Singularities of moduli spaces of sheaves on K3 surfaces

In the eighties Mukai proved that the singularities of the moduli space of sheaves on a K3 surface are contained in the locus of strictly semistable sheaves, that is not empty just if the polarization is non generic or if the Mukai vector is non primitive. In the first case, Kaledin, Lehn and Sorger conjectured that the dg-algebra that controls deformations of sheaves on a K3 is formal. This would give a complete description of the singularities of the moduli space. The conjecture was proved in some cases by Kaledin-Lehn and Zhang. The tecniques they used are similar and they consist in pulling back the sheaves on the K3 to the twistor family and to apply Kaledin's theorem of formality in families. In a work in progress with Manfred Lehn we aim to complete the proof of the conjecture. We proved the conjecture of the remaining case and we are trying to extend our ad hoc construction to a general proof.