Intrinsic Mirror Symmetry

Abstract

We associate a ring R to a log Calabi-Yau pair (X,D) or a degeneration of Calabi-Yau manifolds X->B. The vector space underlying R is determined by the tropicalization of (X,D) or X->B, while the product rule is defined using punctured Gromov-Witten invariants, defined in joint work with Abramovich and Chen. In the log Calabi-Yau case, if D is maximally degenerate, then we propose that Spec R is the mirror to X\D, while in the Calabi-Yau degeneration case, if the degeneration is maximally unipotent, the mirror is expected to be Proj R. The main result in this paper is that R as defined is an associative, commutative ring with unit, with associativity the most difficult part.

175 pages. Accepted version, to appear in Journal of the AMS

References in corpus (5)

• The tropicalization of the moduli space of curves
• Functorial desingularization of quasi-excellent schemes in characteristic zero: the non-embedded case
• The Evaluation Space of Logarithmic Stable Maps
• Punctured logarithmic maps
• The canonical embedding of an unramified morphism in an étale morphism

Cited by in corpus (11)

• The canonical wall structure and intrinsic mirror symmetry
• Punctured logarithmic maps
• Mirror Symmetry and smoothing Gorenstein toric affine 3-folds
• Strong positivity for quantum theta bases of quantum cluster algebras
• The Higher Dimensional Tropical Vertex
• A Gromov-Witten theory for simple normal-crossing pairs without log geometry
• Intrinsic mirror symmetry and categorical crepant resolutions
• Algebraic and symplectic viewpoint on compactifications of two-dimensional cluster varieties of finite type
• An extension of polar duality of toric varieties and its consequences in Mirror Symmetry
• Mirror symmetry for the Tate curve via tropical and log corals
• Towards the Doran-Harder-Thompson conjecture via the Gross-Siebert program