## Journal of Differential Geometry

### Localized mirror functor for Lagrangian immersions, and homological mirror symmetry for $\mathbb{P}^1_{a,b,c}$

#### Abstract

This paper gives a new way of constructing Landau–Ginzburg mirrors using deformation theory of Lagrangian immersions motivated by the works of Seidel, Strominger –Yau–Zaslow and Fukaya–Oh–Ohta–Ono. Moreover, we construct a canonical functor from the Fukaya category to the mirror category of matrix factorizations. This functor derives homological mirror symmetry under some explicit assumptions.

As an application, the construction is applied to spheres with three orbifold points to produce their quantum-corrected mirrors and derive homological mirror symmetry. Furthermore, we discover an enumerative meaning of the (inverse) mirror map for elliptic curve quotients.

#### Article information

Source
J. Differential Geom., Volume 106, Number 1 (2017), 45-126.

Dates
Received: 9 March 2015
First available in Project Euclid: 26 April 2017

Permanent link to this document
https://projecteuclid.org/euclid.jdg/1493172094

Digital Object Identifier
doi:10.4310/jdg/1493172094

Mathematical Reviews number (MathSciNet)
MR3640007

Zentralblatt MATH identifier
1369.53062

#### Citation

Cho, Cheol-Hyun; Hong, Hansol; Lau, Siu-Cheong. Localized mirror functor for Lagrangian immersions, and homological mirror symmetry for $\mathbb{P}^1_{a,b,c}$. J. Differential Geom. 106 (2017), no. 1, 45--126. doi:10.4310/jdg/1493172094. https://projecteuclid.org/euclid.jdg/1493172094