Stability conditions for Slodowy slices and real variations of stability (Q2801873): Difference between revisions

This article constructs Bridgeland stability conditions for the derived category \(\mathcal{C}\) of coherent sheaves on the Springer resolution of the Slodowy slice associated to the regular nilpotent orbit.NEWLINENEWLINEBy the work of \textit{R. Bezrukavnikov} and \textit{S. Riche} [Ann. Sci. Éc. Norm. Supér. (4) 45, No. 4, 535--599 (2012; Zbl 1293.20044)], one can attach a \(t\)-structure in \(\mathcal{C}\) to an alcove in the real regular weight space \(\mathfrak{h}^*_{\mathrm{reg}}\). The authors construct central charges on a certain subset of \(\mathfrak{h}^*_{\mathrm{reg}}\). These data give locally finite Bridgeland stability conditions on \(\mathcal{C}\). The natural actions of affine Weyl group on the set of alcoves and the space of stability conditions are compatible.NEWLINENEWLINEThe central charge constructed is given by a kind of exponential map which is ubiquitous in the known examples of Bridgeland stability conditions, but also has a strange looking. The importance of this construction is that it gives a bunch of examples for higher dimensional base varieties of dimension greater than \(2\), which is difficult for pure algebraic geometry.NEWLINENEWLINEThis article also explains the conjecture of Bezrukavnikov and Okounkov concerning about quantum cohomology of conical symplectic resolutions and the space of stability conditions, and gives a proof of the conjecture for the Springer resolution case.NEWLINENEWLINEThe text is very much readable and gives a clear account for the motivation and the idea of the main theorem. The unique difficult point is that readers must have some knowledge on both the geometric representation theory and Bridgeland stability conditions. However, the reviewer recommends this article for both algebraic geometers and representation theorists.