# Chern classes of automorphic vector bundles, II

Hélène Esnault and Michael Harris

###### Abstract

We prove that the $\ell$-adic Chern classes of canonical extensions of
automorphic vector bundles, over toroidal compactifications of Shimura
varieties of Hodge type over $\bar{\mathbb{Q}}_{p}$, descend to classes in the
$\ell$-adic cohomology of the minimal compactifications. These are invariant
under the Galois group of the $p$-adic field above which the variety and the
bundle are defined.

  *     * Keywords. Automorphic bundles, Chern classes, Shimura varieties, perfectoids

2010 Mathematics Subject Classification. 11G18; 14G35

[Français]

Titre. Classes de Chern des fibrés vectoriels automorphes, II Résumé. Nous
démontrons que, sur les compactifications toroïdales des variétés de Shimura
de type Hodge sur $\bar{\mathbb{Q}}_{p}$, les classes de Chern $\ell$-adiques
des extensions canoniques des fibrés vectoriels automorphes descendent en des
classes de cohomologie $\ell$-adique sur les compactifications minimales.
Elles sont invariantes sous l’action du groupe de Galois du corps $p$-adique
sur lequel la variété et le fibré sont définis.

###### Contents

  1. Introduction
  2. 1\. Generalities on automorphic bundles
  3. 2\. Chern classes for compactified Shimura varieties
  4. 3\. Construction of perfectoid covers
  5. 4\. Construction of $\ell$-adic Chern classes
  6. 5\. Comparison of torsors
  7. Appendix A. Geometry
  8. References

## Introduction

Let $S$ be a Shimura variety. It is defined over a number field [Del71,
Corollaire 5.5], called the reflex field $E$, and carries a family of
automorphic vector bundles $\mathcal{E}$, defined (collectively) over the same
number field ([Har85, Theorem 4.8]; the rationality conventions of [EH17, §1]
are recalled in §1 below). The Shimura variety has a minimal compactification
$S\hookrightarrow S^{\rm min}$ which in dimension $\geq 2$ is singular. This
is minimal in the sense that each of the toroidal compactifications $S^{\rm
tor}_{\Sigma}$, subject to the extra choice of a fan $\Sigma$, admits a
birational proper morphism $S^{\rm tor}_{\Sigma}\to S^{\rm min}$, which is an
isomorphism on $S$ and (under additional hypotheses on the choice of fan and
level subgroup, that we will make systematically) is a desingularization of
$S^{\rm min}$. The automorphic bundle $\mathcal{E}$ admits a canonical
extension $\mathcal{E}^{\rm can}$ on $S^{\rm tor}_{\Sigma}$ constructed by
Mumford [Mum77, Theorem 3.1] and more generally in [Har89]; $\mathcal{E}^{\rm
can}$ is locally free and defined over a finite extension of the reflex field.
On $S^{\rm min},$ on the other hand, there is generally no locally free
extension of $\mathcal{E}$, except in dimension $1$ when $S^{\rm
tor}_{\Sigma}\to S^{\rm min}$ is an isomorphism. In [EH17] we studied the
continuous $\ell$-adic Chern classes of $\mathcal{E}$ in case $S$ is proper.
In particular, we proved that the higher Chern classes in continuous
$\ell$-adic cohomology die if $\mathcal{E}$ is flat [EH17, Theorem 0.2]. A key
ingredient in the proof is the study of the action of Hecke algebra on
continuous $\ell$-adic cohomology, and the fact that Chern classes lie in the
eigenspace for the volume character [EH17, Corollary 1.18]. When $S$ is not
proper, the Hecke algebra does not act on any cohomology of $S^{\rm
tor}_{\Sigma}$ . Thus we cannot apply the methods of loc.cit. to prove the
analogous theorem for the canonical extensions of flat automorphic bundles on
$S^{\rm tor}$.

On the other hand, the Hecke algebra does act on the $\ell$-adic cohomology of
$S^{\rm min}$. In addition, over the field of complex numbers $\mathbb{C}$,
Goresky-Pardon [GP02, Theorem] used explicit estimates on differential forms
to construct classes $c_{n}(\mathcal{E})^{GP}\in H^{2n}(S^{\rm
min},\mathbb{C})$ which descend the Chern classes of $\mathcal{E}^{\rm can}$
in Betti cohomology $H^{2n}(S^{\rm tor}_{\Sigma},\mathbb{Q}(n))$. They do not
come from classes in $H^{2n}(S^{\rm min},\mathbb{Q}(n))$; precisely this fact
enabled Looijenga [Loo17, Theorem 5.1] to construct mixed Tate extensions in
the Hodge category on the Siegel modular variety (see also the result
announced by Nair in [Nai14, §0.4]).

Our main theorem gives an $\ell$-adic version of the Goresky-Pardon
construction.

######  Theorem 0.1 (Theorem 4.13, Theorem 4.29)

Let $S$ be a Shimura variety of Hodge type. Let $\ell$ be a prime number,
$E_{w}$ the completion of $E$ at a place $w$ dividing the prime $p$ different
from $\ell$, and $E_{w}\hookrightarrow\bar{\mathbb{Q}}_{p}$ be an algebraic
closure. Then the Chern classes of $\mathcal{E}^{\rm can}$ in $H^{2n}(S^{\rm
tor}_{\Sigma,\bar{\mathbb{Q}}_{p}},\mathbb{Q}_{\ell}(n))$ descend to well
defined classes $c_{n}(\mathcal{E})$ in $H^{2n}(S_{\bar{\mathbb{Q}}_{p}}^{\rm
min},\mathbb{Q}_{\ell}(n))$. The $c_{n}(\mathcal{E})$ are contravariant for
the change of level and of Shimura data, verify the Whitney product formula,
lie in the eigenspace for the volume character of the Hecke algebra, and are
invariant under the action of the Galois group of $E_{w}$ on
$H^{2n}(S_{\bar{\mathbb{Q}}_{p}}^{\rm min},\mathbb{Q}_{\ell}(n))$. Their
pullback under

$H^{2n}\left(S_{\bar{\mathbb{Q}}_{p}}^{\rm min},\mathbb{Q}_{\ell}(n)\right)\to
H^{2n}\left(S_{\bar{\mathbb{Q}}_{p}}^{\rm
tor},\mathbb{Q}_{\ell}(n)\right)=H^{2n}\left(S_{\Sigma,\bar{\mathbb{Q}}}^{\rm
tor},\mathbb{Q}_{\ell}(n)\right)$

are the Chern classes of $\mathcal{E}^{\rm can}$ which are invariant under the
Galois group of $E$.

Here, as in [EH17, Lemma 1.15], the volume character is the character by which
the Hecke algebra acts on the constant functions. The theorem can easily be
extended to Shimura varieties of abelian type (see Remarks 4.47) but we leave
the precise statement to the reader.

As the classes of $\mathcal{E}^{\rm can}$ are even defined in continuous
$\ell$-adic cohomology of $S^{\rm tor}_{\bar{\mathbb{Q}}}$, where
$E\hookrightarrow\bar{\mathbb{Q}}$ is an algebraic closure, they are in
particular invariant under the Galois group of $E$. The classes we construct
rely on $p$-adic geometry and have no reason a priori not to depend on the
chosen $p$, let alone to lie in $H^{2n}(S_{\bar{\mathbb{Q}}}^{\rm
min},\mathbb{Q}_{\ell}(n))^{G_{E}}$ via the isomorphism
$H^{2n}(S_{\bar{\mathbb{Q}}_{p}}^{\rm
min},\mathbb{Q}_{\ell}(n))=H^{2n}(S_{\bar{\mathbb{Q}}}^{\rm
min},\mathbb{Q}_{\ell}(n))$.

We now describe the method of proof. The main tool used is the existence of a
perfectoid space $P^{\rm min}$ above111We are using a simplified notation here
for the space denoted ${}_{K^{p}}{\mathcal{S}}(G,X)^{\rm min}$ in the body of
the text. Automorphic vector bundles are also denoted $[{\mathcal{E}}]$ rather
than ${\mathcal{E}}$, and the superscript tor is absent in most cases. $S^{\rm
min}$ which has been constructed by Peter Scholze in [Sch15, Theorem 4.1.1]
for Shimura varieties of Hodge type, together with a Hodge-Tate period map
$\pi_{HT}:P^{\rm min}\to\hat{\mathcal{X}}$ (see [CS17, Theorem 2.3] for the
final form) with values in the adic space of the compact dual variety defined
over the completion of $\bar{\mathbb{Q}}_{p}$. This allows us to define vector
bundles $\pi_{HT}^{*}(\mathcal{E})$ on $P^{\rm min}$ where $\mathcal{E}$ are
equivariant vector bundles on $\hat{\mathcal{X}}$ defining the automorphic
bundles. So while the bundles $\mathcal{E}$ do not extend to $S^{\rm min}$,
they do on Scholze’s limit space $P^{\rm min}$.

On the other hand, suppose $S$ is the Siegel modular variety. Then Pilloni and
Stroh [PS16, Corollaire 1.6] have constructed a perfectoid space $P^{\rm tor}$
above $S^{\rm tor}$. On $P^{\rm tor}$ one has the pullback of the bundles
$\pi_{HT}^{*}(\mathcal{E})$ and the pullback via the tower defining the
perfectoid space of the canonical extensions $\mathcal{E}^{\rm can}$. Theorem
2.38 asserts that the two pullbacks on $P^{\rm tor}$ are the same. This
argument generalizes to Shimura varieties satisfying a certain technical
Hypothesis 2.30. We expect Hypothesis 2.30 to hold in general; however, for
general Shimura varieties of Hodge type we provide a more indirect route in §3
to the same comparison (Theorem 3.22), following a suggestion of Bhargav
Bhatt.

Descent for cohomology with torsion coefficients enables one to construct the
classes as claimed in Theorem 0.1, see Appendix Appendix A. Geometry.

The classes $c_{n}(\mathcal{E})\in H^{2n}(S_{\bar{\mathbb{Q}}_{p}}^{\rm
min},\mathbb{Q}_{\ell}(n))$ map in particular to well-defined classes in
$\ell$-adic intersection cohomology $IH^{2n}(S_{\bar{\mathbb{Q}}_{p}}^{\rm
min},\mathbb{Q}_{\ell}(n))$. On intersection cohomology of $S^{\rm min}$ over
$\bar{\mathbb{Q}}$, we identify the eigenspace under the volume character with
the cohomology of the compact dual $\hat{X}$ of $S$ (Proposition 4.36). If the
classes in $IH^{2n}(S_{\bar{\mathbb{Q}}_{p}}^{\rm min},\mathbb{Q}_{\ell}(n))$,
identified with $IH^{2n}(S_{\bar{\mathbb{Q}}}^{\rm
min},\mathbb{Q}_{\ell}(n))$, descended to continuous $\ell$-adic intersection
cohomology, we could try to apply the method developed in [EH17] to show that
the classes of $\mathcal{E}^{\rm can}$ in continuous $\ell$-adic cohomology on
$S^{\rm tor}$ over $E$ die when $\mathcal{E}$ if flat. This would be in
accordance with [EV02, Theorem 1.1] where it is shown that the higher Chern
classes of $\mathcal{E}^{\rm can}$ in the rational Chow groups on the Siegel
modular variety vanish when $\mathcal{E}$ is flat. However, we are not yet
able to work with continuous $\ell$-adic intersection cohomology. We prove in
Lemma 4.46 the vanishing $c_{n}(\mathcal{E})=0$ for all $n>0$ when
$\mathcal{E}$ is flat.

While we were writing the present note, it was brought to our attention that
Nair in the unpublished manuscript [Nai14] independently mentioned the
possibility of using Scholze’s Hodge-Tate morphism to construct Chern classes
in the cohomology of minimal compactifications, see loc. cit. §0.4.

Acknowledgements: This note is based on Peter Scholze’s construction of a
perfectoid space above the minimal compactification of Shimura varieties of
Hodge type together with the Hodge-Tate period map. We thank him profoundly
for the numerous discussions on his theory, for his precise and generous
answers to our questions. We thank Vincent Pilloni for explaining to us the
content of [PS16, Corollaire 1.6] which plays an important role in our note,
and for pointing out some oversights in an earlier version of this note. We
thank David Hansen for providing the keys to navigating the rapidly expanding
literature on perfectoid geometry; we thank Hansen again, as well as Laurent
Fargues, for their patient and detailed explanations that allowed us to
correct misunderstandings that one of the referees pointed out in an earlier
version of this paper. We thank Mark Goresky for helpful discussions, Simon
Pepin Lehalleur who brought the reference [Nai14, §0.4] to our attention, and
Ana Caraiani, who helped us to understand the issues that forced us to take an
unexpected detour in the proof of Proposition 3.11. We are deeply indebted to
the two referees who helped us to improve our article with their precise and
helpful comments. We apologize for the work we unnecessarily gave them through
our numerous typos. Finally, we are grateful to Bhargav Bhatt for a number of
crucial suggestions, notably for the construction of $Z_{s}$ in (3.9).

