Stability conditions on noncommutative crepant resolutions of 3-dimensional isolated singularities

TL;DR

This paper explores the structure of stability conditions and mutation cones in the derived category of 3-dimensional Gorenstein singularities, linking wall-crossing phenomena to autoequivalences and mutations of maximal modifying modules.

Contribution

It introduces the mutation cone structure, the tilting-noetherian property, and establishes a connection between stability conditions and mutation graphs for these singularities.

Findings

Constructed the mutation cone ${ m Cone}(M)$ for maximal modifying modules.

Proved the equivalence between tilting-noetherian property and mutation connectivity.

Established a covering map from stability conditions to the mutation cone.

Abstract

Let $R$ be a 3-dimensional complete local Gorenstein isolated singularity. For a basic maximal modifying $R$-module $M$, we construct a wall-and-chamber structure, denoted by ${\sf Cone}(M)$ and called the mutation cone of $M$, in the real Grothendieck group associated to the maximal modification algebra $\Lambda={\rm End}_R(M)$. Each chamber in ${\sf Cone}(M)$ corresponds to a maximal modifying module obtained by iterated (Iyama--Wemyss) mutations of $M$, and a wall-crossing corresponds to the mutation at an indecomposable summand. Moreover, we introduce the notion of tilting-noetherian property of $\Lambda$, and by analysis of wall-and-chamber structure of ${\sf Cone}(M)$, we prove that this property holds for $\Lambda$ if and only if all maximal modifying $R$-modules are connected by iterated mutations. We then consider the finite length subcategory $\mathscr{D}_M\subset {\rm D}^{\rm b}({\rm mod}\,\Lambda)$ and introduce a full-dimensional connected subspace ${\rm Stab}^{\rm mdf}\mathscr{D}_M\subset{\rm Stab}\mathscr{D}_M$ of Bridgeland stability conditions on $\mathscr{D}_M$. We prove that there is a regular covering map from ${\rm Stab}^{\rm mdf}\mathscr{D}_M$ to the complexification ${\sf Cone}(M)_{\mathbb{C}}$ of the mutation cone of $M$, where the Galois group is the subgroup of ${\rm Auteq} \mathscr{D}_M$ consisting of compositions of equivalences associated to mutations of maximal modifying modules. Finally, using the results on stability conditions, we describe the group of autoequivalences of $\mathscr{D}_M$ that preserve the subspace ${\rm Stab}^{\rm mdf}\mathscr{D}_M$.