Stability conditions on \(A_n\)-singularities
type: Item
instance of: scholarly article
title: Stability conditions on \(A_n\)-singularities
zbMATH Open document ID: 1198.14020
author: Akira Ishii
author: Kazushi Ueda
author: Hokuto Uehara
published in: Journal of Differential Geometry
publication date: 2010-06-22T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/0609551
review text: 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 …
reviewed by: Pawel Sosna
Mathematics Subject Classification ID: 14F05
Mathematics Subject Classification ID: 18E30
Mathematics Subject Classification ID: 14B05
Mathematics Subject Classification ID: 53D37
zbMATH DE Number: 5724749
zbMATH Keywords: derived categories
zbMATH Keywords: stability conditions
zbMATH Keywords: \(A_n\)-singularities
zbMATH Keywords: homological mirror symmetry
zbMATH Keywords: McKay correspondence
Wikidata QID: Q115171387
MaRDI profile type: Publication
OpenAlex ID: W2604272396
arXiv ID: math/0609551
DOI: 10.4310/JDG/1271271794
description: scientific article