##  1\. Generalities on automorphic bundles

If $G$ is a reductive algebraic group over ${\mathbb{Q}}$, by an admissible
irreducible representation of $G({\mathbf{A}})$ we mean an irreducible
admissible $({\mathfrak{g}},K)\times G({{\mathbf{A}}_{f}})$-module, where
${\mathfrak{g}}$ is the complexified Lie algebra of $G$, $K\subset
G({\mathbb{R}})$ is a connected subgroup generated by the center of
$G({\mathbb{R}})$ and a maximal compact connected subgroup, $G({\mathbf{A}})$
($G({{\mathbf{A}}_{f}})$) is the group of (finite) adèles of $G$. If $\pi$ is
such a representation then we write

$\pi\simeq\pi_{\infty}\otimes\pi_{f}$

where $\pi_{\infty}$ is an irreducible admissible $({\mathfrak{g}},K)$-module
and $\pi_{f}$ is an irreducible admissible representation of
$G({{\mathbf{A}}_{f}})$. Let $(G,X)$ be a Shimura datum: $G$ is a reductive
algebraic group over ${\mathbb{Q}}$, and $X$ is a (finite union of) hermitian
symmetric spaces, endowed with a transitive action of $G({\mathbb{R}})$
satisfying a familiar list of axioms. The compact dual ${\hat{X}}$ of $X$ is a
(projective) flag variety for $G$. Thus one can speak of the category ${\rm
Vect}_{G}({\hat{X}})$ of $G$-equivariant vector bundles on ${\hat{X}}$; the
choice of a base point $h\in X\subset{\hat{X}}$ determines an equivalence of
categories ${\rm Rep}(P_{h})\simeq{\rm Vect}_{G}({\hat{X}}),$ where
$P_{h}\subset G$ is the stabilizer of $h$, a maximal parabolic subgroup of
$G$, and for any algebraic group $H$, ${\rm Rep}(H)$ denotes the tensor
category of its representations. Let $K_{h}$ be the Levi quotient of $P_{h}$;
the group $K_{h,\infty}:=K_{h}({\mathbb{C}})\cap G({\mathbb{R}})$ can be
identified with a maximal Zariski connected subgroup of $G({\mathbb{R}})$ that
is compact modulo the center of $G$. If $K\subset G({{\mathbf{A}}_{f}})$ is an
open compact subgroup, we let ${}_{K}S(G,X)$ denote the Shimura variety
attached to $(G,X)$ at level $K$; it has a canonical model over the reflex
field $E(G,X)$ which is a number field (see [Del71, Corollaire 5.5]). We
always assume that $K$ is neat; then ${}_{K}S(G,X)$ is a smooth quasi-
projective variety. Furthermore, we use the notation and conventions of [EH17,
§1.1–1.2], specifically $\mathcal{E}\in{\rm Vect}(\hat{X})$ the notation for
the vector bundle on the compact dual, $[\mathcal{E}]_{K}\in{\rm
Vect}({}_{K}S(G,X))$ for the automorphic vector bundle associated to the
underlying representation of the compact (mod center) group
$K_{h,\infty}\subset G(\mathbb{R})$, the stabilizer of a chosen point $h\in
X$. As in [EH17], we always assume that $K_{h}$ is defined over a CM field and
that every irreducible representation of $K_{h}$ has a model rational over the
CM field $E^{\prime}_{h}$; in particular, $P_{h}$ is also defined over
$E^{\prime}_{h}$. For the purposes of constructing Chern classes, we need only
consider semisimple representations of $P_{h}$, which necessarily factor
through representations of $K_{h}$. The action of the Hecke algebra
$\mathcal{H}_{K}=\\{T(g),g\in G({{\mathbf{A}}_{f}})\\}$ is recalled in Section
4. The Chern character

$ch_{\hat{X}}:{\rm Vect}_{G}({\hat{X}}){\leavevmode\nobreak\
\rightarrow\leavevmode\nobreak\
}CH({\hat{X}})_{\mathbb{Q}}\to\oplus_{i}H^{2i}\left({\hat{X}},{\mathbb{Q}}(i)\right)$

induces isomorphisms

$K_{0}\left({\rm Rep}(K_{h})\right)\otimes_{K_{0}\left({\rm
Rep}(G)\right)}{\mathbb{Q}}{\leavevmode\nobreak\
\overset{\sim}{\longrightarrow}\leavevmode\nobreak\
}CH({\hat{X}})_{{\mathbb{Q}}}{\leavevmode\nobreak\
\overset{\sim}{\longrightarrow}\leavevmode\nobreak\
}\oplus_{i}H^{2i}\left({\hat{X}},{\mathbb{Q}}(i)\right)$ (1.1)

and analogously in $\ell$-adic cohomology. On the other hand, the construction
of automorphic vector bundles gives rise to a homomorphism

$ch_{K}:\left\\{\begin{array}[]{rcl}{\rm
Vect}_{G}({\hat{X}})&{\leavevmode\nobreak\
\rightarrow}\nobreak\leavevmode&CH\left({}_{K}S(G,X)\right)_{{\mathbb{Q}},v}\\\
\mathcal{E}&\mapsto&ch([\mathcal{E}]_{K})\end{array}\right.$ (1.2)

where v indicates the eigenvectors for the volume character. If
$\mathcal{E}\in{\rm Vect}_{G}({\hat{X}})$ is in the image of ${\rm Rep}(G)$
via the factorization

${\rm Rep}(G)\to{\rm Rep}(K_{h}){\leavevmode\nobreak\
\rightarrow\leavevmode\nobreak\ }{\rm Vect}_{G}({\hat{X}}),$

then $[\mathcal{E}]$ is endowed with a natural flat connection, so its higher
Chern classes in $\oplus_{i}H^{2i}({}_{K}S(G,X),{\mathbb{Q}}(i))$ are equal to
$0$. We obtain a morphism

$K_{0}({\rm Rep}(K_{h}))\otimes_{K_{0}({\rm
Rep}(G))}{\mathbb{Q}}\to\oplus_{i}H^{2i}\left({}_{K}S(G,X),{\mathbb{Q}}(i)\right)_{v}\subset\oplus_{i}H^{2i}\left({}_{K}S(G,X),{\mathbb{Q}}(i)\right),$
(1.3)

analogous to (1.1), where the subscript v denotes a certain eigenspace for the
action of the unramified Hecke algebra.

##  2\. Chern classes for compactified Shimura varieties

###  2.A. Toroidal and minimal compactifications, and canonical extensions of
automorphic vector bundles

Henceforward we assume the Shimura variety ${}_{K}S(G,X)$ is not projective;
equivalently, the derived subgroup $G^{{\rm der}}$ of $G$ is isotropic over
${\mathbb{Q}}$. In this case the automorphic theory naturally gives
information about Chern classes of canonical extensions on toroidal
compactifications on the one hand; on the other hand, the $v$-eigenspace most
naturally appears in intersection cohomology of the minimal compactification.
This has been worked out in detail in the $\mathcal{C}^{\infty}$ and the
$L_{2}$ theory by Goresky and Pardon in [GP02]. In what follows, we let
$j_{K}:{}_{K}S(G,X)\hookrightarrow{}_{K}S(G,X)^{\rm min}$ denote the minimal
(Baily-Borel-Satake) compactification. The minimal compactification is
functorial, thus if $K^{\prime}\subset K,$ there is a unique morphism
${}_{K^{\prime}}S(G,X)^{\rm min}{\leavevmode\nobreak\
\rightarrow\leavevmode\nobreak\ }{}_{K}S(G,X)^{\rm min}$ extending the natural
map of open Shimura varieties, and if $g\in G({{\mathbf{A}}_{f}}),$ then the
Hecke correspondence $T(g)$ extends canonically to a correspondence on
${}_{K}S(G,X)^{\rm min}\times{}_{K}S(G,X)^{\rm min}$ where the product is
taken over the reflex field. In particular, for any cohomology theory $H$ as
above, we have Hecke operators

$T(g)\in{\rm End}\big{(}H^{*}({}_{K}S(G,X)^{\rm min})\big{)}.$

The minimal compactification is always singular, except when $G^{ad}$ is a
product of copies of $\mathrm{PGL}(2)$, and the automorphic vector bundles do
not in general extend as bundles to ${}_{K}S(G,X)^{\rm min}$. Thus the
classical theory does not automatically attach Chern classes to automorphic
vector bundles on non-proper Shimura varieties in some cohomology theory
$H^{*}_{?}({}_{K}S(G,X)^{\rm min},*)$. On the other hand, there is a large
collection of toroidal compactifications
${}{}_{K}S(G,X)\hookrightarrow{}{}_{K}S(G,X)_{\Sigma}$ indexed by
combinatorial data $\Sigma$ (see [AMRT75, III, §6, Main Theorem] for details;
the adelic construction is in [Har89, Pin90]). The set of $\Sigma$ is adapted
to the level subgroup $K$. It is partially ordered by refinement: if
$\Sigma^{\prime}$ is a refinement of $\Sigma$, then there is a natural proper
morphism

$p_{\Sigma^{\prime},\Sigma}:{}{}_{K}S(G,X)_{\Sigma^{\prime}}{\leavevmode\nobreak\
\rightarrow\leavevmode\nobreak\ }{}{}_{K}S(G,X)_{\Sigma}$

extending the identity map on the open Shimura variety. Any two $\Sigma$ and
$\Sigma^{\prime}$ can be simultaneously refined by some
$\Sigma^{\prime\prime}$. Further, the open embedding
${}{}_{K}S(G,X)\hookrightarrow{}{}_{K}S(G,X)_{\Sigma},$ is completed to a
diagram

$\textstyle{{}{}_{K}S(G,X)_{\Sigma}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi_{\Sigma}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{}_{K}S(G,X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$
$\scriptstyle{j_{K,\Sigma}}$
$\scriptstyle{j_{K}}$$\textstyle{{}{}_{K}S(G,X)^{\rm min}}$ (2.5)

where $\varphi_{\Sigma}$ is a proper morphism, which is an isomorphism on
${}{}_{K}S(G,X)$. Now assume that $K$ is a neat open compact subgroup of
$G({{\mathbf{A}}_{f}})$, in the sense of [Har89] or [Pin90, 0.6]. In that
case, we can choose $\Sigma$ so that ${}_{K}S(G,X)_{\Sigma}$ is smooth and
projective, and the boundary divisor has normal crossings. We do so unless we
specify otherwise. With these conventions, $\varphi_{\Sigma}$ is always a
desingularization of the minimal compactification, which is constructed as in
[AMRT75], loc.cit.; moreover, for any refinement $\Sigma^{\prime}$ of
$\Sigma$, ${}_{K}S(G,X)_{\Sigma^{\prime}}$ is again smooth and projective.
Mumford proved in [Mum77, Theorem 3.1] that, if ${\mathcal{E}}\in{\rm
Vect}^{\rm ss}_{G}({\hat{X}})$ (recall from [EH17] that the upper index ss
stands for semi-simple) the automorphic vector bundle $[{\mathcal{E}}]{}_{K}$
on ${}_{K}S(G,X)$ admits a canonical extension $[{\mathcal{E}}]{}_{K}^{\rm
can}$ to ${}_{K}S(G,X)_{\Sigma}$; we write $[{\mathcal{E}}]{}_{K}^{\Sigma}$ if
we want to emphasize the toroidal data. The adelic construction is carried out
in Section 4 of [Har89], where Mumford’s result was generalized to arbitrary
${\mathcal{E}}\in{\rm Vect}_{G}({\hat{X}})$. In particular it was shown there
that if ${\mathcal{E}}=(G\times W)/P_{h}$, where $W$ is the restriction to
$P_{h}$ of a representation of $G$, the action of $P_{h}$ is diagonal, and we
have identified $G/P_{h}$ with ${\hat{X}}$, then $[{\mathcal{E}}]{}_{K}$ is a
vector bundle on ${}_{K}S(G,X)$ with a flat connection, and its canonical
extension $[{\mathcal{E}}]{}_{K}^{\rm can}$ on ${}_{K}S(G,X)_{\Sigma}$ is
exactly Deligne’s canonical extension. In particular, the connection has
logarithmic poles along ${}_{K}S(G,X)_{\Sigma}\setminus{}_{K}S(G,X)$.
Moreover, if $\Sigma^{\prime}$ is a refinement of $\Sigma$, then

$p^{*}_{\Sigma^{\prime},\Sigma}[{\mathcal{E}}]{}_{K}^{\Sigma}=[{\mathcal{E}}]{}_{K}^{\Sigma^{\prime}}$

(see [Har90, Lemma 4.2.4]), which in particular implies

$p_{\Sigma^{\prime},\Sigma,*}[{\mathcal{E}}]{}_{K}^{\Sigma^{\prime}}=Rp_{\Sigma^{\prime},\Sigma,*}[{\mathcal{E}}]{}_{K}^{\Sigma^{\prime}}=[{\mathcal{E}}]{}_{K}^{\Sigma}.$

Finally, for any fixed $\Sigma$,

${\mathcal{E}}{\leavevmode\nobreak\ \rightarrow\leavevmode\nobreak\
}[{\mathcal{E}}]{}_{K}^{\rm can}$

is a monoidal functor from ${\rm Vect}_{G}({\hat{X}})$ to the category of
vector bundles on ${}_{K}S(G,X)_{\Sigma}$. Unfortunately, the $g\in
G({{\mathbf{A}}_{f}})$ generally permute the set of $\Sigma$ and thus the
Hecke correspondences $T(g)$ in general do not extend to correspondences on a
given ${}_{K}S(G,X)_{\Sigma}$. Thus the arguments of [EH17], which are based
on the study of the $v$-eigenspace for the Hecke operators in the cohomology
of ${}_{K}S(G,X)$, cannot be applied directly to prove vanishing of higher
Chern classes of $[{\mathcal{E}}]{}_{K}^{\rm can}$ in continuous $\ell$-adic
cohomology of ${}_{K}S(G,X)_{\Sigma}$ over its reflex field, when
$[{\mathcal{E}}]_{K}$ is a flat automorphic vector bundle. One can obtain
information about the classes of $[{\mathcal{E}}]_{K}$ on the open Shimura
variety, but these lose information. From the standpoint of automorphic forms,
the natural target of the topological Chern classes of automorphic vector
bundles should be the intersection cohomology of the minimal compactification.
We fix cohomological notation: $H^{*}(Z,\mathbb{Q})$, resp.
$IH^{*}(Z,\mathbb{Q})$ denotes Betti resp. intersection cohomology of a
complex variety $Z$, while $H^{*}(Z,\mathbb{Q}_{\ell})$, resp.
$IH^{*}(Z,\mathbb{Q}_{\ell})$ denote $\mathbb{Q}_{\ell}$-étale cohomology.

###### Remark 2.6

The action of Hecke correspondences on Betti intersection cohomology
$IH({}_{K}S(G,X),\mathbb{Q})$ are defined analytically by reference to
Zucker’s conjecture. For a purely geometric construction of the action of
Hecke correspondences on $IH({}_{K}S(G,X),\mathbb{Q})$ and thus on
$IH({}_{K}S(G,X),\mathbb{Q}_{\ell})$ see [GM03, (13.3)] (the argument applies
more generally to weighted cohomology as defined there).

The following statement generalizes Proposition 1.20 of [EH17] and is proved
in the same way.

