Quasi-smoothness and microlocal geometry over the sphere

Motivation over the complex numbers

Working in algebraic geometry over $\mathbb{C}$, imagine $X$ is a “derived enhancement” of a smooth projective variety $X_0$. This means $X$ is a derived stack. This means

• There’s some stackiness. For example, we can find a way to exhibit $X_0$ as the space of objects ${\mathcal{G}}^0$ of a groupoid stack ${\mathcal{G}}^{\bullet }$ - this means to specify a way to identify points pairs of points in $X_0$, and ways to identify these identifications, and all the involved coherences. Then we can take $X$ to be the classifying space of this groupoid:

$$|{\mathcal{G}}^{\bullet }|=[X_0/\text{identifications}]$$

which receives a quotient map from $X_0$.

• There’s some derived-ness: For example, we can build out of $X_0$ a stack that is “the most non-transverse possible intersection involving $X_0$” by intersecting $X_0$ with itself. This is the loop stack, whose functions compute Hochschild homology of $X_0$.

Both of these properties simplify if we pull back data along a point $x:Spec(\mathbb{C})\to X_0$.

• In our stacky example, if we only care about symmetries near a point $x$, we can restrict to the component of $|{\mathcal{G}}^{\bullet }|$ containing composition

$$Spec(\mathbb{C})\stackrel{x}{\to }{X}_0\stackrel{\pi }{\to }|{\mathcal{G}}^{\bullet }|$$

this will now be a pointed 1-connective object in presheaves, which is equivalent, by toposic Koszul duality, to the data of a $E_1$ group presheaf.

• In our derived example, the derived loop space pulls back to pointed loop space ${\Omega }_x{X}_0$. Composition of loops equips this with the structure of a $E_1$-group presheaf.

Namely, we see that the question becomes one about $E_1$-group objects.

A foundational insight of the Toen-Vezzosi tradition of Derived Algebraic Geometry is that these properties are reflected succinctly in the Postnikov-amplitude of the derived tangent complexes. Namely

• the usual/geometric tangent vectors sit in degree zero
• stacky objects will have tangent vectors in negative degrees
• derived objects will have tangent vectors in positive degrees (Or the other way around, depends on if the derived algebraic geometer using homological or cohomological grading).

There’s a way to explain how this comes to be. Let $\hat{X}$ denote some notion of formal completion of $X_0$ at $x$.

We saw above that the phenomenon we’re intersted in can be seen point-wise, so we can always pick an affine neighborhood $f:Spec(R)\to X_0$ of $x$, so we’ll for now just be interested in the local geometry of the ring $R$.

The point $x$ is now an augmentation on the ring $R$. This augmented commutative ring gives rise to a sequence of formal $E_n$ stacks

$Sp{f}^{(n)}(R):=Ma{p}_{Al{g}_{aug}^{(n)}}(R,-)$ The tangent vectors of this object at the base point are the same relative derivations, and derivations are classified by the cotangent complex. Namely, we find that

$Sp{f}^{(n)}(R)(\mathbb{C}\oplus \mathbb{C})\simeq {\Omega }^{\infty }ma{p}_{\mathbb{C}}(\mathbb{L}(R;\mathbb{C}),\mathbb{C})$ The right hand side is the mapping anima in a stable $\infty$-category, hence it says that the left hand side is the underlying space of a spectrum. That is, this equivalence equips the space of tangent vectors with an $E_{\infty }$-operation (addition of velocities) as well as a lift to a non-connective spectrum (“deloopings/categorifications”).

Remark Observe if we took $Sp{f}^{(n)}(R)$ not as $\infty$-stacks but as functors valued in sets, we wouldn’t have been able to encode the delooping data given by the RHS.

Remark (Quantitate version) One can give more careful and precise relationship between degrees using the Baez-Dolan stability estimates for $E_n$-algebras proved by Shaul Barkan.

Remark (Riemann Hilbert) This setting is incidentally where $X$ has an equivalent description in terms of the holomorphic complex analytic geometry on $X(\mathbb{C})$, via the Riemann-Hilbert correspondence. This suggests there’s a dual way to geometrize our considerations - we conjecture that this dual is the analytic picture of some of these ideas suggested in upcoming work of Scholze.