2024/12/12 by Yankı Lekili, Lekili, Yanki, Jenia Tevelev +1 · 1 citation
Mathematics · Physics and Astronomy · #Advanced Topics in Algebra #Nonlinear Waves and Solitons #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.2412.09724
As observed by Kawamata, a \ℚ-Gorenstein smoothing of a Wahl\nsingularity gives rise to a one-parameter flat degeneration of a matrix\nalgebra. A similar result holds for a general smoothing of any two-dimensional\ncyclic quotient singularity, where the matrix algebra is replaced by a\nhereditary algebra. From a categorical perspective, these one-parameter\nfamilies of finite-dimensional algebras "absorb" the singularities of the\nthreefold total spaces of smoothings. These results were established using\nabstract methods of birational geometry, making the explicit computation of the\nfamily of algebras challenging. Using mirror symmetry for genus-one fibrations,\nwe identify a remarkable immersed Lagrangian with a bounding cochain in the\npunctured torus. The endomorphism algebra of this Lagrangian in the relative\nFukaya category corresponds to this flat family of algebras. This enables us to\ncompute Kawamata's matrix order explicitly.\n