Arithmetic mirror symmetry for the 2-torus

Abstract

This paper explores a refinement of homological mirror symmetry which relates exact symplectic topology to arithmetic algebraic geometry. We establish a derived equivalence of the Fukaya category of the 2-torus, relative to a basepoint, with the category of perfect complexes of coherent sheaves on the Tate curve over the "formal disc" Spec Z[[q]]. It specializes to a derived equivalence, over Z, of the Fukaya category of the punctured torus with perfect complexes on the curve y^2+xy=x^3 over Spec Z, the central fibre of the Tate curve; and, over the "punctured disc" Spec Z((q)), to an integral refinement of the known statement of homological mirror symmetry for the 2-torus. We also prove that the wrapped Fukaya category of the punctured torus is derived-equivalent over Z to bounded complexes of coherent sheaves on the central fiber of the Tate curve.

95 pages, 5 figures

References in corpus (10)

• Uniqueness of enhancement for triangulated categories
• Cyclic symmetry and adic convergence in LagrangianFloer theory
• Homological Mirror Symmetry for Calabi-Yau hypersurfaces in projective space
• Symplectic cohomology and duality for the wrapped Fukaya category
• Fukaya categories and deformations
• Deformation theory via differential graded Lie algebras
• Fukaya categories of the torus and Dehn surgery
• Quantization on Curves
• Seidel's Mirror Map for Abelian Varieties
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Cited by in corpus (13)

• The HOMFLYPT skein algebra of the torus and the elliptic Hall algebra
• Auslander orders over nodal stacky curves and partially wrapped Fukaya categories
• Homological mirror symmetry for log Calabi-Yau surfaces
• The combinatorics of Morse theory with boundary
• Mirror symmetry and Fukaya categories of singular hypersurfaces
• Formal groups and quantum cohomology
• Homological mirror symmetry for invertible polynomials in two variables
• A modular compactification of ${\mathcal{M}}_{1,n}$ from $A_{\infty }$-structures
• Moduli of curves with nonspecial divisors and relative moduli of $A_{\infty }$-structures
• $A_{\infty }$-structures associated with pairs of 1-spherical objects and noncommutative orders over curves
• A symplectic prolegomenon
• Elliptic Hall algebra on ${\mathbb{F}}_1$
• Maurer-Cartan deformation of Lagrangians