###### Proposition 2.7

There is a canonical isomorphism of algebras

$\displaystyle H^{*}({\hat{X}},{\mathbb{Q}}){\leavevmode\nobreak\
\overset{\sim}{\longrightarrow}\leavevmode\nobreak\ }IH^{*}({}_{K}S(G,X)^{\rm
min},{\mathbb{Q}})_{v}$ $\displaystyle
H^{*}({\hat{X}},{\mathbb{Q}}_{\ell}){\leavevmode\nobreak\
\overset{\sim}{\longrightarrow}\leavevmode\nobreak\ }IH^{*}({}_{K}S(G,X)^{\rm
min},{\mathbb{Q}}_{\ell})_{v}.$

  * Proof.

The second statement is deduced directly from the first one by the comparison
isomorphism [BBD82, Section 6]. As in the proof of the analogous fact in
[EH17], it suffices to prove the corresponding statement over ${\mathbb{C}}$.
By Zucker’s Conjecture [Loo88, SS90], $IH^{*}({}_{K}S(G,X)^{\rm
min},{\mathbb{C}})$ is computed using square-integrable automorphic forms and
Matsushima’s formula [Bor83, 3.6, formula (1)]. Say the space
${\mathcal{A}}_{(2)}(G)$ of square-integrable automorphic forms on
$G({\mathbb{Q}})\backslash G({\mathbf{A}})$ decomposes as the direct sum

${\mathcal{A}}_{(2)}(G)=\oplus_{\pi}m(\pi)\pi$

where $\pi$ runs over irreducible admissible representations of
$G({\mathbf{A}})$ and $m(\pi)$ is a non-negative integer, which is positive
for a countable set of $\pi$ 222Automorphic forms are understood to be
$C^{\infty}$ and $K_{h}$-finite, and to satisfy the remaining conditions
introduced by Harish-Chandra, hence no completion is needed.. Then

$IH^{i}({}_{K}S(G,X)^{\rm min},{\mathbb{C}}){\leavevmode\nobreak\
\overset{\sim}{\longrightarrow}\leavevmode\nobreak\
}\oplus_{\pi}m(\pi)(H^{i}({\mathfrak{g}},K_{h};\pi_{\infty})\otimes\pi_{f}^{K}).$

This implies

$IH^{i}({}_{K}S(G,X),{\mathbb{C}})_{v}{\leavevmode\nobreak\
\overset{\sim}{\longrightarrow}\leavevmode\nobreak\
}\oplus_{\pi}m(\pi)(H^{i}({\mathfrak{g}},K_{h};\pi_{\infty})\otimes(\pi_{f}^{K})_{v}),$
(2.8)

where $(\pi_{f}^{K})_{v}$ is the eigenspace in $\pi_{f}^{K}$ for the volume
character of ${\mathcal{H}}{}_{K}$. Write
$\pi_{f}=\otimes^{\prime}_{q}\pi_{q}$, where $q$ runs over rational primes.
Now if $q$ is unramified for $K$ then $\pi_{f}^{K}=0$ unless $\pi_{q}$ is
spherical. Now the trivial representation of $G({\mathbb{Q}}_{q})$ has the
property that it equals its spherical subspace and the corresponding
representation of the local Hecke algebra is the (local) volume character. By
the Satake parametrization the trivial representation is the only spherical
representation of $G({\mathbb{Q}}_{q})$ with this property. Thus
$(\pi_{f}^{K})_{v}\neq 0$ implies that $\pi_{q}$ is the trivial representation
for all $q$ that are unramified for $K$. It then follows from weak
approximation that $\pi$ is in fact the trivial representation. Thus for all
$i$,

$IH^{i}({}_{K}S(G,X),{\mathbb{C}})_{v}{\leavevmode\nobreak\
\overset{\sim}{\longrightarrow}\leavevmode\nobreak\
}H^{i}({\mathfrak{g}},K_{h};{\mathbb{C}}).$ (2.9)

But this is equal to
$H^{i}({\hat{X}},{\mathbb{C}})=IH^{i}({\hat{X}},{\mathbb{C}})$ by a standard
calculation; see [GP02, Remark 16.6]. The rest of the proof follows as in the
proof of [EH17, Proposition 1.20]. $\Box$

The point of the preceding proposition is that the Chern classes of
automorphic vector bundles on the non-proper Shimura variety ${}_{K}S(G,X)$
are represented by square-integrable automorphic forms, more precisely, by the
differential forms on $G({\mathbb{Q}})\backslash G({\mathbf{A}})$ that are
invariant under the action of $G({\mathbf{A}})$. In other words, the only
$\pi$ that contributes is the space of constant functions on
$G({\mathbb{Q}})\backslash G({\mathbf{A}})$, which are square integrable
modulo the center of $G({\mathbb{R}})$. Thus $IH^{2*}({}_{K}S(G,X)^{\rm
min},\mathbb{Q}(*))_{v}$ can be viewed as $L_{2}$-Chern classes of automorphic
vector bundles. In addition $IH^{2*+1}({}_{K}S(G,X)^{\rm
min},\mathbb{Q}(*))_{v}=0.$ Although most automorphic vector bundles do not
extend as bundles to the minimal compactification ${}_{K}S(G,X)^{\rm min}$, it
was proved by Goresky and Pardon in [GP02, Main Theorem] that under the
natural homomorphism

$H^{*}({}_{K}S(G,X)^{\rm min},{\mathbb{C}})_{v}{\leavevmode\nobreak\
\rightarrow\leavevmode\nobreak\ }IH^{*}({}_{K}S(G,X)^{\rm min},{\mathbb{C}}).$
(2.10)

the classes in $IH^{*}({}_{K}S(G,X)^{\rm min},{\mathbb{C}})_{v}$ lift
canonically, as differential forms, to ordinary singular cohomology
$H^{*}({}_{K}S(G,X)^{\rm min},{\mathbb{C}})_{v}$. We use the notation of the
diagram (2.5). Let $K$ be a neat open compact subgroup of
$G({{\mathbf{A}}_{f}})$, as above. Let $[{\mathcal{E}}]_{K}$ be the
automorphic vector bundle on ${}_{K}S(G,X)$ attached to the homogeneous vector
bundle ${\mathcal{E}}\in{\rm Vect}_{G}({\hat{X}}).$ Let

$c_{n}([{\mathcal{E}}]^{\rm can}_{K})\in
H^{2n}({}_{K}S(G,X)_{\Sigma},\mathbb{Q}(n)),\ c_{n}([{\mathcal{E}}]_{K})\in
H^{2n}({}_{K}S(G,X),\mathbb{Q}(n))$

be the Chern classes in Betti cohomology. The following theorem summarizes the
main results of the article [GP02].

###### Theorem 2.11

  * (i)

There are canonical classes $c_{n}([{\mathcal{E}}]_{K})^{GP}\in
H^{2n}({}_{K}S(G,X)^{\rm min},{\mathbb{C}})$ such that

$\varphi_{\Sigma}^{*}(c_{n}([{\mathcal{E}}]_{K})^{GP})=c_{n}([{\mathcal{E}}]^{\rm
can}_{K})\in H^{2n}({}_{K}S(G,X)_{\Sigma},{\mathbb{Q}}(n)).$

  * (ii)

In particular

$\displaystyle
j_{K}^{*}(c_{n}([{\mathcal{E}}]_{K})^{GP})=c_{n}([{\mathcal{E}}]_{K})\in
H^{2n}({}_{K}S(G,X),{\mathbb{Q}}(n)).$

  * (iii)

The classes $c_{n}([{\mathcal{E}}])^{GP}$ (we drop the subscript $K$) are
represented by square integrable automorphic forms. More precisely, the image
of $c_{n}([{\mathcal{E}}])^{GP}$ under the morphism of (2.10) is contained in
the subspace of intersection cohomology that corresponds, under the
isomorphism (2.8), to the relative Lie algebra cohomology of $\oplus\pi$ with
trivial archimedean component $\pi_{\infty}$.

  * (iv)

Suppose $G^{\rm der}({\mathbb{R}})=\prod_{i}G_{i}$ is a product of simple Lie
groups, all of which are of the form $\mathrm{Sp}(2n,{\mathbb{R}})$,
$SU(p,q)$, $\mathrm{SO}^{*}(2n)$, or $\mathrm{SO}(p,2)$ with $p$ odd 333Some
of the $G_{i}$ can also be of the form $\mathrm{SO}(2,2)$, as in the statement
of [GP02].. The $\mathbb{Q}$-subalgebra

$H^{2*}_{{\rm Chern}}({}_{K}S(G,X)^{\rm min})\subset H^{2*}({}_{K}S(G,X)^{\rm
min},{\mathbb{C}})$

generated by the $c_{n}([{\mathcal{E}}])^{GP}$, as ${\mathcal{E}}$ varies over
${\rm Vect}_{G}({\hat{X}}),$ is endowed with a naturally defined surjective
homomorphism

$H^{2*}_{\rm Chern}({}_{K}S(G,X)^{\rm
min})\overset{h}{\to}H^{2*}({\hat{X}},{\mathbb{Q}}(*))$

such that for any $n$ the diagram

$\displaystyle\begin{CD}H^{2n}_{{\rm Chern}}({}_{K}S(G,X)^{\rm min})@>{}>{\rm
natural\ map}>IH^{2n}({}_{K}S(G,X)^{\rm min},{\mathbb{Q}}(n))_{v}\\\
@V{h}V{}V@A{{\leavevmode\nobreak\
\overset{\sim}{\longrightarrow}\leavevmode\nobreak\ }}A{{\rm
Prop}.\ref{isomQ}}A\\\
H^{2n}({\hat{X}},{\mathbb{Q}}(n))@>{=}>{}>H^{2n}({\hat{X}},{\mathbb{Q}}(n))\end{CD}$

commutes. In other words, the isomorphism of Proposition 2.7 tensor
${\mathbb{C}}$ factors through

$\displaystyle H^{2n}({}_{K}S(G,X)^{\rm min},{\mathbb{C}})\xrightarrow{{\rm
natural\ map}}IH^{2n}({}_{K}S(G,X)^{\rm min},{\mathbb{C}}).$

In the next sections, we use Peter Scholze’s perfectoid geometry and his
Hodge-Tate morphism to prove an analogue of the Goresky-Pardon theorem for
$\ell$-adic cohomology of Shimura varieties of abelian type.

####  2.A.a. Even-dimensional quadrics

Theorem 2.11 excludes the case where $G^{\rm der}({\mathbb{R}})$ contains a
factor isomorphic to $\mathrm{SO}(2k-2,2)$ with $k>2$. Assuming $G^{\rm der}$
is ${\mathbb{Q}}$-simple, there is then a totally real field $F$, with
$[F:{\mathbb{Q}}]=d$, say, such that $\hat{X}_{{\mathbb{C}}}$ is isomorphic to
a product $Q_{n}^{d}$ of $d$ smooth projective complex quadrics $Q_{n}$, each
of dimension $n=2k-2$. The reason for this exclusion is explained in §16.6 of
[GP02]. Following §16.5 of [BH58], we consider the cohomology algebra
$A:=H^{*}(Q_{n},C)$ of a complex quadric
$Q_{n}=\mathrm{SO}(n+2)/\mathrm{SO}(n)\times\mathrm{SO}(2)$ of dimension $n$,
with $C={\mathbb{C}}$ or ${\mathbb{Q}}_{\ell}$. Then $A$ contains a subalgebra
$A^{+}$ isomorphic to $C[c_{1}]/(c_{1}^{n+1})$, with $c_{1}\in H^{2}(Q_{n},C)$
given by the Chern class of the line bundle corresponding to the standard
representation of $\mathrm{SO}(2)$. (There is a misprint in [GP02]; the total
dimension of $A^{+}$ as $C$-vector space is $n+1$, not $n$.) Moreover there is
an isomorphism

$A=A^{+}[e]/(e^{2}-c_{1}^{n})$ (2.12)

with $e$ the Euler class of the the vector bundle arising from the standard
representation of $\mathrm{SO}(n)$. In [BH58], the class $c_{1}$ is denoted
$x_{1}$ and the class $e$ is denoted $\prod_{i=2}^{n}x_{i}$; the equation
$e^{2}=c_{1}^{n}$ then follows immediately for formula (6) of [BH58]. More
generally, if $\hat{X}_{{\mathbb{C}}}$ is isomorphic to $Q_{n}^{d}$ as above,
we denote by $c_{1,r}\in H^{2}(\hat{X}_{{\mathbb{C}}},C(1))$ the class of the
line bundle defined above corresponding to the $r$-th factor of $Q_{n}^{d}$,
$r=1,\dots,d$, and let $e_{r}$ correspond to the Euler class in the $r$-th
factor. The isomorphism of Proposition 2.7 is valid in all cases, and Goresky
and Pardon showed that the image in $IH^{2j}({}_{K}S(G,X)^{\rm
min},{\mathbb{Q}}(j))_{v}$ of the classes $c_{1,r}^{j}\in
H^{2j}(\hat{X}_{{\mathbb{C}}},C(j))$ lift canonically to the cohomology of the
minimal compactification (the twist $j$ here is unnecessary for
$C=\mathbb{C}$, we write it for the case $C=\mathbb{Q}_{\ell}$). However, when
$n=2k-2>2$, they were unable to show that the $e_{r}$ lift. We can extend the
statement of Theorem 2.11 with the following definition.

