Mirror symmetry for log Calabi-Yau surfaces I

2011/06/24 by Mark Gross, Gross, Mark, Paul Hacking +3 · 5 citations
Computer Science · Mathematics · #14J33 #58K60 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Commutative Algebra and Its Applications #FOS: Mathematics #Polynomial and algebraic computation

paper · pdf · doi:10.48550/arxiv.1106.4977

openalex publication_date 2011/06/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

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.

Citations

• Mirror symmetry and T-duality in the complement of an anticanonical divisor
• Double affine Hecke algebras of rank 1 and affine cubic surfaces
• Wall-crossing, Hitchin Systems, and the WKB Approximation

Cited by

• Fano compactifications of mutation algebras
• Lectures on stabilized ellipsoid embeddings
• Degenerations of cluster type varieties
• Non-archimedean cylinder counts are logarithmic Gromov-Witten invariants
• Equivariant deformation of minimally elliptic singularities

Related

• Mirror Symmetry for log Calabi-Yau Surfaces II
• Explicit equations for mirror families to log Calabi-Yau surfaces
• Rational curves on Calabi-Yau manifolds: verifying predictions of Mirror Symmetry
• Intrinsic mirror symmetry and punctured Gromov-Witten invariants
• The mirror of the cubic surface