Interactions between autoequivalences, stability conditions, and moduli problems (Q436166)

Let \(\mathcal T\) be a triangulated category and let \((Z, \mathcal{P})\) be a Bridgeland's stability condition on \(\mathcal T\). An autoequivalence \(\Phi\) of \(\mathcal T\) is compatible with \((Z, \mathcal{P})\) if it sends semistable objects to semistable one and it preserves all Harder--Narasimhan filtrations. The main object of the paper is to prove a criterion on when a given autoequivalence is compatible with a given stability condition.NEWLINENEWLINEAs an application the author describes the group of autoequivalences compatible with a given stability condition on the derived category of coherent sheaves on a cycle \(E_n\) of \(n>1\) projective lines intersecting transversally. Then he describes the closure of the moduli space of stable objects in the coarse moduli space of semistable objects on \(E_n\) with a fixed phase. This result was known for a nodal cubic ``\(n=1\) case'', considered by \textit{I. Burban} and \textit{B. Kreußler} [Compos. Math. 142, No. 5, 1231--1262 (2006; Zbl 1103.14007)] and for \(n=2\) [\textit{D. Hernández Ruipérez} et al., Int. Math. Res. Not. 2009, No. 23, 4428--4462 (2009; Zbl 1228.14009)].