###### Definition 2.13

(i) Suppose $G^{\rm der}$ is ${\mathbb{Q}}$-simple and $G^{\rm
der}({\mathbb{R}})\simeq\mathrm{SO}(2k-2,2)^{d}$ for some integer $d$, with
$k>2$. Suppose $C={\mathbb{C}},{\mathbb{Q}}_{\ell}$, or ${\mathbb{Q}}$. Define

$\displaystyle H^{*}(\hat{X},C)^{+}$ $\displaystyle\subset H^{*}(\hat{X},C);$
(2.14) $\displaystyle IH^{*}({}_{K}S(G,X)^{\rm min},{\mathbb{Q}}(n))^{+}_{v}$
$\displaystyle\subset IH^{*}({}_{K}S(G,X)^{\rm min},{\mathbb{Q}}(n))_{v}$

to be the subalgebras generated respectively by the classes $c_{1,r}$,
$r=1,\dots,d$, and their images under the isomorphism of Proposition 2.7. We
similarly define

${}[K_{0}({\rm Rep}(K_{h}))\otimes_{K_{0}({\rm Rep}(G))}C]^{+}\subset
K_{0}({\rm Rep}(K_{h}))\otimes_{K_{0}({\rm Rep}(G))}C$ (2.15)

to be the inverse image of $H^{*}(\hat{X},C)^{+}$ under the isomorphism (1.1).
(ii) If $G$ satisfies condition (iv) of Theorem 2.11, we let

$H^{*}(\hat{X},C)^{+}=H^{*}(\hat{X},C);$ $IH^{*}({}_{K}S(G,X)^{\rm
min},{\mathbb{Q}}(n))^{+}_{v}=IH^{*}({}_{K}S(G,X)^{\rm
min},{\mathbb{Q}}(n))_{v};$ $[K_{0}({\rm Rep}(K_{h}))\otimes_{K_{0}({\rm
Rep}(G))}C]^{+}=K_{0}({\rm Rep}(K_{h}))\otimes_{K_{0}({\rm Rep}(G))}C.$

(iii) In either case, we define

${\rm Vect}_{G}(\hat{X})^{+}=ch_{\hat{X}}^{-1}(H^{*}(\hat{X},C)^{+})\subset
K_{0}({\rm Vect}_{G}(\hat{X})).$

We continue to use the notation $\mathcal{E}$ to denote (virtual) bundles in
$Vect_{G}(\hat{X})^{+}$.

###### Theorem 2.16

Suppose $G$ satisfies either (i) or (ii) of Definition 2.13. Then the
conclusions of Theorem 2.11 hold with $H^{*}(\hat{X},C)$, ${\rm
Vect}_{G}(\hat{X})$, and $IH^{*}({}_{K}S(G,X)^{\rm min},{\mathbb{Q}}(n))_{v}$
replaced by the versions with superscript +, and for $\mathcal{E}\in
Vect_{G}(\hat{X})^{+}$.

The theorem, and its application in Proposition 4.36 below, naturally extend
to groups $G$ such that $G^{\rm der}$ is a product of groups of type (i) and
(ii) in Definition 2.13. We omit the details.

###  2.B. Perfectoid Shimura varieties and the Hodge-Tate morphism

The results of the present section are entirely due to Scholze [Sch15], then
to Caraiani-Scholze [CS17] and Pilloni-Stroh [PS16]. Fix a level subgroup
$K\subset G({{\mathbf{A}}_{f}})$. Let $G({{\mathbf{A}}_{f}})\to
G({{\mathbb{Q}}_{p}}),\ k\mapsto k_{p}$ be the projection. Denote by $K_{p}$
its projection to $G({{\mathbb{Q}}_{p}})$. For $r\geq 0$ we let
$K_{p,r}\subset K_{p}$ be a decreasing family of subgroups of finite index,
with $K_{p,r}\supset K_{p,r+1}$ for all $r$, and such that
$\bigcap_{r}K_{p,r}=\\{1\\}$. Let $K_{r}=\\{k\in K\leavevmode\nobreak\
|\leavevmode\nobreak\ k_{p}\in K_{p,r}\\}$, and let $K^{p}=\cap_{r}K_{r}$. We
identify $K^{p}$ with its projection to the prime-to-$p$ adèles
$G(\mathbf{A}_{f}^{p})$; then $K^{p}$ is an open compact subgroup of
$G(\mathbf{A}^{p}_{f})$ called a “tame level subgroup". We assume that the
Shimura datum $(G,X)$ is of Hodge type. Thus, up to replacing $K$ by a
subgroup of finite index, ${}_{K}S(G,X)$ admits an embedding of Shimura
varieties in a Siegel modular variety of some level attached to the Shimura
datum $(\mathrm{GSp}(2g),X_{2g})$ for some $g$, where $X_{2g}$ is the union of
the Siegel upper and lower half-spaces. We let $\hat{X}_{2g}$ denote the
compact dual flag variety of $X_{2g}$. Let $C$ denote the completion of an
algebraic closure ${\overline{\mathbb{Q}}_{p}}$ of ${\mathbb{Q}}_{p}$. Denote
by ${}_{K_{r}}{\mathcal{S}}(G,X)$, resp. ${}_{K_{r}}{\mathcal{S}}(G,X)^{\rm
min}$ the adic space over $C$ attached to ${}_{K_{r}}S(G,X)$, resp.
${}_{K_{r}}S(G,X)^{\rm min}$. We assume that $K^{p}$ is contained in the
principal congruence subgroup of level $N$ for some $N\geq 3$, in the sense
explained in §4 of [Sch15]. We restate one of the main theorems of Scholze’s
article. In §4.1 of _loc. cit._ it is shown that there is a level
$K^{\prime}_{r}$ for $(\mathrm{GSp}(2g),X_{2g})$ such that
$K_{r}=K^{\prime}_{r}\cap G({{\mathbf{A}}_{f}})$, and such that the scheme
theoretic image of ${}_{K_{r}}S(G,X)^{\rm min}$ in
${}_{K^{\prime}_{r}}S(\mathrm{GSp}(2g),X_{2g})^{\rm min}$ does not depend on
the choice of $K^{\prime}_{r}$. We denote it by
$\overline{{}_{K_{r}}S(G,X)^{\rm min}}$, and by
$\overline{{}_{K_{r}}{\mathcal{S}}(G,X)^{\rm min}}$ its associated $C$-adic
space.

######  Theorem 2.17 (Theorem 4.1.1 in [Sch15])

There is a commutative diagram of morphisms of adic spaces 444The projective
limits on the right-hand side are not literally adic spaces; see Discussion
2.23 for the precise meaning of this statement.

$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}_{K^{p}}{\mathcal{S}}(G,X)\
\ \ \ \ \ \ \
\sim}$$\scriptstyle{j}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\varprojlim_{r}{}_{K_{r}}{\mathcal{S}}(G,X)}$$\scriptstyle{\varprojlim_{r}j_{r}}$$\textstyle{\overline{{}_{K^{p}}{\mathcal{S}}(G,X)}^{\rm
min}\ \ \ \ \
\sim}$$\textstyle{\varprojlim_{r}\overline{{}_{K_{r}}{\mathcal{S}}(G,X)}^{\rm
min}.}$ (2.22)

Here, on the left side, $j$ is an open embedding of perfectoid Shimura
varieties over $C$, on the right side, $\varprojlim_{r}j_{r}$ denotes the
formal inverse system of open embeddings of adic spaces over $C$, and the
notation $\sim$ is defined precisely in [SW13, Definition 2.4.1]. Moreover,
there is a $G({{\mathbb{Q}}_{p}})$-equivariant Hodge-Tate morphism

$\pi_{HT}:\overline{{}_{K^{p}}{\mathcal{S}}(G,X)}^{\rm
min}{\leavevmode\nobreak\ \rightarrow\leavevmode\nobreak\
}{\hat{\mathcal{X}}}_{2g}$

which is compatible with change of tame level subgroup $K^{p}$.

###### Discussion 2.23

Here we use the notation $\hat{\mathcal{X}}_{2g}$ for the adic space over $C$
attached to the flag variety $X_{2g}$. In loc. cit. the notation
$\mathcal{X}^{*}_{K^{p}}$ is used for the perfectoid Shimura variety
${}_{K^{p}}{\mathcal{S}}(G,X)^{\rm min}.$ The notation $\sim$ indicates that
the right hand side of (2.22) is not to be viewed as a projective limit in the
category of adic spaces, which in general does not make sense. Rather, what is
meant is that, for each $r$, there is a commutative diagram

$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}_{K^{p}}{\mathcal{S}}(G,X)\
\ignorespaces\ignorespaces\ignorespaces\ignorespaces\
}$$\scriptstyle{j}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}_{K_{r}}{\mathcal{S}}(G,X)}$$\scriptstyle{j_{r}}$$\textstyle{\overline{{}_{K^{p}}{\mathcal{S}}(G,X)}^{\rm
min}\ \ \ignorespaces\ignorespaces\ignorespaces\ignorespaces\
}$$\textstyle{\overline{{}_{K_{r}}{\mathcal{S}}(G,X)}^{\rm min};}$ (2.28)

that these are compatible with the natural maps from level $K_{r+1}$ to level
$K_{r}$; and that the objects on the left have the properties indicated in
[SW13, Definition 2.4.1].

The target of the Hodge-Tate morphism was clarified in [CS17]. Let
$\hat{\mathcal{X}}\subset\hat{\mathcal{X}}_{2g}$ be the closed embedding of
$C$-adic spaces corresponding to the closed embedding
$\hat{X}\subset\hat{X}_{2g}$ of the compact duals of $X$ and $X_{2g}$.

######  Theorem 2.29 (Theorem 2.1.3 in [CS17])

The Hodge-Tate morphism $\pi_{HT}$ in Theorem 2.17 factors through the
inclusion $\hat{\mathcal{X}}\subset\hat{\mathcal{X}}_{2g}$, yielding a
$G(\mathbb{Q}_{p})$\- equivariant Hodge-Tate morphism

$\pi_{HT}:\leavevmode\nobreak\ \leavevmode\nobreak\
\overline{{}_{K^{p}}{\mathcal{S}}(G,X)}^{\rm min}{\leavevmode\nobreak\
\rightarrow\leavevmode\nobreak\ }\hat{\mathcal{X}}.$

  * Proof.

The existence of $\pi_{HT}$ is stated for the open perfectoid Shimura variety
${}_{K^{p}}{\mathcal{S}}(G,X)$ with values in $\hat{\mathcal{X}}$. Since
$\hat{\mathcal{X}}$ is closed in $\hat{\mathcal{X}}_{2g}$, the extension to
the boundary then follows from Theorem 2.17 by continuity, as in [C+6, Theorem
3.3.4]. $\Box$

Assume for the moment that $(G,X)=(\mathrm{GSp}(2g),X_{2g})$. Fix
$K_{g}\subset\mathrm{GSp}(2g,{{\mathbf{A}}_{f}})$ a neat compact open
subgroup, with $K_{p}=\mathrm{GSp}(2g,{{\mathbb{Z}}_{p}})$ and write
${}_{K_{g}}{\mathcal{A}}_{g}$ instead of
${}_{K_{g}}S(\mathrm{GSp}(2g),X_{2g})$ for the Siegel modular variety of genus
$g$ and level $K_{g}$, viewed as an adic space over $C$. Let $K_{g,r}\subset
K_{g}$ be the principal congruence subgroup of $K_{g}$ of level $p^{r}$. Let
${}_{K_{g}}{\mathcal{A}}_{g}^{\rm tor}={}_{K_{g}}{\mathcal{A}}_{g,\Sigma_{g}}$
be a smooth projective toroidal compactification of
${}_{K_{g}}{\mathcal{A}}_{g}$ for some combinatorial datum $\Sigma_{g}$ as
above. Following Pilloni-Stroh in [PS16], §1.3, but with a change of notation,
we let ${}_{K_{g,r}}{\mathcal{A}}_{g}^{\rm tor}$ denote the corresponding
toroidal compactification of ${}_{K_{g,r}}{\mathcal{A}}_{g}$ for each $r$,
with the same $\Sigma_{g}$. The ${}_{K_{g,r}}{\mathcal{A}}_{g}^{\rm tor}$ form
a projective system of adic spaces over $C$. The authors construct a
projective system of normal models
${}_{K_{g,r}}\mathcal{A}_{g,\mathcal{O}_{C}}^{\rm tor}$ over ${\rm
Spec}(\mathcal{O}_{C})$ for each $r$, and define a perfectoid space
${}_{K^{p}_{g}}\mathcal{A}_{g}^{\rm tor}$ which is the generic fiber of the
projective limit of the ${}_{K_{g,r}}\mathcal{A}_{g,\mathcal{O}_{C}}^{\rm
tor}$, in the sense of [SW13, Section 2.2]. See [PS16, Section A.12 and
Corollaire A.19] for the statement that the projective limit, which they
denote $\mathcal{X}(p^{\infty})^{\rm tor-mod}$, is indeed perfectoid. There is
a natural map

$\mathfrak{q}_{g}:_{K^{p}_{g}}\mathcal{A}_{g}^{\rm tor}{\leavevmode\nobreak\
\rightarrow\leavevmode\nobreak\ }_{K^{p}_{g}}\mathcal{A}_{g}^{\rm min}.$

Let $(G,X)$ be any Shimura datum of Hodge type, with
$\iota:(G,X)\hookrightarrow(\mathrm{GSp}(2g),X_{2g})$ a fixed symplectic
embedding. In the remainder of this section we make the following hypothesis,
which (as we have seen) is true for the Shimura datum
$(\mathrm{GSp}(2g),X_{2g})$:

###### Hypothesis 2.30

The projective system of ${}_{K_{r}}S(G,X)$ has a compatible projective system
of toroidal compactifications ${}_{K_{r}}S(G,X)_{\Sigma}$ such that, if
${}_{K_{r}}{\mathcal{S}}(G,X)_{\Sigma}$ is the associated adic space over $C$,
then there is a perfectoid space ${}_{K^{p}}{\mathcal{S}}(G,X)_{\Sigma}$ such
that

${}_{K^{p}}{\mathcal{S}}(G,X)_{\Sigma}\leavevmode\nobreak\
\sim\leavevmode\nobreak\ \varprojlim{}_{K_{r}}{\mathcal{S}}(G,X)_{\Sigma}$

in the sense of [SW13, Section 2.2].

