Some remarks about deformation theory and formality conjecture (Q6592701)

The topic of this paper is related to the Kaledin-Lehn formality conjecture, proved by \textit{N. Budur} and \textit{Z. Zhang} [Compos. Math. 155, No. 5, 902--911 (2019; Zbl 1452.14009)], which states that the differential graded algebra \(R\operatorname{Hom}(\mathcal{F},\mathcal{F})\) of derived endomorphisms of a polystable sheaf \(\mathcal{F}\) on a \(K3\) surface is formal. The authors prove the (Lie version of the) formality conjecture for objects \(F\) in the bounded derived category of a \(K3\) surface with linearly reductive automorphism group and which are universally gluable, i.e., such that \(\operatorname{Ext}^i(F,F)=0\) for all \(i <0\), using the criterion of [\textit{R. Bandiera} et al., Compos. Math. 157, No. 2, 215--235 (2021; Zbl 1455.14075)].\N\NThey also obtain a formality statement for polystable objects in certain Enriques categories that arise as semiothogonal components in the bounded derived category of some Fano varieties. More precisely, they prove that if \(X\) is a quartic double solid, a Gushel-Mukai three- or fivefold, \(\sigma\) is a Serre-invariant stability condition on the Kuznetsov component \(\operatorname{Ku}(X)\), the DG-Lie algebra \(R\operatorname{Hom}(E,E)\) is formal for every \(\sigma\)-polystable object \(E \in \operatorname{Ku}(X)\).