This post is on the recent paper [1] of Toën on vanishing of Chern classes for crystals along derived foliations in the sense of [2] and [3]. See my previous post on [2] for background and definitions.

To motivate the argument Toën gives for vanishing, consider the following proof that Chern classes of flat bundles are torsion in characteristic $0$. Specifically, let $X/\bC$ be a smooth scheme and let $E$ be a flat $\bC$-bundle on $X$. Each $c_i(E)\in\H^{2i}(X,\bZ)$ is torsion.

To prove this, it is enough to show that the complexified Chern classes in $\H^{2i}(X,\bC)$ vanish. As $E$ is flat, it arises via pullback along $X\rightarrow X_{\dR}$ of a vector bundle on $X_\dR$. Here, $X_\dR$ is Simpson’s de Rham stack, the prestack with $X_\dR(S)=X(S_{\mathrm{red}})$ for $S$ a commutative $\bC$-algebra.

By looking at the cohomology of $\mathrm{BGL}_n,$ one finds that the complexified Chern classes of any vector bundle are in $\F^i\H^{2i}(X,\bC)$. In particular, those of the vector bundle on $X_\dR$ corresponding to the flat bundle $E$ are in $\F^i\H^{2i}(X_\dR,\bC)$. However, $X_\dR$ is formally étale so $\F^i\H^{2i}(X_\dR,\bC)=0$, and hence the complexified Chern classes vanish as well.

Another perspective on the previous paragraph is that $\R\Gamma(X_\dR,\Oscr)\we\R\Gamma_\dR(X/\bC)$ and de Rham cohomology in characteristic $0$ is idempotent, so that $\R\Gamma_\dR(X_\dR/\bC)\we\R\Gamma(X,\Oscr)$ and the higher filtration weights vanish.

This result predates the de Rham stack; it can also be proved by direct calculations in Chern–Weil theory.

One case where something similar is known in mixed characteristic is for crystalline Chern classes of crystals on smooth $k$-schemes where $k$ is a perfect field of characteristic $p>0$. That these vanish integrally (not just rationally) is a theorem of Esnault and Shiho, which I understand has also been proven (and generalized) by Bhatt and Lurie in forthcoming work using the stack-theoretic approach to crystalline cohomology of Drinfeld.

# Derived foliations in mixed characteristics

The setting of Toën’s paper is derived foliations, which I recall are complete filtered derived commutative rings $\dR_\Fscr$ which on associated grades are free over $\gr^0\dR_\Fscr$ on $\gr^1\dR_\Fscr=:\L_\Fscr[-1]$. The crucial subtlety to realize in mixed characteristics is that there are really (more than!) two notions of filtered derived commutative rings. This distinction appears in Raksit’s thesis and I am writing up a general treatment.

It suffices here to note that Toën’s derived foliations are what I call crystalline filtered derived commutative rings and that the standard example of such an object is Hodge-filtered derived de Rham cohomology. The other natural example arises from HKR-filtered Hochcshild homology and leads to infinitesimal cohomology.

Both notions support a notion of foliation. An infinitesimal derived foliation is in particular a complete filtered derived commutative ring with $\gr^\star\dR_\Fscr\we\mathrm{LSym}^\star_{\gr^0\dR_\Fscr}(\L_\Fscr[-1])$ whereas a crystalline derived foliation has $\gr^\star\dR_\Fscr\we(\Lambda^\star_{\gr^0\dR_\Fscr}\L_\Fscr)[-\star].$ In a precise sense, these are one shear off’’ where the shearing functor is the symmetric monoidal endofunctor $\mathrm{GrD}(\bZ)$ which takes a graded object $M(\star)$ to $M(\star)[2\star]$.

Only crystalline derived foliations are considered in the remainder of this post.

# Chern classes of perfect complexes

Suppose that $k$ is a commutative ring, $S=\Spec k$, and $X$ is a derived $k$-scheme. Fix $\Fscr$ a $k$-linear foliation on $X$. If $E$ on $X$ is a perfect complex on $X$, then Toën shows there are Chern classes $c_i(E)\in\H^{2i}_\dR(X/S)$. In fact, $c_i(E)\in\F^i_\H\H^{2i}_\dR(X/S),$ the $i$th piece of the Hodge filtration, as follows in the end from the computation of the Hodge cohomology of $\mathrm{BGL}_n$.

# The vanishing theorem

The main idea that makes Toën’s argument work is to use that if $E$ is a perfect complex on $X$ which admits the structure of an $\Fscr$-crystal, then $c_i(E)$ vanishes in the image of $\H^{2i}_\dR(X/S)\rightarrow\H^{2i}(X/\Fscr),$ where the right-hand side is defined to be the cohomology of the global sections of the de Rham complex of $\Fscr$ (or equivalently the derived global sections of $\F^0\dR_\Fscr$).

