Mirror symmetry for log Calabi-Yau surfaces I

Abstract

We give a canonical synthetic construction of the mirror family to a pair (Y,D) of a smooth projective surface with an anti-canonical cycle of rational curves, as the spectrum of an explicit algebra defined in terms of counts of rational curves on Y meeting D in a single point. In the case D is contractible, the family gives a smoothing of the dual cusp, and thus a proof of Looijenga's 1981 cusp conjecture.

144 pages, 3 figures, Second version significantly shorter, 109 pages. The first version has a lot of material (particularly in the introduction and material on cyclic quotient singularities) which does not appear in the new version. Download version 1 if this material is desired. Third and final version, small changes from Version 2, to appear in Publ. IHES

References in corpus (6)

• Mirror symmetry and T-duality in the complement of an anticanonical divisor
• Fukaya categories and deformations
• Moduli of surfaces with an anti-canonical cycle
• Stable logarithmic maps to Deligne--Faltings pairs II
• Double affine Hecke algebras of rank 1 and affine cubic surfaces
• An invitation to toric degenerations

Cited by in corpus (8)

• Moduli of surfaces with an anti-canonical cycle
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• Curved String Topology and Tangential Fukaya Categories II
• Compactification of the moduli of polarized abelian varieties and mirror symmetry
• On Landau-Ginzburg models for quadrics and flat sections of Dubrovin connections