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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.

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

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  • $\begingroup$ I'm confused; if $Y$ is smooth, shouldn't $\text{Sing}(Y)$ be empty, rather than equal to all of $Y$? $\endgroup$
    – Vidit Nanda
    Commented Dec 6, 2022 at 23:55
  • $\begingroup$ @ViditNanda It's the total space of $H^{-1}(\mathbf{T}^*_Y)$, which when $Y$ is smooth is the zero vector bundle, whose total space is just $Y$. $\endgroup$
    – Pulcinella
    Commented Dec 7, 2022 at 0:23
  • $\begingroup$ Let me verify: is the later basically characterizing the ULAness of the constructible sheaf or the regular holonomic D-module along the direction given by the co-vector? $\endgroup$
    – Z. M
    Commented Dec 7, 2022 at 9:58
  • $\begingroup$ @Z.M I don't know what ULA is, but I mean the notion in HTT/Kashiwara Schapiro, e.g. the support of the associated graded of the D module, or the "microlocal directions in which the constructible sheaf is not locally constant". $\endgroup$
    – Pulcinella
    Commented Dec 7, 2022 at 10:52
  • $\begingroup$ OK, I was thinking about the relative case (a reference is [Hansen–Scholze,§3]), and when the base is a point, it is essentially being locally constant. $\endgroup$
    – Z. M
    Commented Dec 7, 2022 at 19:22

1 Answer 1

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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: $$\mathrm{Coh}(\mathbb{T}_X[-1])^{\mathbb{G}_m} \simeq \mathrm{Coh}(\mathbb{T}^*_X)^{\mathbb{G}_m}$$ 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. I have a follow-up paper 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.

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