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

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 $\mathsf{C}\mathsf{o}\mathsf{n}\mathsf{e}(M)$ and called the mutation cone of $M$, in the real Grothendieck group associated to the maximal modification algebra $\mathrm{\Lambda }={\mathrm{E}\mathrm{n}\mathrm{d}}_R(M)$. Each chamber in $\mathsf{C}\mathsf{o}\mathsf{n}\mathsf{e}(M)$ 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 $\mathrm{\Lambda }$, and by analysis of wall-and-chamber structure of $\mathsf{C}\mathsf{o}\mathsf{n}\mathsf{e}(M)$, we prove that this property holds for $\mathrm{\Lambda }$ if and only if all maximal modifying $R$-modules are connected by iterated mutations. We then consider the finite length subcategory ${\mathcal{D}}_M\subset {\mathrm{D}}^{\mathrm{b}}(\mathrm{m}\mathrm{o}\mathrm{d}\mathrm{\Lambda })$ and introduce a full-dimensional connected subspace ${\mathrm{S}\mathrm{t}\mathrm{a}\mathrm{b}}^{\mathrm{m}\mathrm{d}\mathrm{f}}{\mathcal{D}}_M\subset \mathrm{S}\mathrm{t}\mathrm{a}\mathrm{b}{\mathcal{D}}_M$ of Bridgeland stability conditions on ${\mathcal{D}}_M$. We prove that there is a regular covering map from ${\mathrm{S}\mathrm{t}\mathrm{a}\mathrm{b}}^{\mathrm{m}\mathrm{d}\mathrm{f}}{\mathcal{D}}_M$ to the complexification $\mathsf{C}\mathsf{o}\mathsf{n}\mathsf{e}(M{)}_{\mathbb{C}}$ of the mutation cone of $M$, where the Galois group is the subgroup of $\mathrm{A}\mathrm{u}\mathrm{t}\mathrm{e}\mathrm{q}{\mathcal{D}}_M$ 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 ${\mathcal{D}}_M$ that preserve the subspace ${\mathrm{S}\mathrm{t}\mathrm{a}\mathrm{b}}^{\mathrm{m}\mathrm{d}\mathrm{f}}{\mathcal{D}}_M$.