# Formality of some inner 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. Jan 18 '18 at 5:51

• if one forgets about the algebra structure there's a necessary and sufficient condition for $$\mathbb{R}\underline{Hom}_{\mathcal O_Y}(\iota_*\mathcal O_X,\iota_*\mathcal O_X)$$ to be formal as a sheaf of $$\mathcal O_X$$-modules. To view it as a sheaf of $$\mathcal O_X$$-modules, we rather consider $$\mathbb{R}\underline{Hom}_{\mathcal O_X}(\iota^*\iota_*\mathcal O_X,\iota_*\mathcal O_X)$$, where $$\iota$$ is the embedding of $$X$$ into $$Y$$. This is a result of Arikin and Calararu (see https://arxiv.org/pdf/1007.1671.pdf), and the sufficient condition is that the normal bundle extends to a bundle on the first infinitesimal neighborhood $$X^{(1)}$$ of $$X$$ into $$Y$$. It seems believed (but not proven) that the formality always holds as sheaves of $$k$$-modules, and even as sheaves of $$\mathcal O_Y$$-modules.
• the only result I know about the formality as a sheaf of $$k$$-algebras is for the diagonal embedding $$\Delta:X\to X\times X$$. This is proven in a paper of mine together with Van den Bergh: https://arxiv.org/pdf/0708.2725.pdf (note that this can't hold over $$\mathcal O_X$$ as the homotopy for the commutativity of the product in the Hochschild cochain complex is not $$\mathcal O_X$$-linear). It is conjectured (but not known) that the result is more generally true for the case of the inclusion of a fixed point locus $$X=Y^\sigma$$ of an involution $$\sigma:Y\to Y$$.
• I don't know about any formality result in the case of $$X$$ lagrangian in a symplectic $$Y$$. In this case one knows, after Behrend and Fantechi, that the Ext algebra comes equipped with a BV structure. This additionnal structure might be useful for approaching the question.