On SYZ mirror transformations

Abstract

In this expository paper, we discuss how Fourier-Mukai-type transformations, which we call SYZ mirror transformations, can be applied to provide a geometric understanding of the mirror symmetry phenomena for semi-flat Calabi-Yau manifolds and toric Fano manifolds. We also speculate the possible applications of these transformations to other more general settings.

v2: 22 pages; revised according to the updated version of the preprint arXiv:0801.2830; to appear in Advanced Studies in Pure Mathematics, "New Developments in Algebraic Geometry, Integrable Systems and Mirror Symmetry"

References in corpus (10)

• Mirror symmetry and T-duality in the complement of an anticanonical divisor
• Mirror symmetry for Del Pezzo surfaces: Vanishing cycles and coherent sheaves
• Homogeneous coordinate rings and mirror symmetry for toric varieties
• Mirror symmetry for toric Fano manifolds via SYZ transformations
• Real and integral structures in quantum cohomology I: toric orbifolds
• T-Duality and Homological Mirror Symmetry of Toric Varieties
• Mirror symmetry for P^2 and tropical geometry
• Lagrangian Floer theory on compact toric manifolds II : Bulk deformations
• Potential functions via toric degenerations
• Homological mirror symmetry is T-duality for ${\mathbb{P}}^n$

Cited by in corpus (5)

• Mirror symmetry for toric Fano manifolds via SYZ transformations
• The Coherent-Constructible Correspondence and Homological Mirror Symmetry for Toric Varieties
• T-Duality and Homological Mirror Symmetry of Toric Varieties
• Mirror maps equal SYZ maps for toric Calabi-Yau surfaces
• Holomorphic line bundles on projective toric manifolds from Lagrangian sections of their mirrors by SYZ transformations