Introduction to stability conditions (Q2875831)

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, due to Bridgeland, is that a connected component of the space of locally finite stability conditions is locally homeomorphic to the affine space of stability functions. This fourth part may look technical for beginners, but the reviewer think the presentation is perfect and no comparable explanation can be found in other introductory literature. This part also gives a detailed explanation of the support condition, which nowadays is common to be included in the definition of the stability condition.NEWLINENEWLINEThe fifth part gives a detailed discussion on the space of stability conditions for \(K3\) surfaces. The main fact here is that the connected component of the stability conditions containing the one given in the third part (called the distinguished component) is a covering space of certain set of stability functions, and the deck transformation group is given by a subgroup of the autoequivalence group. The article also explains the famous conjecture by Bridgeland that the distinguished connected component is simply connected. It also gives a restatement of this conjecture in terms of stacky fundamental group, which is interesting and stirs our imagination.NEWLINENEWLINEThe final sixth part gives a brief citation of further results for non-compact cases (CY3 categories) and compact cases (non-projective \(K3\) surfaces and higher dimensional cases).NEWLINENEWLINEAs mentioned in the article, more advanced topics such as wall-chamber structures and Donaldson-Thomas type invariants are not treated. Representation theoretic topics are also excluded. Nevertheless the reviewer recommends this article for those who have the background of algebraic geometry or of representation theory and want to get an outlook of Bridgeland's theory.NEWLINENEWLINEFor the entire collection see [Zbl 1286.14002].