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

Abstract

For many compound $A_n$ ($cA_n$) singularities $R_f=\mathbb{C}[u,v,x,y]/(uv-f(x,y))$ with crepant resolutions $Y_f$, their mirrors are affine $A_n$ plumbings $W^\circ_f$ of $3$-dimensional lens spaces along circles. We prove two versions of homological mirror symmetry for these Stein $3$-folds. (i) The uncompleted version: there is an equivalence $D^\mathit{perf}\mathcal{W}(W_f^\circ)\simeq D^b\mathit{Coh}(Y^\circ_f)$ between the derived wrapped Fukaya category and the bounded derived category of coherent sheaves on some divisor complement $Y^\circ_f=Y_f\setminus D$. (ii) The completed version: there is an equivalence $D^\mathit{perf}\widehat{\mathcal{W}}(W^\circ_f)\simeq D^b\mathit{Coh}(\widehat{Y}_f)$, where $\widehat{\mathcal{W}}(W^\circ_f)$ is the completion of $\mathcal{W}(W^\circ_f)$ with respect to the word-length filtration of Hamiltonian chords, and $\widehat{Y}_f$ is the complete local version of $Y_f$. As an application of (i), we show that certain infinitely generated subgroup of the pure braid group $\mathit{PBr}_{n+2}$ split injects into the compactly supported symplectic mapping class group of $W^\circ_f$ as long as $R_f$ is isolated, generalizing the work of Keating-Smith in the case of a conifold smoothing. Applying categorical localization to (ii), we obtain an equivalence between the (uncompleted) derived wrapped Fukaya category of the corresponding (non-affine) $A_n$ plumbing $W_f$ of lens spaces along circles and the relative singularity category of $Y_f$. This generalizes the result of Smith-Wemyss in the case of double bubble plumbings and partially answers their realization question.