Deformations of Kalck--Karmazyn algebras via Mirror Symmetry

TL;DR

This paper connects mirror symmetry and algebraic geometry to explicitly compute deformations of Kalck--Karmazyn algebras associated with surface singularities, revealing their structure through Lagrangian Floer theory.

Contribution

It introduces a novel approach using mirror symmetry to explicitly determine deformations of certain matrix algebras linked to surface singularities.

Findings

Explicit computation of Kawamata's matrix order.

Identification of a special Lagrangian in the punctured torus.

Connection between Fukaya categories and algebra deformations.

Abstract

As observed by Kawamata, a $\mathbb{Q}$-Gorenstein smoothing of a Wahl singularity gives rise to a one-parameter flat degeneration of a matrix algebra. A similar result holds for a general smoothing of any two-dimensional cyclic quotient singularity, where the matrix algebra is replaced by a hereditary algebra. From a categorical perspective, these one-parameter families of finite-dimensional algebras "absorb" the singularities of the threefold total spaces of smoothings. These results were established using abstract methods of birational geometry, making the explicit computation of the family of algebras challenging. Using mirror symmetry for genus-one fibrations, we identify a remarkable immersed Lagrangian with a bounding cochain in the punctured torus. The endomorphism algebra of this Lagrangian in the relative Fukaya category corresponds to this flat family of algebras. This enables us to compute Kawamata's matrix order explicitly.