It is likely the Hypothesis 2.30 is valid, but its proof is probably as
elaborate as the construction of integral toroidal compactifications in
[Lan13] 555Ana Caraiani has informed us that a proof of Hypothesis 2.30, for
the case of a Shimura variety attached to a quasi-split unitary similitude
group of even dimension, will appear in her forthcoming joint work with
Scholze.. Thus we will provide an alternative proof of our main theorems, that
does not depend on this hypothesis, in section §3. However, the proof assuming
Hypothesis 2.30 is considerably simpler, and it is valid, by [PS16], in the
Siegel modular case. Under Hypothesis 2.30, there is automatically a map

$q:{}_{K^{p}}{\mathcal{S}}(G,X)_{\Sigma}{\leavevmode\nobreak\
\rightarrow\leavevmode\nobreak\ }{}_{K^{p}}{\mathcal{S}}(G,X)^{\rm min}$

as well as the maps

$q_{r}:{}_{K_{r}}{\mathcal{S}}(G,X)_{\Sigma}{\leavevmode\nobreak\
\rightarrow\leavevmode\nobreak\ }{}_{K_{r}}{\mathcal{S}}(G,X)^{\rm min}$

at finite level (the existence of which does not depend on the hypothesis). By
[Har85, Section 3.4], there is a correspondence

---
$\textstyle{{}_{K}I(G,X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{b^{\prime}}$$\scriptstyle{a^{\prime}}$$\textstyle{\hat{X}}$$\textstyle{{}_{K}S(G,X)}$
(2.35)

where $b^{\prime}$ is a family of $G$-torsors, functorial with respect to
inclusions $K^{\prime}\subset K$ and translation by elements $g\in
G({\mathbf{A}}_{f})$, and $a^{\prime}$ is a $G$-equivariant morphism. For any
$G$-equivariant vector bundle $\mathcal{E}$ over $\hat{X}$, the pullback
$a^{\prime*}(\mathcal{E})$ is $G$-equivariant, and thus descends to the
automorphic vector bundle $[\mathcal{E}]$ over ${}_{K}S(G,X)$. The
$G-\rm{torsor}$ ${}_{K}I(G,X)$ is constructed as the moduli space
$\mathrm{Isom}^{\otimes}([V_{\rho}],V)$ of trivializations of a flat
automorphic vector bundle $[V_{\rho}]$ attached to a faithful representation
$\rho:G{\leavevmode\nobreak\ \rightarrow\leavevmode\nobreak\ }\mathrm{GL}(V)$;
the superscript ⊗ indicates that the isomorphisms respect absolute Hodge
cycles. Thus the construction requires a priori the existence of the
automorphic vector bundle $[V_{\rho}]$. When ${}_{K}S(G,X)$ is of Hodge type,
one can take $\rho$ to be a symplectic embedding $\rho:G{\leavevmode\nobreak\
\rightarrow\leavevmode\nobreak\ }\mathrm{GSp}(V)$, and define $[V_{\rho}]$ to
be the pullback to ${}_{K}S(G,X)$ of the (dual of the) relative de Rham
$H^{1}$ of the universal abelian scheme over the Siegel modular variety
attached to $\mathrm{GSp}(V)$. Then the morphism $a$ is defined by
transferring the Hodge filtration on $[V_{\rho}]$ to the constant vector
bundle defined by $V$. That this construction is canonically independent of
the choice of symplectic embedding $\rho$ is explained in [Har85, Remark
4.9.1]; see also [CS17, Lemma 2.3.4]. Choose a base point $h\in
X\subset{\hat{X}}$ as in Section 1; we may as well assume $h$ to be a $CM$
point. Recall that $P_{h}$ denotes the stabilizer of $h$ and $K_{h}$ its Levi
quotient (in [CS17] this group is denoted $M_{\mu}$). Any faithful
representation $\alpha:K_{h}{\leavevmode\nobreak\
\rightarrow\leavevmode\nobreak\ }\mathrm{GL}(W_{h})$ defines by pullback to
$P_{h}$ a $G$-equivariant vector bundle $\mathcal{W}_{h}$ over $\hat{X}$, and
thus an automorphic vector bundle $[\mathcal{W}_{h}]$ over ${}_{K}S(G,X)$ that
varies functorially in $K$. We can then define a family (depending on $K$) of
$K_{h}$-torsors $b:T={}_{K}T_{h}(G,X){\leavevmode\nobreak\
\rightarrow\leavevmode\nobreak\ }{}_{K}S(G,X)$ with a $K_{h}$-equivariant
morphism $a:{}_{K}T_{h}(G,X){\leavevmode\nobreak\
\rightarrow\leavevmode\nobreak\ }\hat{X}$ as the moduli space of
trivializations of $[\mathcal{W}_{h}]$ as above. More precisely, letting
$R_{u}P_{h}$ denote the unipotent radical of $P_{h}$, the natural morphism
$\hat{X}_{h}:=G/R_{u}P_{h}{\leavevmode\nobreak\
\rightarrow\leavevmode\nobreak\ }\hat{X}$ is canonically a $G$-equivariant
$K_{h}$-torsor, whose pullback $a^{\prime-1}\hat{X}_{h}$ descends to a
$K_{h}$-torsor over ${}_{K}S(G,X)$, over the reflex field of the CM point $h$,
that is naturally identified with $T$ defined above. Moreover, the
construction is canonically independent of the choice of base point. This is
also constructed in Section 2.3, especially Lemma 2.3.5, of [CS17]. The
pullback of $T$ via the ringed space morphism $\hat{\mathcal{X}}\to\hat{X}$ is
denoted by $\mathcal{T}$. The $K_{h}$-torsor $b$ has a Mumford extension
$T^{\rm can}\to{}_{K}S(G,X)_{\Sigma}$ as a $G$-torsor. See [HZ94],
specifically Lemma 4.4.2 and pages 320-321, where the Mumford extension of the
$G$-torsor $b^{\prime}$ in (2.35) is constructed; the $K_{h}$-torsor is
constructed in the same way. One denotes by $\mathcal{M}^{\rm
can}={}_{K}\mathcal{M}^{\rm can}$ the pullback of $T^{\rm can}$ via the ringed
space morphism ${}_{K}{\mathcal{S}}(G,X)_{\Sigma}\to{}_{K}S(G,X)_{\Sigma}.$ We
define the $K_{h}$-torsor

$\displaystyle\mathcal{M}_{p}:=\pi_{HT}^{*}\mathcal{T}$

on ${}_{K^{p}}{\mathcal{S}}(G,X)^{\rm min}$, which coincides by definition
with the $K_{h}$-torsor defined in [CS17, Lemma 2.3.8]. We define the
$K_{h}$-torsor

$\displaystyle\mathcal{M}_{dR,\Sigma,K}:=\pi_{K,K^{p}}^{*}\left({}_{K}\mathcal{M}^{\rm
can}\right)$ (2.36)

on ${}_{K^{p}}{\mathcal{S}}(G,X)_{\Sigma}$.

######  Proposition 2.37 (Proposition 2.3.9 in [CS17])

For any neat level subgroup $K$, there is a canonical isomorphism

$j^{*}{\mathcal{M}}_{p}{\leavevmode\nobreak\
\overset{\sim}{\longrightarrow}\leavevmode\nobreak\
}j_{\Sigma}^{*}{\mathcal{M}}_{dR,{\Sigma},K}$

of $K_{h}$-torsors over ${}_{K^{p}}{\mathcal{S}}(G,X)$.

Although the article [CS17] is written for compact Shimura varieties, the
argument developed there for this point is valid for any Shimura variety of
Hodge type. Strictly speaking, as explained in [CS17], the torsors
${\mathcal{M}}_{p}$ and ${\mathcal{M}}_{dR,{\Sigma},K}$ have natural
extensions to $G$-torsors by pullback to torsors for opposite parabolics,
followed by pushforward to $G$, so the comparison only applies to semisimple
automorphic vector bundles. The following theorem is essentially due to
Pilloni and Stroh.

###### Theorem 2.38

Assume Hypothesis 2.30. Then, for any neat level subgroup $K$, the isomorphism
of Proposition 2.37 extends to a canonical isomorphism

$\displaystyle q^{*}\mathcal{M}_{p}{\leavevmode\nobreak\
\overset{\sim}{\longrightarrow}\leavevmode\nobreak\
}\mathcal{M}_{dR,\Sigma,K}$

of $K_{h}$-torsors over ${}_{K^{p}}{\mathcal{S}}(G,X)_{\Sigma}$. As $K^{p}$
varies, these isomorphisms are equivariant under the action of
$G(\mathbf{A}_{f}^{p})$.

In [PS16] this is proved for the Siegel modular variety, although it is not
stated in this form. In Section 5 we explain their result and show how to
obtain Theorem 2.38 for general Shimura varieties of Hodge type, under
Hypothesis 2.30. In the following section, we show how to dispense with
Hypothesis 2.30 and prove Theorem 3.22 as an alternative to Theorem 2.38. In
Section 4 we construct $\ell$-adic Chern classes using either Theorem 2.38
(when it is available) or Theorem 3.22 for general Shimura varieties of Hodge
type.

##  3\. Construction of perfectoid covers

Let $(G,X)$ be any Shimura datum of Hodge type, with
$\iota:(G,X)\hookrightarrow(\mathrm{GSp}(2g),X_{2g})$ a fixed symplectic
embedding. Choose a combinatorial datum $\Sigma$ for ${}_{K}S(G,X)$ that is
compatible with the fixed $\Sigma_{g}$ chosen above, so that there is a
morphism

$[\iota]:{}_{K}S(G,X)_{\Sigma}\to{}_{K_{g}}{\mathcal{A}}_{g,\Sigma_{g}}$

which factors through

$\displaystyle\overline{{}_{K}S(G,X)_{\Sigma}}={}_{K_{g}}{\mathcal{A}}_{g,\Sigma_{g}}\times_{{}_{K_{g}}{\mathcal{A}}^{\rm
min}_{g}}\overline{{}_{K}{\mathcal{S}}(G,X)}^{\rm min}.$ (3.1)

We denote the corresponding $C$-adic spaces by
${}_{K}{\mathcal{S}}(G,X)_{\Sigma}$ and
$\overline{{}_{K}{\mathcal{S}}(G,X)_{\Sigma}}$. If we admitted Hypothesis 2.30
then we would be able to replace
$\overline{{}_{K}{\mathcal{S}}(G,X)_{\Sigma}}$ by
${}_{K}{\mathcal{S}}(G,X)_{\Sigma}$ in what follows; however, we will not
assume Hypothesis 2.30 in the remainder of this section. By [Sch12, Prop.
6.18] the fibre product

$\displaystyle\overline{{}_{K^{p}}{\mathcal{S}}(G,X)_{\Sigma}}:={}_{K^{p}_{g}}\mathcal{A}_{g}^{\rm
tor}\times_{{}_{K_{g}^{p}}\mathcal{A}_{g}^{\rm
min}}\overline{{}_{K^{p}}{\mathcal{S}}(G,X)}^{\rm min}$ (3.2)

exists in the category of perfectoid spaces. We denote by

$\displaystyle\bar{q}:\overline{{}_{K^{p}}{\mathcal{S}}(G,X)_{\Sigma}}\to\overline{{}_{K^{p}}{\mathcal{S}}(G,X)}^{\rm
min}$ (3.3)

the projection on the second factor, and by

$\bar{q}_{r}:\overline{{}_{K^{p}}{\mathcal{S}}(G,X)_{\Sigma}}\to\overline{{}_{K_{r}}{\mathcal{S}}(G,X)}^{\rm
min}$

the composition with the canonical map of
$\overline{{}_{K^{p}}{\mathcal{S}}(G,X)}^{\rm min}$ to
$\overline{{}_{K_{r}}{\mathcal{S}}(G,X)}^{\rm min}$. By definition, in the
sense of [SW13, Definition 2.4.1], one has

${}_{K^{p}_{g}}\mathcal{A}_{g}^{\rm
tor}\sim\varprojlim_{r}{}_{K_{g,r}}{\mathcal{A}}_{g}^{\rm tor},\ \
{}_{K^{p}_{g}}\mathcal{A}_{g}^{\rm
min}\sim\varprojlim_{r}{}_{K_{g,r}}{\mathcal{A}}_{g}^{\rm min},\ \
\overline{{}_{K^{p}}{\mathcal{S}}(G,X)}^{\rm
min}\sim\varprojlim_{r}\overline{{}_{K_{r}}{\mathcal{S}}(G,X)}^{\rm min}.$

One deduces

$\displaystyle\overline{{}_{K^{p}}{\mathcal{S}}(G,X)_{\Sigma}}\sim\varprojlim_{r}\overline{{}_{K_{r}}{\mathcal{S}}(G,X)_{\Sigma}}.$
(3.4)

In particular, by [Sch12, Corollary 7.1.8], one has the relation

$\displaystyle
H^{*}\left(\overline{{}_{K^{p}}{\mathcal{S}}(G,X)_{\Sigma}},\mathbb{Z}/\ell^{n}\right)=\varinjlim_{r}H^{*}\left(\overline{{}_{K_{r}}{\mathcal{S}}(G,X)_{\Sigma}},\mathbb{Z}/\ell^{n}\right).$
(3.5)

On the other hand, Scholze’s Hodge-Tate map yields the composite map

$\displaystyle\overline{{}_{K^{p}}{\mathcal{S}}(G,X)_{\Sigma}}\to{}_{K_{g}^{p}}\mathcal{A}_{g}^{\rm
min}\to\hat{\mathcal{X}}_{2g}$

which by definition is the same as the composite map

$\displaystyle\overline{{}_{K^{p}}{\mathcal{S}}(G,X)_{\Sigma}}\to\overline{{}_{K^{p}}{\mathcal{S}}(G,X)}^{\rm
min}\xrightarrow{\pi_{HT}}\hat{\mathcal{X}}\to\hat{\mathcal{X}}_{2g}.$

This defines a map

$\displaystyle\overline{{}_{K^{p}}{\mathcal{S}}(G,X)_{\Sigma}}\to\hat{\mathcal{X}}$
(3.6)

in the category of $C$-adic spaces. It is $G(\mathbb{Q}_{p})$-equivariant. In
what follows the idea to use the space denoted $\mathbb{P}^{N,{\rm perf}}$ to
define a perfectoid space above ${}_{K_{s}}{\mathcal{S}}(G,X)_{\Sigma}$ is due
to Bhargav Bhatt. We fix a natural number $s$. Associated to it we have the
normalization morphism
$\nu_{s}:{}_{K_{s}}{\mathcal{S}}(G,X)_{\Sigma}\to\overline{{}_{K_{s}}{\mathcal{S}}(G,X)_{\Sigma}}.$
It is a finite morphism, thus it factors as

$\nu_{s}:{}_{K_{s}}{\mathcal{S}}(G,X)_{\Sigma}\xrightarrow{\eta_{s}}\overline{{}_{K_{s}}{\mathcal{S}}(G,X)_{\Sigma}}\times_{C}\mathbb{P}^{N}\xrightarrow{\rm
projection}\overline{{}_{K_{s}}{\mathcal{S}}(G,X)_{\Sigma}}$

for some natural number $N$, where $\eta_{s}$ is a closed embedding. We set

$Y_{s}={\rm Image}(\eta_{s}).$

By definition $\eta_{s}$ induces an isomorphism between
${}_{K_{s}}{\mathcal{S}}(G,X)_{\Sigma}$ and $Y_{s}$. For any $\rho\geq 0$ we
also define

