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

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 $Λ={\rm End}_R(M)$. 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 $Λ$, and by analysis of wall-and-chamber structure of ${\sf Cone}(M)$, we prove that this property holds for $Λ$ if and only if all maximal modifying $R$-modules are connected by iterated mutations. We then consider the finite length subcategory $\mathscr{D}_M\subset {\rm D}^{\rm b}({\rm mod}\,Λ)$ 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 ${\rm Stab}^{\rm mdf}\mathscr{D}_M$.

Theorems & Definitions (205)

• Theorem 1.1
• Theorem 1.2
• Conjecture 1.3
• Theorem 1.4
• Theorem 1.5
• Corollary 1.6
• Theorem 1.7
• Definition 2.1
• Lemma 2.2
• Lemma 2.3