Geometric realisations of type $\tilde{A}_n$ preprojective algebras in homological mirror symmetry

Abstract

The type $A_n$-singularity $\mathbb{C}^2/\mathbb{Z}_{n+1}$ can be resolved by hyper-Kähler manifolds $X_ζ$ with underlying smooth manifolds diffeomorphic to the resolution of singularities $X_{\text{res}}$, whose hyper-Kähler structure depends on a parameter $ζ\in H_2(X_{\text{res}};\mathbb{R})$. The structure as a complex manifold of each such hyper-Kähler manifold is equivalent to the resolution of singularities at the poles and the structure of a Milnor fibre with roots determined by $ζ$ elsewhere; the symplectic structure is exact along the equator and is deformed by areas depending on $ζ$ on the exceptional $(-2)$-spheres away from the equator. We show that removing suitable divisors $D_u$ from a fixed $X_ζ$ varying with $u$ in the underlying upper hemisphere of the $S^2$-family of Kähler-structures yields a log Calabi--Yau hyper-Kähler family (in particular a family of log Calabi--Yau submanifolds), and that mirror symmetry is satisfied (partly conjectural in one direction) for this family by hyper-Kähler rotation, in particular by interchanging the structures over the equator and the pole. We furthermore show homological mirror symmetry after adding the missing divisors, which is related to attaching stops and computing singularity categories of certain Landau--Ginzburg potentials on the $A$-side and $B$-side, respectively. More concretely: we compute wrapped Fukaya categories and compare them with (previous and new) computations of derived categories of coherent sheaves and derived categories of singularities in algebraic geometry. We show that the relevant categories (with two exceptions) are triangulated equivalent to module categories over the additive and the multiplicative preprojective algebras of type $\tilde{A}_n$, or to deformations of these algebras depending on the parameters $ζ$.

Figures (38)

• Figure 1: Example of quivers whose underlying graphs are of type $\tilde{A}_3$ respectively $D_4$.
• Figure 2: The doubled quivers corresponding to the quivers in Figure \ref{['quiversexample']}, which are used to define the preprojective algebras.
• Figure 3: Examples of quivers used to define the Ginzburg algebra, obtained from quivers with underlying graphs $\tilde{A}_3$ and $D_4$ respectively; compare with Figure \ref{['quiversexample']} and Figure \ref{['quiversdoubled']}.
• Figure 4: The classical Dynkin diagrams of type $A_n$, $D_n$ and $E_{6,7,8}$ shown in filled edges. If we add the dashed edges we get the extended Dynkin diagrams $\tilde{A}_n$, $\tilde{D}_n$ and $\tilde{E}_{6,7,8}$.
• Figure 5: Cyclically oriented $\tilde{A}_{n}$.

Theorems & Definitions (120)

• Theorem 1.1.1: Theorem \ref{['formal']}
• Theorem 1.1.2: Theorem \ref{['stopsect']}, Lemma \ref{['chords']} and Lemma \ref{['curves']}
• Theorem 1.1.3: Theorem \ref{['surgerycon']}
• Theorem 1.1.4: Theorem \ref{['alggen']}
• Theorem 1.1.5: Theorem \ref{['cocoregen']}
• Proposition 1.1.6: Proposition \ref{['ginzcalc']}
• Theorem 1.1.7: Theorem \ref{['mainresult']}
• Proposition 1.1.8: Proposition \ref{['bsidecycplumb']}
• Proposition 1.1.9: Proposition \ref{['multloc']}
• Proposition 1.1.10: Section \ref{['sectrivialpot']}