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

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.