## Journal of Differential Geometry

### Hodge-theoretic mirror symmetry for toric stacks

#### Abstract

Using the mirror theorem [15], we give a Landau–Ginzburg mirror description for the big equivariant quantum cohomology of toric Deligne–Mumford stacks. More precisely, we prove that the big equivariant quantum $D$-module of a toric Deligne–Mumford stack is isomorphic to the Saito structure associated to the mirror Landau–Ginzburg potential. We give a Gelfand–Kapranov–Zelevinsky (GKZ) style presentation of the quantum $D$-module, and a combinatorial description of quantum cohomology as a quantum Stanley–Reisner ring. We establish the convergence of the mirror isomorphism and of quantum cohomology in the big and equivariant setting.

#### Article information

Source
J. Differential Geom., Volume 114, Number 1 (2020), 41-115.

Dates
First available in Project Euclid: 28 December 2019

https://projecteuclid.org/euclid.jdg/1577502022

Digital Object Identifier
doi:10.4310/jdg/1577502022

Mathematical Reviews number (MathSciNet)
MR4047552

Zentralblatt MATH identifier
07147343

#### Citation

Coates, Tom; Corti, Alessio; Iritani, Hiroshi; Tseng, Hsian-Hua. Hodge-theoretic mirror symmetry for toric stacks. J. Differential Geom. 114 (2020), no. 1, 41--115. doi:10.4310/jdg/1577502022. https://projecteuclid.org/euclid.jdg/1577502022