$Y_{s+\rho}=Y_{s}\times_{\overline{{}_{K_{s}}{\mathcal{S}}(G,X)_{\Sigma}}}\overline{{}_{K_{s+\rho}}{\mathcal{S}}(G,X)_{\Sigma}}.$
(3.7)

We fix coordinates $(x_{0}:\ldots:x_{N})$ of $\mathbb{P}^{N}$ and define for
$\rho\in\mathbb{N}$ the finite flat morphisms
$\phi_{\rho}:\mathbb{P}^{N}\to\mathbb{P}^{N}$ by
$\phi_{\rho}(x_{0}:\ldots:x_{N})=(x_{0}^{p^{\rho}}:\ldots:x_{N}^{p^{\rho}})$.
Over $C$, a perfectoid space $\mathbb{P}^{N,{\rm perf}}$ is associated to
$(\mathbb{P}^{N},\phi_{\rho})$ [Sch14, Section 17], such that

$\mathbb{P}^{N,{\rm perf}}\sim\varprojlim_{\rho}\phi_{\rho}.$

Then the fiber product

${}_{K^{p}}{\mathcal{S}}(G,X)_{\Sigma,N}:=\overline{{}_{K^{p}}{\mathcal{S}}(G,X)_{\Sigma}}\times_{C}\mathbb{P}^{N,{\rm
perf}}$

exists as a perfectoid space, by [Sch12, Proposition 6.18], and we have

${}_{K^{p}}{\mathcal{S}}(G,X)_{\Sigma,N}\sim\varprojlim_{\rho}\overline{{}_{K_{s+\rho}}{\mathcal{S}}(G,X)_{\Sigma}}\times_{C}\mathbb{P}^{N},$
(3.8)

where the map

$\overline{{}_{K_{s+\rho}}{\mathcal{S}}(G,X)_{\Sigma}}\times_{C}\mathbb{P}^{N}{\leavevmode\nobreak\
\rightarrow\leavevmode\nobreak\
}\overline{{}_{K_{s}}{\mathcal{S}}(G,X)_{\Sigma}}\times_{C}\mathbb{P}^{N}$

is given by the natural projection in the first factor and $\phi_{\rho}$ in
the second factor.

###### Proposition 3.9

There is a closed perfectoid subspace

$Z_{s}\subset{}_{K^{p}}{\mathcal{S}}(G,X)_{\Sigma,N}$

such that

$Z_{s}\sim\varprojlim_{\rho\geq 0}Y_{s+\rho}.$

  * Proof.

We define $Z_{s}\subset{}_{K^{p}}{\mathcal{S}}(G,X)_{\Sigma,N}$ by the
pullback of the ideal of $Y_{s}$ in
$\overline{{}_{K_{s}}{\mathcal{S}}(G,X)_{\Sigma}}\times_{C}\mathbb{P}^{N}$ via
(3.8)

${}_{K^{p}}{\mathcal{S}}(G,X)_{\Sigma,N}\to\overline{{}_{K_{s}}{\mathcal{S}}(G,X)_{\Sigma}}\times_{C}\mathbb{P}^{N}.$

It follows from [Bha17, Proposition 9.4.1] that $Z_{s}$ is perfectoid. $\Box$

By construction, $Z_{s}$ maps to
$\overline{{}_{K^{p}}{\mathcal{S}}(G,X)_{\Sigma}}$ which itself maps to
$\overline{{}_{K^{p}}{\mathcal{S}}(G,X)}^{\rm min}$, and to
${}_{K_{s}}{\mathcal{S}}(G,X)_{\Sigma}$, but not necessarily to
${}_{K_{r}}{\mathcal{S}}(G,X)_{\Sigma}$ for $r>s$. On the other hand,
${}_{K_{s}}{\mathcal{S}}(G,X)_{\Sigma}$ naturally maps to
${}_{K_{s}}{\mathcal{S}}(G,X)^{\rm min}$, and thus we have a composite diagram

$Z_{s}\overset{\phi}{\to}{}_{K_{s}}{\mathcal{S}}(G,X)_{\Sigma}\overset{q_{s}}{\to}{}_{K_{s}}{\mathcal{S}}(G,X)^{\rm
min}\overset{u_{s}}{\to}\overline{{}_{K_{s}}{\mathcal{S}}(G,X)}^{\rm min}$
(3.10)

We let $Q$ denote the composite of the three maps, and let
$\underline{q}_{s}=u_{s}\circ q_{s}$ denote the composite of the last two
maps.

###### Proposition 3.11

The perfectoid space $Z_{s}$ has the following properties.

  * (i)

There are morphisms

$Q:Z_{s}\xrightarrow{\psi}\overline{{}_{K^{p}}{\mathcal{S}}(G,X)_{\Sigma}}\xrightarrow{\bar{q}}\overline{{}_{K^{p}}{\mathcal{S}}(G,X)}^{\rm
min}$ $\phi:Z_{s}{\leavevmode\nobreak\ \rightarrow\leavevmode\nobreak\
}{}_{K_{s}}{\mathcal{S}}(G,X)_{\Sigma}$

such that the following diagram commutes

$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces
Z_{s}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{Q_{s}}$$\scriptstyle{\phi}$$\scriptstyle{\psi}$$\textstyle{\overline{{}_{K^{p}}{\mathcal{S}}(G,X)_{\Sigma}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\bar{q}_{s}}$$\textstyle{Y_{s}={}_{K_{s}}{\mathcal{S}}(G,X)_{\Sigma}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\underline{q}_{s}}$$\textstyle{\overline{{}_{K_{s}}{\mathcal{S}}(G,X)}^{\rm
min}}$ (3.16)

  * (ii)

The homomorphism on cohomology

$\phi^{*}:H^{\bullet}(Y_{s},\mathbb{F}_{\ell})\to
H^{\bullet}(Z_{s},\mathbb{F}_{\ell}),$

induced from (i), is injective, where $\ell$ is a prime different from $p$.
The injectivity holds also true with $\mathbb{Q}_{\ell}$-coefficients when
$s=0$.

  * Proof.

In what follows we assume $G^{ad}$ contains no ${\mathbb{Q}}$-simple factor
isomorphic to $\mathrm{PGL}(2)$ – in other words, that $S(G,X)$ contains no
factor isomorphic to the elliptic modular curve – since in that case the
minimal and toroidal compactifications coincide and there is nothing to prove.
The point (i) is just a restatement of the above construction. We prove (ii).
By Proposition 3.9 together with [Sch12, Corollary 7.18], we have

$H^{\bullet}(Z_{s},\mathbb{F}_{\ell})=\varinjlim_{\rho\geq
0}H^{\bullet}(Y_{s+\rho},\mathbb{F}_{\ell}).$

On the other hand, the proper maps $Y_{s+\rho}\to Y_{s+\rho-1}$ for $\rho\geq
1$ are finite, of degree a $p$-power for $s\geq 1$. Indeed, the finiteness of
this map is reduced by Definition (3.7) to the corresponding assertion for the
maps

$\overline{{}_{K_{s+\rho}}{\mathcal{S}}(G,X)_{\Sigma}}\rightarrow\overline{{}_{K_{s}}{\mathcal{S}}(G,X)_{\Sigma}}.$

This in turn is reduced by (3.1) to the well known fact that the corresponding
morphisms for change of level in the toroidal compactification
${}_{K_{g}}{\mathcal{A}}_{g,\Sigma_{g}}$ and the compactifications
$\overline{{}_{K}{\mathcal{S}}(G,X)}^{\rm min}$ are finite. Indeed, let
$\omega_{g}$ denote the determinant of the relative cotangent bundle of the
universal abelian scheme over ${}_{K_{g}}{\mathcal{A}}_{g}$; in other words,
$\omega_{g}$ is the line bundle whose global sections are Siegel modular forms
of weight $1$ and level $K_{g}$. Then it is well known (from the theory of the
minimal compactification, and because we have excluded the case of modular
curves) that $\omega_{g}$ extends to an ample line bundle over
${}_{K_{g}}{\mathcal{A}}_{g}^{\rm min}$, and that the pullback of this
extension to ${}_{K}{\mathcal{S}}(G,X)^{\rm min}$ is ample as well. Thus –
although we have been told there is a problem with the injectivity claim in
Corollary 10.2.3 of [SW17] – the map from ${}_{K}{\mathcal{S}}(G,X)^{\rm min}$
to its image in ${}_{K_{g}}{\mathcal{A}}_{g}^{\rm min}$ is finite, and the
morphisms from ${}_{K}{\mathcal{S}}(G,X)^{\rm min}$ to
$\overline{{}_{K}{\mathcal{S}}(G,X)}^{\rm min}$ are finite, and from the
commutative square

$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}_{K}{\mathcal{S}}(G,X)_{s}^{\rm
min}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rm
finite}$$\scriptstyle{\rm
finite}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\overline{{}_{K}{\mathcal{S}}(G,X)}_{s}^{\rm
min}}$$\textstyle{{}_{K}{\mathcal{S}}(G,X)_{r}^{\rm
min}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rm
finite}$$\textstyle{\overline{{}_{K}{\mathcal{S}}(G,X)}_{r}^{\rm min}}$ (3.21)

for $s\geq r$ we conclude that the the level-changing maps
$\overline{{}_{K}{\mathcal{S}}(G,X)}_{s}^{\rm
min}\to\overline{{}_{K}{\mathcal{S}}(G,X)}_{r}^{\rm min}$ are finite. This
implies (ii) for $s\geq 1$ – and also for $s=0$, after replacing
$\mathbb{F}_{\ell}$ by $\mathbb{Q}_{\ell}$ in case the order of the group
$K_{0}/K_{1}$ is divisible by $\ell$. $\Box$

The analogue of Theorem 2.38 is

###### Theorem 3.22

For any neat level subgroup $K$ and any $s\geq 0$, the isomorphism of
Proposition 2.37 extends to a canonical isomorphism

$\displaystyle Q^{*}\mathcal{M}_{p}{\leavevmode\nobreak\
\overset{\sim}{\longrightarrow}\leavevmode\nobreak\
}\phi^{*}\mathcal{M}_{dR,\Sigma,K_{s}}$

of $K_{h}$-torsors over $Z_{s}$. As $K^{p}$ varies, these isomorphisms are
equivariant under the action of $G(\mathbf{A}_{f}^{p})$.

The proof is given in Section 5.

##  4\. Construction of $\ell$-adic Chern classes

As in the previous sections, all level subgroups $K$ are neat and all toroidal
compactifications are smooth and projective. The constructions in the present
section depend in an essential way on on Theorem 3.22 – or on Theorem 2.38, if
we assume Hypothesis 2.30. Proofs are given without assuming Hypothesis 2.30;
to pass from the situation with Hypothesis 2.30, we just formally set $Q=q$.

###  4.A. Construction using $\pi_{HT}$

As in [CS17, Theorem 2.1.3 (2)], the existence of the $K_{h}$-torsor
${\mathcal{M}}_{p}$ over $\overline{{}_{K^{p}}{\mathcal{S}}(G,X)}^{\rm min}$
implies that there is a functor

$\displaystyle\begin{array}[]{rcl}{\rm Vect}^{\rm
ss}_{G}(\hat{X})[\leavevmode\nobreak\ \leavevmode\nobreak\
\overset{r}{\simeq}{\rm Rep}_{K_{h}}]&\to&\text{ \\{Vector bundles over
}\overline{{}_{K^{p}}{\mathcal{S}}(G,X)}^{\rm min}\\}\\\
{\mathcal{E}}&\mapsto&[{\mathcal{E}}]_{p}.\end{array}$ (4.3)

Similarly, the existence of the $K_{h}$ torsor $\mathcal{M}_{dR,\Sigma,K}$
over ${}_{K}{\mathcal{S}}(G,X)_{\Sigma}$ gives rise to a functor

$\displaystyle\begin{array}[]{rcl}{\rm Vect}^{\rm
ss}_{G}(\hat{X})[\leavevmode\nobreak\ \leavevmode\nobreak\
\overset{r}{\simeq}{\rm Rep}_{K_{h}}]&\to&\text{ \\{Vector bundles over
}{}_{K}{\mathcal{S}}(G,X)_{\Sigma}\\}\\\
{\mathcal{E}}&\mapsto&[\mathcal{E}^{\rm can}].\end{array}$ (4.6)

With the notation of Theorem 3.22, we define

$\displaystyle[{\mathcal{E}}]_{p,{\Sigma},K_{s}}=Q^{*}[{\mathcal{E}}]_{p},\
[{\mathcal{E}}]_{dR,{\Sigma},K_{s}}=\phi^{*}[\mathcal{E}^{\rm can}].$ (4.7)

Let
$Z^{o}_{s}={}_{K^{p}}{\mathcal{S}}(G,X)\times_{{}_{K^{p}}{\mathcal{S}}(G,X)^{\rm
min}}Z_{s}$, $\mathfrak{j}_{s}:Z^{o}_{s}\hookrightarrow Z_{s}$ the canonical
map. The isomorphism of torsors in Proposition 2.37 gives rise to a canonical
isomorphism of tensor functors over $Z^{o}_{s}$

$({\mathcal{E}}\mapsto\mathfrak{j}_{s}^{*}([{\mathcal{E}}]_{p,{\Sigma},K_{s}})){\leavevmode\nobreak\
\overset{\sim}{\longrightarrow}\leavevmode\nobreak\
}({\mathcal{E}}\mapsto\mathfrak{j}_{s}^{*}([{\mathcal{E}}]_{dR,{\Sigma},K_{s}})).$
(4.8)

From Theorem 3.22 one immediately obtains the following corollary.

###### Corollary 4.9

There is a canonical isomorphism of tensor functors

$({\mathcal{E}}\mapsto[{\mathcal{E}}]_{p,{\Sigma},K_{s}}){\leavevmode\nobreak\
\overset{\sim}{\longrightarrow}\leavevmode\nobreak\
}({\mathcal{E}}\mapsto[{\mathcal{E}}]_{dR,{\Sigma},K_{s}})$

over $Z_{s}$, lifting the isomorphism (4.8) over $Z_{s}^{o}$.

