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

TL;DR

This paper establishes a deep connection between the symplectic geometry of certain 3D lens space plumbings and algebraic geometry of crepant resolutions of compound A_n singularities, confirming a conjecture and extending previous results.

Contribution

It proves an equivalence between wrapped Fukaya categories of affine plumbings and derived categories of coherent sheaves on crepant resolutions, confirming a conjecture and generalizing prior work.

Findings

Verified the Lekili-Segal conjecture.

Established an equivalence of categories for affine plumbings.

Constructed a faithful braid group representation on symplectic mapping class groups.

Abstract

We prove that the completed derived wrapped Fukaya categories of certain affine $A_n$ plumbings $W_f^\circ$ of $3$-dimensional lens spaces along circles are equivalent to the derived categories of coherent sheaves on crepant resolutions of the corresponding compound $A_n$ ($cA_n$) singularities $\mathbb{C}[\![u,v,x,y]\!]/(uv-f(x,y))$. The proof relies on the verification of a conjecture of Lekili-Segal. After localization, we obtain an equivalence between the derived wrapped Fukaya category of the (non-affine) $A_n$ plumbing $W_f\supset W_f^\circ$ of lens spaces along circles and the relative singularity category of the crepant resolution. This generalizes the result of Smith-Wemyss and partially answers their realization question. As a consequence, we obtain a faithful representation of the pure braid group $\mathit{PBr}_{n+1}$ on the graded symplectic mapping class group of $W_f$ when the corresponding $cA_n$ singularity is isolated.