In fact, more is true. There is a second filtration on $\R\Gamma_\dR(X/S)$ which Toën calls the Hodge filtration, but which I will call here the Gauss–Manin connection because it generalizes Gauss–Manin connections. This filtration $\F^\star_{\Fscr\mathrm{GM}}\R\Gamma_{\dR}(X/S)$ has the property that it gives a complete filtration on $\R\Gamma_\dR(X/S)$ with associated graded pieces $\gr^\star\R\Gamma_\dR(X/S)\we\R\Gamma_\dR(X/\Fscr,\Lambda^\star_{\Oscr_X}\N_\Fscr^\vee)[-\star],$ where $\N_\Fscr^\vee$ is by definition the fiber of $\L_{R/k}\rightarrow\L_\Fscr$, i.e., the conormal bundle of $\Fscr$. This conormal bundle canonically admits the structure of an $\Fscr$-crystal and the cohomology terms on the right are given by the derived global sections of the de Rham complex of the $\Fscr$-crystals $\Lambda^\star_{\Oscr_X}\N_\Fscr^\vee$.

(In fact, this is a complete filtration in complete filtered complexes…)

Main lemma. If $E$ is an $\Fscr$-crystal, then $c_i(E)\in\F^i_{\Fscr\mathrm{GM}}\H^{2i}_\dR(X/S).$

The proof follows by functoriality of the Gauss–Manin connection. Specifically, $E$ defines a morphism of stacks $X\rightarrow\mathbf{Perf}$ and the $\Fscr$-crystal structure on $E$ promotes $X\rightarrow\mathbf{Perf}$ into a morphism of derived foliations $(X,\Fscr)\rightarrow(\mathbf{Perf},\mathbf{0})$, where $\mathbf{0}$ is the initial foliation, corresponding to the trivial de Rham complex $\Oscr_{\mathbf{Perf}}$. The lemma follows by functoriality of Chern classes and the fact that the Gauss–Manin connection on the de Rham cohomology of $\mathbf{Perf}$ corresponding to the initial foliation $\mathbf{0}$ is nothing other than the Hodge filtration.

Main theorem. Suppose that $E$ is an $\Fscr$-crystal and that $\N_\Fscr^\vee$ is a locally free $\Oscr_X$-module of rank $d$. If $f(x_1,\ldots)$ is a polynomial of degree $q>d$ (where $x_i$ has degree $2i$), then $f(c_1(E),\ldots)=0$ in $\H^{2q}_{\dR}(X/S)$.

This follows immediately from the main lemma and the fact that $\F^i_{\Fscr\mathrm{GM}}\R\Gamma_\dR(X/S)=0$ for $i>d$ by the completeness of the Gauss–Manin filtration and the identification of the associated graded pieces: $\Lambda^i_{\Oscr_X}\N_\Fscr^\vee=0$ for $i>d$.

Example. If $X\rightarrow Y\rightarrow S$ is any factorization of the structure morphism where $Y$ is smooth of rank relative dimension $d$ over $S$, then for any $q>d$ and any crystal $E$ for the relative de Rham complex $\dR_{X/Y}$, the Chern class $c_q(E)=0$ vanishes in $\H^{2q}_{\dR}(X/S)$.

The theorem gives a rigorous corollary to the following non-rigorous intuition. If the conormal bundle of $\Fscr$ is locally free of dimension $d$, then the “space of leaves” of $\Fscr$ behaves like a $d$-dimensional smooth scheme over $S$ and any $\Fscr$-crystal descends along the map from $X$ to the space of leaves. Vanishing of the Chern classes follows now from functoriality and the dimension of the space of leaves.

The paper [1] contains a few other related results, especially on the vanishing of high-degree Chern classes in crystalline cohomology of conormal bundles for derived foliations which admits `lifts’ to characteristic $0$. These are easy corollaries of the main theorem.

# References

[1] Toën, Classes caractéristiques des schémas feuilletés, arXiv:2008.10489.

[2] Toën and Vezzosi, Algebraic foliations and derived geometry: the Riemann-Hilbert correspondence, arXiv:2001.05450.

[3] Toën and Vezzosi, Algebraic foliations and derived geometry II: the Grothendieck-Riemann-Roch theorem, arXiv:2007.09251.

[4] Esnault and Shiho, Chern classes of crystals, Trans. AMS 371(2) 2019, 1333–1358, arXiv:1511.06874.