Arinkin-Gaitsgory have defined the notion of singular support for any quasismooth $Y$
$$\text{SS}(\mathcal{F})\ \subseteq\ \text{Sing}(Y)$$
and $\mathcal{F}$ any ind-coherent sheaf, where $\text{Sing}(Y)$ is the category of singularities of $Y$ (the classical truncation of the minus one shifted cotangent complex). Thus when $Y$ is a smooth classical scheme, $\text{Sing}(Y)=Y$.


On the other hand, there is a classical notion of singular support of a D module/$\ell$ adic sheaf/constructible sheaf... on any (at least smooth) $Y$
$$\text{SS}(\mathcal{F})\ \subseteq\ T^\*Y.$$
See e.g. section 2.2 of [D-Modules, Perverse Sheaves, and Representation Theory](https://personal.math.ubc.ca/%7Ecautis/dmodules/hottaetal.pdf).


What precisely is the relation between these two notions of singular support?



Two notions of singular support?
When $X$ is a smooth scheme, the derived loop space (i.e. odd tangent bundle) $\mathcal{L}X\simeq\mathbb{T}\_X[-1]$ has $\mathrm{Sing}(\mathbb{T}\_X[-1]) = T^\*X$. Furthermore, there is a Koszul duality:
$$\mathrm{Coh}(\mathbb{T}\_X[-1])^{B\mathbb{G}\_a \rtimes \mathbb{G}\_m} \simeq F\mathcal{D}(X)$$
where the right-hand side is filtered $\mathcal{D}$-modules. Forgetting the $B\mathbb{G}\_a$-action corresponds to taking the associated graded, and this Koszul duality becomes the linear Koszul duality of [Mirkovic--Riche](https://arxiv.org/abs/0804.0923):
$$\mathrm{Coh}(\mathbb{T}\_X[-1])^{\mathbb{G}\_m} \simeq \mathrm{Coh}(\mathbb{T}^\*\_X)^{\mathbb{G}\_m}$$
and the notion of singular support on the left corresponds to the classical support on the right (i.e. singular support of the corresponding $\mathcal{D}$-module).


This was first written up by [Ben-Zvi--Nadler](https://arxiv.org/abs/1002.3636). I have a follow-up [paper](https://arxiv.org/abs/2301.06949) for stacks and there are some references in the intro. I should say that I think none of us write up the compatibility of singular support, but one can use for example the point-wise characterization in [Section 6 of Arinkin--Gaitsgory](https://arxiv.org/abs/1201.6343).


