## Journal of Differential Geometry

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

#### Abstract

For a toric Calabi–Yau (CY) orbifold $\mathcal{X}$ whose underlying toric variety is semi-projective, we construct and study a non-toric Lagrangian torus fibration on $\mathcal{X}$, which we call the Gross fibration. We apply the Strominger–Yau–Zaslow (SYZ) recipe to the Gross fibration of $\mathcal{X}$ to construct its mirror with the instanton corrections coming from genus $0$ open orbifold Gromov–Witten (GW) invariants, which are virtual counts of holomorphic orbi-disks in $\mathcal{X}$ bounded by fibers of the Gross fibration. We explicitly evaluate all these invariants by first proving an open/closed equality and then employing the toric mirror theorem for suitable toric (parital) compactifications of $\mathcal{X}$. Our calculations are then applied to

(1) prove a conjecture of Gross-Siebert on a relation between genus $0$ open orbifold GW invariants and mirror maps of $\mathcal{X}$—this is called the open mirror theorem, which leads to an enumerative meaning of mirror maps, and

(2) demonstrate how open (orbifold) GW invariants for toric CY orbifolds change under toric crepant resolutions—an open analogue of Ruan’s crepant resolution conjecture.

#### Article information

Source
J. Differential Geom., Volume 103, Number 2 (2016), 207-288.

Dates
First available in Project Euclid: 16 May 2016

https://projecteuclid.org/euclid.jdg/1463404118

Digital Object Identifier
doi:10.4310/jdg/1463404118

Mathematical Reviews number (MathSciNet)
MR3504949

Zentralblatt MATH identifier
1344.53071

#### Citation

Chan, Kwokwai; Cho, Cheol-Hyun; Lau, Siu-Cheong; Tseng, Hsian-Hua. Gross fibrations, SYZ mirror symmetry, and open Gromov–Witten invariants for toric Calabi–Yau orbifolds. J. Differential Geom. 103 (2016), no. 2, 207--288. doi:10.4310/jdg/1463404118. https://projecteuclid.org/euclid.jdg/1463404118