Formality of some inner RHom

Question

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 $Y$.

Answer 1

I don't think that a full answer to this question is known. Here are the results I'm aware of:

• 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.