A Note on Formality and Singularities of Moduli Spaces

This paper studies formality of the differential graded algebra $\mathrm{RHom}^\bullet(E,E)$, where $E$ is a semistable sheaf on a K3 surface. The main tool is Kaledin's theorem on formality in families. For a large class of sheaves $E$, this DG algebra is formal, therefore we have an explicit description of the singularity type of the moduli space of semistable sheaves at the point represented by $E$. This paper also explains why Kaledin's theorem fails to apply in the remaining case.