Mini-walls for Bridgeland stability conditions on the derived category of sheaves over surfaces (Q407361)

This paper studies Bridgeland stability conditions for the derived category of coherent sheaves on general smooth projective surfaces. The result consists of two parts. The first is on the local finiteness of mini-walls in the space of stability conditions. The second is on the large volume limit.NEWLINENEWLINEThe concept of stability conditions for triangulated categories was introduced by \textit{T. Bridgeland} [Duke Math. J. 141, No. 2, 241--291 (2008; Zbl 1138.14022)], who also proved the locally finiteness of walls in the space of stability conditions for the derived category of coherent sheaves on a \(K3\) or abelian surface. After Bridgeland's seminal work, there appeared many results on the structure of moduli spaces of semistable objects with the help of the analysis of wall-crossing in the space of stability conditions. The paper under review is one of the first papers which treat the case of general projective surface, and is very important and helpful.NEWLINENEWLINELet us explain the first part of the article. The authors introduce mini-walls and mini-chambers in the space of stability conditions for the derived category of coherent sheaves on smooth projective surfaces, and show the local finiteness of mini-walls. The main idea of the proof is to find an upper bound for the rank of the object defining a mini-wall and to apply the proof of the local finiteness of walls for the polynomial Bridgeland stability given by \textit{A. Bayer} [Geom. Topol. 13, No. 4, 2389--2425 (2009; Zbl 1171.14011)].NEWLINENEWLINEThe next part is on the classification of polynomial Bridgeland semistable objects in terms of Gieseker-Simpson semistable objects. The result is that under the large volume limit condition on the stability condition, the moduli space of polynomial Bridgeland semistable objects are isomorphic to the moduli space of Gieseker-Simpson moduli spaces. The authors also show that under a special condition the moduli space is isomorphic to the Uhlenbeck compactification space.