Modularity of Landau-Ginzburg models

Abstract

For each Fano threefold, we construct a family of Landau-Ginzburg models which satisfy many expectations coming from different aspects of mirror symmetry; they are log Calabi-Yau varieties with proper potential maps; they admit open algebraic torus charts on which the potential function $w$ restricts to a Laurent polynomial satisfying a deformation of the Minkowski ansatz; the general fibres of $w$ are Dolgachev-Nikulin dual to the anticanonical hypersurfaces in $X$. To do this, we study the deformation theory of Landau-Ginzburg models in arbitrary dimension, following the third-named author, Kontsevich, and Pantev, specializing to the case of Landau-Ginzburg models obtained from Laurent polynomials. Our proof of Dolgachev-Nikulin mirror symmetry is by detailed case-by-case analysis, refining work of Cheltsov and the fifth-named author.

116 pages, final version

References in corpus (22)

• Mirror symmetry and T-duality in the complement of an anticanonical divisor
• An integral structure in quantum cohomology and mirror symmetry for toric orbifolds
• PALP: A Package for Analyzing Lattice Polytopes with Applications to Toric Geometry
• G_2-manifolds and associative submanifolds via semi-Fano 3-folds
• Quantum Periods for 3-Dimensional Fano Manifolds
• Mirror Symmetry and the Classification of Orbifold del Pezzo Surfaces
• Quantum Cohomology and Periods
• Asymptotically cylindrical Calabi-Yau 3-folds from weak Fano 3-folds
• Toric Degenerations of Fano Threefolds Giving Weak Landau-Ginzburg Models
• K3 surfaces and equations for Hilbert modular surfaces
• Deformations and obstructions of pairs (X,D)
• Picard group of hypersurfaces in toric 3-folds
• Hilbert modular surfaces for square discriminants and elliptic subfields of genus 2 function fields
• Calabi-Yau compactifications of toric Landau-Ginzburg models for smooth Fano threefolds
• Laurent Inversion
• Calabi-Yau Threefolds Fibred by Mirror Quartic K3 Surfaces
• Katzarkov-Kontsevich-Pantev Conjecture for Fano threefolds
• Toric Landau-Ginzburg models
• Hodge-Tate conditions for Landau-Ginzburg models
• Compactifications of spaces of Landau-Ginzburg models
• Calabi-Yau threefolds fibred by high rank lattice polarized K3 surfaces
• Modularity of equations D3 and the Iskovskikh classification