Mirror Symmetry and Fano Manifolds

Abstract

We consider mirror symmetry for Fano manifolds, and describe how one can recover the classification of 3-dimensional Fano manifolds from the study of their mirrors. We sketch a program to classify 4-dimensional Fano manifolds using these ideas.

16 pages, 1 figure. To appear in the Proceedings of the 6th European Congress of Mathematics

References in corpus (3)

• On G-Fano threefolds
• Upper Bounds for Mutations of Potentials
• Spectra and strains

Cited by in corpus (38)

• Minkowski Polynomials and Mutations
• Algorithms for minimal Picard-Fuchs operators of Feynman integrals
• Smoothing toroidal crossing spaces
• Maximally Mutable Laurent Polynomials
• Landau-Ginzburg models -- old and new
• Mutations of fake weighted projective planes
• Laurent Inversion
• Calabi-Yau Threefolds Fibred by Mirror Quartic K3 Surfaces
• String topology with gravitational descendants, and periods of Landau-Ginzburg potentials
• On toric geometry and K-stability of Fano varieties
• On deformations of toric Fano varieties
• On the quantum periods of del Pezzo surfaces with $\frac{1}{3}(1,1)$ singularities
• Formal groups and quantum cohomology
• Two poset polytopes are mutation-equivalent
• Minimality and mutation-equivalence of polygons
• Lagrangian torus fibration models of Fano threefolds
• Projecting Fanos in the mirror
• Deformations of Smooth Complete Toric Varieties: Obstructions and the Cup Product
• Homogeneous deformations of toric pairs
• Deformations of Dimer Models
• Reconstruction of singularities on orbifold del Pezzo surfaces from their Hilbert series
• Fano mirror periods from the Frobenius structure conjecture
• Mutations of Fake Weighted Projective Spaces
• Databases of quantum periods for Fano manifolds
• K-moduli of Fano 3-folds can have embedded points
• Newton-Okounkov bodies of flag varieties and combinatorial mutations
• The Doran-Harder-Thompson conjecture for toric complete intersections
• Polarized rigid del Pezzo surfaces in low codimension
• Some examples of non-smoothable Gorenstein Fano toric threefolds
• Non-abelian Abel's theorems and quaternionic rotation
• A Reconstruction Theorem for Quantum Cohomology of Fano Bundles on Projective Space
• Apéry extensions
• The proper Landau--Ginzburg potential, intrinsic mirror symmetry and the relative mirror map
• Four dimensional Fano quiver flag zero loci (with an appendix by T. Coates, E. Kalashnikov, and A. Kasprzyk)
• Degenerations, transitions and quantum cohomology
• Biregular models of log Del Pezzo surfaces with rigid singularities
• On some Fano 4-folds with Lefschetz defect 3
• Combinatorial mutations of Newton-Okounkov polytopes arising from plabic graphs