heart of canonical wall structures
• Fock-Goncharov dual cluster varieties and Gross-Siebert mirrors
• Quivers and curves in higher dimension
• Mirror symmetry for the Tate curve via tropical and log corals
• Newton--Okounkov bodies and minimal models for cluster varieties
• Scattering diagrams for generalized cluster algebras
• The proper Landau--Ginzburg potential, intrinsic mirror symmetry and the relative mirror map
• The local/logarithmic correspondence and the degeneration formula for quasimaps