A Note on Formality and Singularities of Moduli Spaces

TL;DR

This paper investigates the formality of the differential graded algebra associated with semistable sheaves on K3 surfaces, providing insights into the singularity structure of their moduli spaces and analyzing the applicability of Kaledin's theorem.

Contribution

It establishes conditions under which the DG algebra is formal and describes the singularity types of moduli spaces, extending understanding of sheaf moduli on K3 surfaces.

Findings

DG algebra is formal for a large class of sheaves

Explicit description of singularity types at moduli space points

Kaledin's theorem fails in certain cases

Abstract

This paper studies formality of the differential graded algebra $RHom(E,E)$, 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.