Kwokwai Chan

SYZ mirror symmetry for toric Calabi-Yau manifolds

We investigate mirror symmetry for toric Calabi-Yau manifolds from the perspective of the SYZ conjecture. Starting with a non-toric special Lagrangian torus fibration on a toric CalabiYau manifold X, we construct a complex manifold y X using Tduality modified by quantum corrections. These corrections are encoded by Fourier transforms of generating functions of certain open Gromov-Witten invariants. We conjecture that this complex manifold y X, which belongs to the Hori-Iqbal-Vafa mirror family, is inherently written in canonical flat coordinates. In particular, we obtain an enumerative meaning for the (inverse) mirror maps, and this gives a geometric reason for why their Taylor series ex(

Mirror symmetry for toric Fano manifolds via SYZ transformations

Abstract We construct and apply Strominger–Yau–Zaslow mirror transformations to understand the geometry of the mirror symmetry between toric Fano manifolds and Landau–Ginzburg models.

Gross fibrations, SYZ mirror symmetry, and open Gromov–Witten invariants for toric Calabi–Yau orbifolds

Given a toric Calabi-Yau orbifold X whose underlying toric variety is semi- projective, we construct and study a non-toric Lagrangian torus bration on X , which we call the Gross bration. We apply the Strominger-Yau-Zaslow recipe to the Gross bration of (a toric modication of) X to construct its instanton-corrected mirror, where the instanton corrections come from genus 0 open orbifold Gromov-Witten invariants, which are virtual counts of holomorphic orbi-disks inX bounded by bers of the Gross bration. We explicitly evaluate all these invariants by rst proving an open/closed equality and then employing the toric mirror theorem for suitable toric (parital) compactications of X. Our calculations are then applied to (1) prove a conjecture of Gross-Siebert on a relation between genus 0 open orbifold Gromov- Witten invariants and mirror maps ofX { this is called the open mirror theorem, which leads to an enumerative meaning of mirror maps, and (2) demonstrate how open (orbifold) Gromov-Witten invariants for toric Calabi-Yau orb- ifolds change under toric crepant resolutions { this is an open analogue of Ruans crepant resolution conjecture.

A formula equating open and closed Gromov–Witten invariants and its applications to mirror symmetry

We prove that open Gromov‐Witten invariants for semi-Fano toric manifolds of the form XD P.KY OY/, where Y is a toric Fano manifold, are equal to certain 1-pointed closed Gromov‐Witten invariants of X. As applications, we compute the mirror superpotentials for these manifolds. In particular, this gives a simple proof for the formula of the mirror superpotential for the Hirzebruch surface F2.

Open Gromov–Witten Invariants and Superpotentials for Semi-Fano Toric Surfaces

In this paper, we compute the open Gromov-Witten invariants for every compact toric surface X which is semi-Fano (i.e. the anticanonical line bundle is nef). Unlike the Fano case, this involves non-trivial obstructions in the corresponding moduli problem. As a consequence, an explicit formula for the Lagrangian Floer superpotential W is obtained, which in turn gives an explicit presentation of the small quantum cohomology ring of X. We also provide a computational verification of the conjectural ring isomorphism between the small quantum cohomology of X and the Jacobian ring of W.

Lagrangian Torus Fibrations and Homological Mirror Symmetry for the Conifold

We discuss homological mirror symmetry for the conifold from the point of view of the Strominger–Yau–Zaslow conjecture.

Homological mirror symmetry for An-resolutions as a T-duality

We study Homological Mirror Symmetry (HMS) for

Dual torus fibrations and homological mirror symmetry for

A_n

A_n

-resolutions from the SYZ viewpoint. Let

-singularities

X\to\bC^2/\bZ_{n+1}