We return to this theme in the next paragraph. Recall briefly how the Hecke
algebra acts on automorphic bundles on ${}_{K}S(G,X)$. See [EH17, Section
1.2]. Fix $K\subset G({{\mathbf{A}}_{f}})$. Let $h\in G({{\mathbf{A}}_{f}})$
and consider $K_{h}=K\cap hKh^{-1}\subset K$. Right multiplication by $h$
defines an isomorphism $r_{h}:{}_{hKh^{-1}}S(G,X){\leavevmode\nobreak\
\overset{\sim}{\longrightarrow}\leavevmode\nobreak\ }{}_{K}S(G,X).$ One
defines

$\displaystyle
T(h)[\mathcal{E}]_{K}=r_{h*}\circ\pi_{hKh^{-1},K_{h}*}\circ\pi_{K,K_{h}}^{*}[\mathcal{E}]_{K}.$
(4.10)

The projection formula implies

$\displaystyle
T(h)[\mathcal{E}]_{K}=[\mathcal{E}]_{K}\otimes\pi_{K,K_{h}*}\mathcal{O}_{{}_{K_{h}}S(G,X).}$

As one has pullbacks and push-downs for $K_{0,\mathbb{Q}}$ and for $\ell$-adic
cohomology, one has on them an action of the Hecke algebra $\mathcal{H}_{K}$,
spanned by the $T(h)$ for $h\in G({{\mathbf{A}}_{f}})$, with $h$ trivial at
$p$ and at ramified places for $K$. In particular, formula (4.10) implies the
formula

$\displaystyle{\rm class\ of}\ T(h)[\mathcal{E}]_{K}={\rm class\ of}\
[\mathcal{E}]_{K}^{[K:K_{h}]}\ {\rm in}\ K_{0}({}_{K}S(G,X))_{\mathbb{Q}.}$
(4.11)

Recall that the volume character of the Hecke algebra spanned by the $T(h)$
with values in a ring $R$, is the character sending $T(h)$ to $[K:K_{h}]$
viewed as an element in $R$ [EH17, Lemma 1.15]. Thus (4.11) is saying that
classes of automorphic bundles in $K_{0}({}_{K}S(G,X))_{\mathbb{Q}}$ are
eigenvectors for the volume character under the action of $\mathcal{H}_{K}$.
Let $\ell\neq p$ be a prime number.

###### Lemma 4.12

The isomorphism $r_{h}$ induces an isomorphism

$\displaystyle r_{h}^{\rm min}:{}_{K}S(G,X)^{\rm min}{\leavevmode\nobreak\
\overset{\sim}{\longrightarrow}\leavevmode\nobreak\ }{}_{hKh^{-1}}S(G,X)^{\rm
min}.$

The finite morphism $\pi_{K,K_{h}}:{}_{K_{h}}S(G,X)\to{}_{K}S(G,X)$ extends to
a finite morphism

$\displaystyle\pi^{\rm min}_{K,K_{h}}:{}_{K_{h}}S(G,X)^{\rm
min}\to{}_{K}S(G,X)^{\rm min}$

and the action of $\mathcal{H}_{K}$ on
$H^{2n}({}_{K}S(G,X),\mathbb{Z}_{\ell}(n))$ extends to an action on
$H^{2n}({}_{K}S(G,X)^{\rm min},\mathbb{Z}_{\ell}(n))$.

  * Proof.

When $G=\mathrm{GL}(2)_{\mathbb{Q}}$ the Shimura varieties are finite unions
of modular curves, and the result in that case is standard. We may thus assume
$G^{\rm ad}$ contains no factor isomorphic to $\mathrm{PGL}(2)_{\mathbb{Q}}$.
Then we know by Theorem 10.14 of [BB66] that $H^{0}({}_{K}S(G,X),\omega^{n})$
is finite dimensional for any $n\geq 0$, where $\omega$ is the dualizing
sheaf. The isomorphism $r_{h}$ induces an algebra isomorphism

$\displaystyle\oplus_{n\in\mathbb{N}}H^{0}({}_{K}S(G,X),\omega^{n}){\leavevmode\nobreak\
\overset{\sim}{\longrightarrow}\leavevmode\nobreak\
}\oplus_{n\in\mathbb{N}}H^{0}(_{hKh^{-1}}S(G,X),\omega^{n}).$

Similarly the finite cover $\pi_{K,K_{h}}$ induces an injective algebra
morphism

$\displaystyle\oplus_{n\in\mathbb{N}}H^{0}({}_{K}S(G,X),\omega^{n})\hookrightarrow\oplus_{n\in\mathbb{N}}H^{0}({}_{K_{h}}S(G,X),\omega^{n}).$

Since ${}_{K}S(G,X)^{\rm min}$ (resp. ${}_{K_{h}}S(G,X)^{\rm min}$) is
$\mathrm{Proj}$ of the left-hand side (resp. right-hand side) of the last
diagram [BB66, Theorem 10.11], the induced map on the $\mathrm{Proj}$ defines
the extensions $r^{\rm min}_{h}$ and $\pi_{K,K_{h}}^{\rm min}$. On the other
hand, pullback on cohomology is defined, while the trace map ${\rm
Tr}:\pi_{K,K_{h}*}\mathbb{Z}_{\ell}\to\mathbb{Z}_{\ell}$ extends to
$j_{K*}\pi_{K,K_{h}*}\mathbb{Z}_{\ell}=\pi^{\rm
min}_{K,K_{h}*}j_{K_{h}*}\mathbb{Z}_{\ell}=\pi^{\rm
min}_{K,K_{h}*}\mathbb{Z}_{\ell}\to
j_{K*}\mathbb{Z}_{\ell}=\mathbb{Z}_{\ell}.$ This proves the second part.
$\Box$

Let $E(G,X)=:E$ be the reflex field. Fix an embedding
$E\hookrightarrow\bar{{\mathbb{Q}}}$. We choose a place $w$ of $E$ dividing
the prime $p$ which we assume to be different from $\ell$. We extend
$E\hookrightarrow\bar{{\mathbb{Q}}}$ to the completion
$E_{w}\hookrightarrow\bar{{\mathbb{Q}}}_{p}$. We denote by $G_{E_{w}},G_{E}$
the Galois groups of $E_{w},E$. We add a subscript on the lower right to
indicate the field over which we regard the varieties considered. We denote by
$H^{i}({}_{K}S(G,X)^{\rm
min}_{\bar{{\mathbb{Q}}}},\mathbb{Q}_{\ell})_{v}=H^{i}({}_{K}S(G,X)^{\rm
min}_{\bar{{\mathbb{Q}}}_{p}},\mathbb{Q}_{\ell})_{v}$ the eigenspace under the
volume character of the action of $\mathcal{H}_{K}$. Through the end of this
section, we set $K_{0}=K$ for the neat subgroup $K\subset
G({{\mathbf{A}}_{f}})$ and use the notation $H^{i}(\Lambda(j))$ for étale
cohomology with coefficients $\Lambda$, which are either $\mathbb{Z}_{\ell}$
or $\mathbb{Q}_{\ell}$.

###### Theorem 4.13

Let $(G,X)$ be a Shimura datum of Hodge type. For any prime $\ell\neq p$, any
${\mathcal{E}}\in{\rm Vect}^{\rm ss}_{G}(\hat{X}_{E})$, we define $\ell$-adic
Chern classes

$\displaystyle c_{n,\ell}([{\mathcal{E}}]_{p}){}_{K_{r}}\in
H^{2n}\left({}_{K_{r}}S(G,X)^{\rm
min}_{\bar{{\mathbb{Q}}}_{p}},\Lambda(n)\right)_{v}^{G_{E_{w}}},\ \ n\ \in\
\mathbb{N}$

where $\Lambda=\mathbb{Q}_{\ell}$ for $r=0$ and $\mathbb{Z}_{\ell}$ for $r\geq
1$, with the following properties:

  * (1)

$j_{K_{r}}^{*}(c_{n,\ell}([{\mathcal{E}}]_{p}){}_{K_{r}})=c_{n,\ell}([{\mathcal{E}}]_{K_{r}})\in
H^{2n}(_{K_{r}}S(G,X)_{\bar{{\mathbb{Q}}}},\Lambda(n)),$ where the right hand
side is the $\ell$-adic Chern class of $[{\mathcal{E}}]$ on
${}_{K_{r}}S(G,X)$.

  * (2)

If $K^{\prime}\subset K$ then $\pi^{\rm
min*}_{K_{r},K^{\prime}_{r}}(c_{n,\ell}([{\mathcal{E}}]_{p}){}_{K_{r}})=c_{n,\ell}([{\mathcal{E}}]_{p}){}_{K_{r}^{\prime}}$.

  * (3)

$r_{h}^{\rm
min*}(c_{n,\ell}([{\mathcal{E}}]_{p}){}_{K_{r}})=c_{n,\ell}([{\mathcal{E}}]_{p})_{hK_{r}h^{-1}}$
for $h\in G({{\mathbf{A}}_{f}})$ with trivial component at $p$ and at places
ramified for $K$.

  * (4)

Whitney product formula: If
$0\to\mathcal{E}_{1}\to\mathcal{E}\to\mathcal{E}_{2}\to 0$ is an exact
sequence in ${\rm Vect}^{\rm ss}_{G}(\hat{X}_{E})$, then

$c_{n,\ell}([{\mathcal{E}}]_{p}){}_{K_{r}})=\oplus_{a+b=n}c_{a,\ell}([{\mathcal{E}}_{1}]_{p}){}_{K_{r}})\cdot
c_{b,\ell}([{\mathcal{E}}_{2}]_{p}){}_{K_{r}}).$

  * (5)

If $(G^{\prime},X^{\prime})\to(G,X)$ is a morphism of Shimura data of Hodge
type, $K^{\prime}\to K$ is a compatible level, with $K_{r}$ and
$K_{r}^{\prime}$ compatibly chosen, and

$f:{}_{K_{r}^{\prime}}S(G^{\prime},X^{\prime})^{\rm min}{\leavevmode\nobreak\
\rightarrow\leavevmode\nobreak\ }{}_{K_{r}}S(G,X)^{\rm min}$

the corresponding morphism of Shimura varieties, then

$f^{*}(c_{n,\ell}([{\mathcal{E}}]_{p}){}_{K_{r}})=(c_{n,\ell}([f^{*}{\mathcal{E}}]_{p}){}_{K_{r}^{\prime}})\in
H^{2n}({}_{K_{r}^{\prime}}S(G^{\prime},X^{\prime})^{\rm
min}_{\bar{{\mathbb{Q}}}_{p}},\Lambda(n))_{v}^{G_{E_{w}}}.$

  * Proof.

We first construct $c_{n,\ell}([{\mathcal{E}}]_{p}){}_{K_{r}}$ without the $v$
invariance. The vector bundle $[{\mathcal{E}}]_{p}$ from (4.3) has Chern
classes

$\bar{c}_{n,\ell}([{\mathcal{E}}]_{p})\in\varprojlim_{m}H^{2n}(\overline{{}_{K^{p}}\mathcal{S}(G,X)}^{\rm
min},\mathbb{Z}/\ell^{m}(n)),\ n\in\mathbb{N}$

by Proposition A.1. On the other hand, $[{\mathcal{E}}]_{p}$ is defined as
$\pi_{HT}^{*}\mathcal{E}$, where $\mathcal{E}$ is viewed as a vector bundle on
the $C$-adic space $\hat{\mathcal{X}}$. But $\mathcal{E}$ is already defined
on $\hat{X}_{E}$. In particular, with $K_{p,r}$ defined as in Section 2.B, the
classes $\bar{c}_{n,\ell}([{\mathcal{E}}]_{p})$ are invariant under the action
of $K_{p,r}\subset G(\mathbb{Q}_{p})$ for all $r$ and by $G_{E_{w}}$. By
Proposition A.4 it follows that the classes
$\bar{c}_{n,\ell}([{\mathcal{E}}]_{p})$ uniquely define classes

$c_{n,\ell}([{\mathcal{E}}]_{p})\in H^{2n}(\overline{{}_{K_{r}}S(G,X)}^{\rm
min}_{\bar{\mathbb{Q}}_{p}},\mathbb{Z}_{\ell}(n))^{G_{E_{w}}}$

for $r\geq 1$. Since $K/K_{1}$ is finite, we can descend to
$\mathbb{Q}_{\ell}$-cohomology at level $K$. We thus have descended classes

$\bar{c}_{n,\ell}([{\mathcal{E}}]_{p}){}_{K_{r}}\in
H^{2n}(\overline{{}_{K}S(G,X)}^{\rm
min}_{\bar{\mathbb{Q}}_{p}},\mathbb{Q}_{\ell}(n))^{G_{E_{w}}}.$

Letting $u_{r}:{}_{K_{r}}S(G,X)^{\rm min}{\leavevmode\nobreak\
\rightarrow\leavevmode\nobreak\ }\overline{{}_{K_{r}}S(G,X)}^{\rm min}$ denote
the natural map, as in (3.10), the classes

$c_{n,\ell}([{\mathcal{E}}]_{p}){}_{K_{r}}:=u_{r}^{*}(\bar{c}_{n,\ell}([{\mathcal{E}}]_{p}){}_{K_{r}})$

are the desired ones. Property (1) is then a direct consequence of Proposition
2.37. Property (2) follows directly from the construction using descent via
Proposition A.4. As for Property (3), using descent again, it is enough to see
that $r_{h}^{\rm min}$ extends to an isomorphism

