Noncommutative crepant resolutions of cA_n singularities via Fukaya categories

2023/07/13 by J. D. Evans, Evans, Jonathan David, Yankı Lekili +1 · 1 citation
Mathematics · #14A22 #14J33 #16S38 #53D37 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic structures and combinatorial models #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Representation Theory (math.RT) #Symplectic Geometry (math.SG)

paper · doi:10.48550/arxiv.2307.06592

openalex publication_date 2023/07/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We compute the wrapped Fukaya category W(T^*S¹, D) of a cylinder relative to a divisor D= \p₁,…, p_n\ of n points, proving a mirror equivalence with the category of perfect complexes on a crepant resolution (over k[t₀,…, t_n]) of the singularity uv=t₀t₁… t_n. Upon making the base-change t_i= f_i(x,y), we obtain the derived category of any crepant resolution of the cA_n singularity given by the equation uv= f₀… f_n. These categories inherit braid group actions via the action on W(T^*S¹,D) of the mapping class group of T^*S¹ fixing D. We also give a geometric model of the derived contraction algebra of a cA_n singularity in terms of the relative Fukaya category of the disc.

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• Plumbings of lens spaces and crepant resolutions of compound A_n singularities

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