# Formality of some innner RHom

Let $Y$ be a scheme and $X$ be a closed subscheme. Consider $\underline{RHom}_{\mathcal O_Y}(\mathcal O_X,\mathcal O_X)$ (inner RHom). We can view this is as a sheaf of dg-algebras on $X$; when both $X$ and $Y$ are (for simplicity) of finite type over a field and smooth then its cohomology is the exterior algebra of the normal bundle to $X$ inside $Y$.

$\mathbf{Question:}$ What kind of results are known about the formality of this algebra in some nice cases? I am especially interested in the case when $Y$ is symplectic and $X$ is Lagrangian in $Y$.

• Do you know papers of Arinkin-Caldararu and Arinkin-Caldararu-Hablicsek? They may contain some useful things. – Dan Petersen Jan 18 '18 at 5:51