Kyungmin Rho, Paderborn University: Matrix factorizations from Fukaya categories of surfaces

Kyungmin Rho, Paderborn University: Matrix factorizations from Fukaya categories of surfaces
Burban-Drozd (2017) classified all indecomposable objects in the category of maximal Cohen-Macaulay (MCM) modules over the non-isolated surface singularity xyz=0, and hence, in the category of matrix factorizations of xyz. Under homological mirror symmetry (HMS), these categories are also equivalent to the Fukaya category of the pair-of-pants surface, as proven by Abouzaid-Auroux-Efimov-Katzarkov-Orlov (2013), and the equivalence is explicitly realized by Cho-Hong-Lau's localized mirror functor (2017). In this talk, we find the objects in the Fukaya category that correspond to the indecomposable MCM modules in Burban-Drozd's classification. They are given by immersed curves in the surface equipped with local systems. Using this geometric description, we derive an explicit canonical form of matrix factorizations of xyz, and present applications to algebraic operations through geometric approaches. This is based on joint works Cho-Jeong-Kim-Rho (2022) and Cho-Rho (2024).