Moduli spaces of sheaves on \(K3\) surfaces (Q326595)

This article is a compact survey on moduli spaces of sheaves and complexes on \(K3\) surfaces.NEWLINENEWLINEThe main text starts with the recollection on Mumford-Gieseker stability and moduli spaces of semistable sheaves in section 2. Hitchin-Kobayashi correspondence is also mentioned in section 2.4.NEWLINENEWLINESection 3 deals with the Chow ring of \(K3\) surface, in particular the conjecture by \textit{K. G. O'Grady} given in [J. Math. Pures Appl. (9) 100, No. 5, 701--718 (2013; Zbl 1325.14054)].NEWLINENEWLINEThe main concern of sections 4 and 5 is the birational geometry of moduli spaces. Section 4 presents some recollections on ample and nef cones, focusing on symplectic varieties. Section 5 explains Bridgeland stability, the work of \textit{A. Bayer} and \textit{E. Macrì} [J. Am. Math. Soc. 27, No. 3, 707--752 (2014; Zbl 1314.14020)]. Some applications, in particular to the existence problem of Lagrangian fibrations on Hilbert schemes of points, are also presented.NEWLINENEWLINEThis article is only 15 pages long but contains various topics. The reviewer recommend it to students who have fundamental knowledge of sheaves on surfaces and want to know the present status of the moduli problems on \(K3\) surfaces.