# MathSciDoc: An Archive for Mathematician ∫

#### Differential Geometrymathscidoc:1702.10002

J. Differential Geom., 90, (2), 177-250, 2012
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 Calabi-Yau manifold $X$, we construct a complex manifold $\check{X}$ using T-duality 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 $\check{X}$, 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 expansions in terms of the K\"ahler parameters of $X$ have integral coefficients. Applying the results in \cite{Chan10} and \cite{LLW10}, we compute the open Gromov-Witten invariants in terms of local BPS invariants and give evidences of our conjecture for several 3-dimensional examples including $K_{\proj^2}$ and $K_{\proj^1\times\proj^1}$.
SYZ, mirror symmetry, toric Calabi-Yau
@inproceedings{kwokwai2012syz,
title={SYZ mirror symmetry for toric Calabi-Yau manifolds},
author={Kwokwai Chan, Siu-Cheong Lau, and Nai-Chung Conan Leung},
url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170213234357902326435},
booktitle={J. Differential Geom.},
volume={90},
number={2},
pages={177-250},
year={2012},
}

Kwokwai Chan, Siu-Cheong Lau, and Nai-Chung Conan Leung. SYZ mirror symmetry for toric Calabi-Yau manifolds. 2012. Vol. 90. In J. Differential Geom.. pp.177-250. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170213234357902326435.