Stability conditions on \(A_n\)-singularities
The concept of a stability condition on a triangulated category \(\mathcal{T}\) was introduced by \textit{T. Bridgeland} [Ann. Math. (2) 166, No. 2, 317--345 (2007; Zbl 1137.18008)]. This notion can be seen as a generalisation of classical \(\mu\)-stability for vector bundles on curves. Bridgeland proved that under some technical assumptions the space of stability conditions is a finite-dimensional complex manifold, denoted by \(\text{Stab}(\mathcal{T})\). In general we do not know much about the geometry of this manifold, for example whether it is connected. However, quite a few (partial) results exist in some specific cases, an important one being the case where \(\mathcal{T}\) is the bounded derived category of a smooth projective \(K3\) surface \(X\) and a distinguished connected component of \(\text{Stab}({ D}^{ b}(X))\) has been identified \textit{T. Bridgeland} [Duke Math. J. 141, No. 2, 241--291 (2008; Zbl 1138.14022)]. An open conjecture, which would provide information about the group of autoequivalences of \({\text D}^{\text b}(X)\), claims (in particular) that this connected component is in fact simply-connected. In the paper under review, the authors investigate stability conditions on \(A_n\)-singularities. To be more precise, let \(f: X \rightarrow Y=\text{Spec}\, \mathbb{C}\left[x,y,z\right]/(xy+z^{n+1})\) be the minimal resolution of the \(A_n\)-singularity. We consider \(\mathcal{D}\), the bounded derived category of coherent sheaves on \(X\) supported at the exceptional set \(Z=C_1 \cup \ldots \cup C_n\), and \(\mathcal{C}\), its full triangulated subcategory consisting of objects \(E\) satisfying \(Rf_*E=0\). The main theorem of the paper states that \(\text{Stab}(\mathcal{C})\) is connected and that \(\text{Stab}(\mathcal{D})\) is connected and simply-connected. This can be seen as evidence for the above mentioned conjecture, since the categories \(\mathcal{C}\) and \(\mathcal{D}\) are local models for derived categories of \(K3\) surfaces. The authors first prove the connectedness of \(\text{Stab}(\mathcal{D})\). The strategy is as follows. Denote by \(U\) the set consisting of stability conditions such that skyscraper sheaves of closed points \(x \in Z\) are stable. One proves that \(U\) is connected. Furthermore, the connected component containing \(U\) is preserved by the action of \(\text{Br}(\mathcal{D})\), the group of autoequivalences generated by the spherical twists associated to the sheaves \(\mathcal{O}_{C_i}(-1)\) and the dualizing sheaf of \(Z\). The authors then show that any other connected component has to contain a certain stability condition \(\sigma\) with the property that \(\Phi \sigma \in U\) for some \(\Phi \in \text{Br}(\mathcal{D})\), and this concludes the proof. In the next step the authors prove that the affine braid group action on \(\mathcal{D}\) is faithful. To be slightly more precise, the authors prove homological mirror symmetry for \(A_n\)-singularities (in particular, the McKay correspondence is used), then establish the faithfulness in characteristic two and finally lift the latter result to any characteristic. By a theorem of \textit{T. Bridgeland} [Int. Math. Res. Not. 2009, No. 21, 4142--4157 (2009; Zbl 1228.14012)] the faithfulness implies the simply-connectedness of \(\text{Stab}(\mathcal{D})\). In the last section the connectedness of \(\text{Stab}(\mathcal{C})\) is established. In particular, homological mirror symmetry enters as the authors use topological arguments on the symplectic side to prove a certain technical statement on the algebraic side. In the appendix it is shown that any autoequivalence of \(\mathcal{D}\) is of Fourier--Mukai type.
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