Stability conditions on triangulated categories
type: Item
instance of: scholarly article
title: Stability conditions on triangulated categories
zbMATH Open document ID: 1137.18008
author: Tom Bridgeland
published in: Annals of Mathematics. Second Series
publication date: 2008-03-12T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/0212237
full work available at URL: https://projecteuclid.org/euclid.annm/1200928839
review text: Motivated by the results obtained by M. R. Douglas for Dirichlet branes in string theory, in the present paper the author introduces a general notion of stability on triangulated categories: If \(\mathcal{D}\) is a triangulated category, a stability condition on \(\mathcal{D}\) is a pair \((Z,\mathcal{P})\), where \(Z:K(\mathcal{D})\to \mathbb{C}\) is a group homomorphism, and \(\mathcal{P}=\{\mathcal{P}(\varphi)\mid \varphi\in\mathbb{R}\}\) is a set of full additive subcategories of \(\mathcal{D}\) which satisfy some natural conditions (e.g., it is proved in Lemma 5.2 that all subcategories \(\mathcal{P}(\varphi)\) are abelian). In the hypothesis that \(\mathcal{D}\) is essentially small, the class of all stability conditions on \(\mathcal{D}\) is a set. In Section 6 the author constructs a natural topology (which is induced by a natural metric, as it is proved in Section 8) on the set \(\text{Stab}(\mathcal{D})\) of locally-finite stability conditions on \(\mathcal{D}\) (this kind of stability conditions are defined in Definition 5.7). The main result of the paper is Theorem 1.2. This theorem states that if \(\mathcal{D}\) is a locally small triangulated category, for each connected component \(\Sigma\subset \text{Stab}(\mathcal{D})\) there are linear spaces \(V(\Sigma)\subset \text{Hom}_{\mathbb{Z}}(K(\mathcal{D}),\mathbb{C})\), with a well defined linear topology, and a local homeomorphism \(\mathcal{Z}:\Sigma\to V(\Sigma)\) which maps a stability condition \((Z,\mathcal{P})\) into \(Z\). This theorem is proved in Proposition 6.3 and Section 7 (Theorem 7.1).
reviewed by: Simion Sorin Breaz
Mathematics Subject Classification ID: 18E30
Mathematics Subject Classification ID: 81T30
Mathematics Subject Classification ID: 14D21
zbMATH DE Number: 5248860
zbMATH Keywords: triangulated category
zbMATH Keywords: stability condition
zbMATH Keywords: quasi-abelian category
MaRDI profile type: Publication
OpenAlex ID: W2110382157
Wikidata QID: Q106870101
arXiv ID: math/0212237
DOI: 10.4007/ANNALS.2007.166.317
description: scientific article