Introduction to stability conditions
type: Item
instance of: scholarly article
zbMATH Open document ID: 1316.14003
author: Daniel Huybrechts
publication date: 2014-08-12T00:00:00Z
full work available at URL: https://arxiv.org/abs/1111.1745
Mathematics Subject Classification ID: 14-02
Mathematics Subject Classification ID: 14F05
Mathematics Subject Classification ID: 18E30
Mathematics Subject Classification ID: 14D22
Mathematics Subject Classification ID: 14J28
zbMATH DE Number: 6329338
zbMATH Keywords: triangulated category
zbMATH Keywords: stability condition
zbMATH Keywords: quasi-abelian category
zbMATH Keywords: t-structure
zbMATH Keywords: \(K3\) surfaces
zbMATH Keywords: deck transformation group
MaRDI profile type: Publication
arXiv ID: 1111.1745
arXiv classification: math.AG
title: Introduction to stability conditions
review text: This article is a brief and very nice introduction to Bridgeland stability conditions on triangulated categories, oriented to the case of derived category of coherent sheaves on projective smooth curves and surfaces. Readers are assumed to have basic knowledge on coherent sheaves on projective varieties, but the requisite is not high. The content mainly follows two seminal papers by \textit{T. Bridgeland}, one of which [Ann. Math. (2) 166, No. 2, 317--345 (2007; Zbl 1137.18008)] deals with the general theory of stability conditions on triangulated categories, and the other [Duke Math. J. 141, No. 2, 241--291 (2008; Zbl 1138.14022)] treats the construction of stability condition for \(K3\) surfaces and give a certain description of the space of stability conditions.NEWLINENEWLINEThe article consists of six parts. The first part starts with Mumford's slope stability for vector bundles on curves, and naturally moves to the torsion theory for abelian categories. Then the t-structure of triangulated category is introduced and compared to the torsion theory, where the vector bundle case appears as a fundamental example and helps the readers' understanding.NEWLINENEWLINEThe second part describes Bridgeland's definition of stability conditions on triangulated category. The autoequivalence group action on the space of stability conditions is also explained briefly. As an example, the stability conditions on coherent sheaves on curves with genus greater than one are completely described.NEWLINENEWLINEThe third part deals with the stability conditions for surfaces, including the explanation of Bridgeland's construction for \(K3\) surfaces. Compared to the curve case, the existence of stability condition is non-trivial and the tilting procedure in the torsion theory plays the key role in the construction. The article gives a careful explanation on this point.NEWLINENEWLINEThe fourth part discusses the topological nature of the space of stability conditions. The main result here…
reviewed by: Shintarou Yanagida
description: scientific article; zbMATH DE number 6329338