Matrix factorizations from Fukaya categories of surfacesBonn symplectic geometry seminar

by Kyungmin Rho (Universität Paderborn)

Thursday Apr 10, 2025, 2:00 PM → 3:00 PM Europe/Berlin

Endenicher Allee 60/1-016 - Lipschitzsaal (Mathezentrum)

Bonn Symplectic Geometry Seminar

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 consequently in the category of matrix factorizations of xyz. Under homological mirror symmetry (HMS), these algebraic categories are also equivalent to the wrapped 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 investigate the objects in the Fukaya category that correspond to the indecomposable MCM modules classified by Burban-Drozd. We show that it is natural to consider all "immersed" Lagrangians equipped with local systems as objects of the Fukaya category. Using this geometric description, we derive an explicit canonical form of matrix factorizations of xyz, and demonstrate how this perspective leads to new insights and applications in algebraic operations via geometric methods. This is based on joint works Cho-Jeong-Kim-Rho (2022) and Cho-Rho (2024).