SYZ transformations for toric varieties (Q2900285)

This is a survey article on the author's program on understanding mirror symmetry for toric varieties from the SYZ viewpoint. After briefly reviewing mirror symmetry and the SYZ conjecture (see [\textit{A. Strominger} et al., Nucl. Phys., B 479, No. 1--2, 243--259 (1996; Zbl 0896.14024)]), the author explains how to apply the SYZ picture and in particular SYZ transformations to construct instanton-corrected mirrors for compact toric manifolds (following \textit{K. Chan} and the author [Adv. Math. 223, No. 3, 797--839 (2010; Zbl 1201.14029)]) and toric Calabi-Yau manifolds (following \textit{K. Chan}, \textit{S.-C. Lau} and the author [J. Differ. Geom. 90, No. 2, 177--250 (2012; Zbl 1297.53061)]). The genus-0 open Gromov-Witten invariants, which are virtual counts of holomorphic disks bounded by fibers of an SYZ fibration, play an important role in both constructions and in understanding the geometric meaning of mirror maps. The author surveys on how to compute these invariants using various techniques such as open/closed equality, flop arguments, degeneration formulas, etc.NEWLINENEWLINEFor the entire collection see [Zbl 1235.00045].