$r^{\rm perf,min}_{h}:{}_{K_{r}}\mathcal{S}(G,X)^{\rm
min}\to{}_{hK_{r}h^{-1}}\mathcal{S}(G,X)^{\rm min}$

of perfectoid spaces, which respects $[{\mathcal{E}}]_{p}$. (See [Sch15,
Theorem 4.1.1 (iv)] for the Siegel modular case; the general case is
identical.) Our notation $[{\mathcal{E}}]_{p}$ does not refer to $K_{p}$. One
should replace the notation $[{\mathcal{E}}]_{p}$ with
$[{\mathcal{E}}]_{p}^{K_{p}}$. Then the compatibility means $r_{h}^{\rm
perf,min*}[{\mathcal{E}}]_{p}^{hK_{p}h^{-1}}=[{\mathcal{E}}]^{K_{p}}_{p}$,
which follows from Theorem 2.17 together with the addendum Theorem 2.29.
Property (4) follows directly from the Whitney product formula for the Chern
classes of $[\mathcal{E}]_{p}$ and Property (5) of the functoriality of the
construction of ${}_{K^{p}}\mathcal{S}(G,X)^{\rm min}$ ([Sch15], loc.cit.,
comes from the construction, even if this is not explicitly mentioned). It
remains to see that $c_{n,\ell}([{\mathcal{E}}]_{p}){}_{K_{r}}\in
H^{2n}({}_{K_{r}}\mathcal{S}(G,X)^{\rm
min}_{\bar{\mathbb{Q}}_{p}},\Lambda(n))_{v}$. By construction, this follows
from

$\bar{c}_{n,\ell}([{\mathcal{E}}]_{p})\in\varprojlim_{m}H^{2n}\left(\overline{{}_{K^{p}}\mathcal{S}(G,X})^{\rm
min},\mathbb{Z}/\ell^{m}(n)\right)_{v},\ n\in\mathbb{N},$

which again follows by the projection formula from the trivial relation

$\displaystyle
T(g)[\mathcal{E}]_{p}=[\mathcal{E}]_{p}\otimes_{\mathcal{O}_{{}_{K^{p}}{\mathcal{S}}(G,X)^{\rm
min}}}\mathcal{O}_{{}_{K_{h}^{p}}{\mathcal{S}}(G,X)^{\rm min}.}$

$\Box$

###### Notation 4.14

The functor

$\displaystyle\left\\{\begin{array}[]{rcl}{\rm
Rep}_{\bar{\mathbb{Q}}}(K_{h})&\to&\\{{\rm Vector\ Bundles\ on}\
{}_{K^{p}}\mathcal{S}(G,X)^{\rm min}\\}\\\
\mathcal{E}&\mapsto&[\mathcal{E}]_{p}\end{array}\right.,$ (4.17)

is a tensor functor, which sends ${\rm Rep}_{\bar{\mathbb{Q}}}(G)$ to trivial
bundles. It induces the Chern character

$K_{0}({\rm Rep}_{\bar{\mathbb{Q}}}(K_{h})\otimes_{{\rm
Rep}_{\bar{\mathbb{Q}}}(G)}\mathbb{Q})\xrightarrow{ch_{\ell}}H^{2*}\left({}_{K}S(G,X)^{\rm
min}_{\bar{\mathbb{Q}}_{p}},\mathbb{Q}_{\ell}(*)\right)^{G(E_{w})}_{v}\subset\left(\varprojlim_{m}H^{2*}\left({}_{K^{p}}\mathcal{S}(G,X)^{\rm
min},\mathbb{Z}/\ell^{m}(*)\right)\right)\otimes{\mathbb{Q}}.$ (4.18)

###  4.B. The image of $H^{*}({}_{K}S(G,X)^{\rm min})$ in
$H^{*}({}_{K}S(G,X)_{\Sigma})$

We now return to the topological setting. In this section, ${}_{K}S(G,X)^{\rm
min}$ and ${}_{K}S(G,X)_{\Sigma}$ are identified with the analytic spaces
underlying their ${\mathbb{C}}$-valued points. We fix $\Sigma$, use the
notation (2.5) and let $\varphi:=\varphi_{\Sigma}$ denote the
desingularization map
$\varphi_{\Sigma}:{}_{K}S(G,X)_{\Sigma}{\leavevmode\nobreak\
\rightarrow\leavevmode\nobreak\ }{}_{K}S(G,X)^{\rm min}$. It is a projective
morphism. Recall from the general theory [dCM05, Theorem 2.8.1] that the
choice of a polarization $\mathcal{L}$ for $\varphi$ induces a factorization

$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces
H^{i}\left({}_{K}S(G,X)^{\rm
min},{\mathbb{Q}}\right)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi^{*}}$$\scriptstyle{\rm
natural\ map}$$\textstyle{IH^{i}\left({}_{K}S(G,X)^{\rm
min},{\mathbb{Q}}\right)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi^{*}_{\mathcal{L}}}$$\textstyle{H^{i}\left({}_{K}S(G,X)_{\Sigma},{\mathbb{Q}}\right)}$
(4.23)

in which all morphisms are compatible with the polarized Hodge structure, and
$\varphi_{\mathcal{L}}^{*}$ is injective (see e.g. [dCM05, Corollary 2.8.2]).
For a prime $\ell\neq p$ and any $s\geq 0$, the diagram (4.23) has an
$\ell$-adic version

$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces
H^{i}\left({}_{K_{s}}S(G,X)^{\rm
min},\mathbb{Q}_{\ell}(j)\right)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi^{*}}$$\scriptstyle{\rm
natural\ map}$$\textstyle{IH^{i}\left({}_{K_{s}}S(G,X)^{\rm
min},\mathbb{Q}_{\ell}(j)\right)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi^{*}_{\mathcal{L}}}$$\textstyle{H^{i}\left({}_{K_{s}}S(G,X)_{\Sigma},\mathbb{Q}_{\ell}(j)\right)}$
(4.28)

Thus by the comparison isomorphism [BBD82, Section 6], we conclude that
$\varphi_{\mathcal{L}}^{*}$ in the diagram (4.28) is injective as well. The
image of $\varphi_{\mathcal{L}}^{*}$ might depend on the choice of
$\mathcal{L}$, see [dCM05, Example 2.9]. Furthermore,
$IH^{i}({}_{K}S(G,X)^{\rm min},{\mathbb{Q}}),$ thus

$\varphi_{\mathcal{L}}^{*}\left(IH^{i}({}_{K}S(G,X)^{\rm
min},{\mathbb{Q}})\right)\subset H^{i}({}_{K}S(G,X)_{\Sigma},{\mathbb{Q}}),$

is pure of weight $i$, and $\varphi^{*}(H^{i}({}_{K}S(G,X)^{\rm
min},{\mathbb{Q}}))\subset H^{i}({}_{K}S(G,X)_{\Sigma},{\mathbb{Q}})$ is the
maximal pure weight $i$ quotient of $H^{i}({}_{K}S(G,X)^{\rm
min},{\mathbb{Q}})$ by [Del74, Proposition 8.2.5].

###### Theorem 4.29

For any ${\mathcal{E}}\in{\rm Vect}^{\rm ss}_{G}(\hat{X})$, any neat level
subgroup $K\subset G({{\mathbf{A}}_{f}})$ and all $s\geq 0$, we have

$\displaystyle\varphi^{*}((c_{n,\ell}([{\mathcal{E}}]_{p}){}_{K_{s}})=c_{n,\ell}(([{\mathcal{E}}]^{\rm
can}){}_{K_{s}})\in
H^{2n}\left({}_{K_{s}}S(G,X)_{\Sigma,\bar{\mathbb{Q}}},\Lambda(n)\right)^{G_{E}}.$

Here $\Lambda=\mathbb{Q}_{\ell}$ for $s=0$ and $\mathbb{Z}_{\ell}$ for $s\geq
1$, as in Theorem 4.13.

  * Proof.

We place ourselves in the situation of Theorem 3.22 which we use; in
particular we fix the natural number $s$. The proof assuming Hypothesis 2.30
is the same without fixing $s$. It suffices to check the claims with
$c_{n,\ell}([{\mathcal{E}}]_{p}){}_{K_{s}}$ replaced by
$\bar{c}_{n,\ell}([{\mathcal{E}}]_{p}){}_{K_{s}}$. Notation is as in Section
3. For $s\geq 0$ we have the diagram with commutative squares

$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces
Z_{s}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{Q}$$\scriptstyle{\phi}$$\scriptstyle{\psi}$$\textstyle{\overline{\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}_{K^{p}}{\mathcal{S}}(G,X)_{\Sigma}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{K_{s},K^{p}}}$$\scriptstyle{\bar{q}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\overline{{}_{K^{p}}{\mathcal{S}}(G,X)}^{\rm
min}}$$\scriptstyle{\pi_{K_{s},K^{p}}^{\rm
min}}$$\textstyle{{}_{K_{s}}{\mathcal{S}}(G,X)_{\Sigma}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\gamma}}$$\textstyle{\overline{{}_{K_{s}}{\mathcal{S}}(G,X)_{\Sigma}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta}$$\textstyle{\overline{{}_{K_{s}}{\mathcal{S}}(G,X)}^{\rm
min}}$ (4.34)

Here $\delta\circ\gamma=\varphi$ and $\delta,\bar{q}$ come from the
definitions (3.1) and (3.2). By Theorem 3.22 one has

$\displaystyle(\pi_{K_{s},K^{p}}^{\rm min}\circ
Q)^{*}\left(\bar{c}_{n,\ell}([\mathcal{E}]_{p}))_{K_{s}}\right)$
$\displaystyle=\phi^{*}(\delta\circ\gamma)^{*}(\bar{c}_{n,\ell}([\mathcal{E}]_{p}))_{K_{s}})$
$\displaystyle=Q^{*}(\bar{c}_{n,\ell}([\mathcal{E}]_{p}))=\phi^{*}(\bar{c}_{n,\ell}([{\mathcal{E}}]^{\rm
can}))$ $\displaystyle\in Q^{*}H^{2n}\left({}_{K^{p}}{\mathcal{S}}(G,X)^{\rm
min},\Lambda(n)\right)\subset H^{2n}(Z_{s},\Lambda(n)).$

Since (4.34) commutes, and since $\phi^{*}$ is injective by Proposition 3.11,
we thus have

$c_{n,\ell}([{\mathcal{E}}]^{\rm
can})=(\delta\circ\gamma)^{*}(\bar{c}_{n,\ell}([\mathcal{E}]_{p}))_{K_{s}})=\varphi^{*}((\bar{c}_{n,\ell}([{\mathcal{E}}]_{p}){}_{K_{s}}),$
(4.35)

as classes in
$H^{2n}({}_{K_{s}}{\mathcal{S}}(G,X)_{\Sigma,\bar{\mathbb{Q}}},\Lambda(n)).$
On the other hand,

$H^{2n}\left({}_{K_{s}}{\mathcal{S}}(G,X)_{\Sigma,\bar{\mathbb{Q}}},\Lambda(n)\right)=H^{2n}\left({}_{K_{s}}S(G,X)_{\Sigma,\bar{\mathbb{Q}}},\Lambda(n)\right)$

by the comparison isomomorphism [Hub98, Theorem 4.2]. In addition, the
automorphic bundles $[{\mathcal{E}}]^{\rm can}$ are defined over $E$. Thus the
classes in (4.35) lie in
$H^{2n}({}_{K_{s}}S(G,X)_{\Sigma,\bar{\mathbb{Q}}},\Lambda(n))^{G_{E}}$. This
finishes the proof. $\Box$

Let

$\displaystyle H^{2*}_{\rm Chern}\left({}_{K}S(G,X)^{\rm
min}_{\bar{\mathbb{Q}}_{p}},\mathbb{Q}_{\ell}(*)\right)\subset
H^{2*}\left({}_{K}S(G,X)^{\rm
min}_{\bar{\mathbb{Q}}_{p}},\mathbb{Q}_{\ell}(*)\right)$

denote the $\mathbb{Q}_{\ell}$-subalgebra generated by the images of
$c_{*,\ell}([{\mathcal{E}}]_{p}){}_{K}$, and

$\displaystyle H^{2*}_{\rm
Chern}({}_{K}S(G,X)_{\Sigma,\bar{\mathbb{Q}}_{p}},\mathbb{Q}_{\ell}(*))\subset
H^{2*}(_{K}S(G,X)_{\Sigma,\bar{\mathbb{Q}}_{p}}\mathbb{Q}_{\ell}(*))$

denote the $\mathbb{Q}_{\ell}$-subalgebra generated by the images of
$c_{*,\ell}([{\mathcal{E}}]^{\rm can}_{K}){}_{K}$. Here, when $G$ satisfies
(i) of Definition 2.13, we take ${\mathcal{E}}$ to belong to ${\rm
Vect}_{G}(\hat{X})^{+}$.

###### Proposition 4.36

The commutative diagram (4.28) restricts to the commutative diagram of
$\mathbb{Q}_{\ell}$-algebras

$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\left[K_{0}({\rm
Rep}_{\overline{\mathbb{Q}}}(K_{h}))\otimes_{K_{0}({\rm
Rep}_{\overline{\mathbb{Q}}}(G))}{\mathbb{Q}}_{\ell}\right]^{+}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$
$\scriptstyle{{\rm ch}_{\ell}\ \sim}$ $\scriptstyle{\sim(\rm
ch\leavevmode\nobreak\
\circ\eqref{01})}$$\textstyle{H^{2*}\left(\hat{X}_{\bar{\mathbb{Q}}_{p}},\mathbb{Q}_{\ell}(*)\right)^{+}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim({\rm
Prop}.\leavevmode\nobreak\
\ref{isomQ})}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces
H^{2*}_{\rm Chern}\left({}_{K}S(G,X)^{\rm
min}_{\bar{\mathbb{Q}}_{p}},{\mathbb{Q}}_{\ell}(*)\right)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi^{*}\
\sim}$$\scriptstyle{\rm natural\ map\
\sim}$$\textstyle{IH^{2*}({}_{K}S(G,X)^{\rm
min}_{\bar{\mathbb{Q}}_{p}},{\mathbb{Q}}_{\ell}(*))_{v}^{+}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi^{*}_{\mathcal{L}}\
\sim}$$\textstyle{H^{2*}_{\rm
Chern}\left({}_{K}S(G,X)_{\Sigma,\bar{\mathbb{Q}}_{p}},{\mathbb{Q}}_{\ell}(*)\right)}$
(4.45)

in which $\varphi_{\mathcal{L}}^{*}$, ${\rm ch}_{\ell}$, the natural map and
$\varphi^{*}$ are isomorphisms of $\mathbb{Q}_{\ell}$-algebras. In particular

$\displaystyle\varphi_{\mathcal{L}}^{*}\left(IH^{2*}({}_{K}S(G,X)^{\rm
min}_{\bar{\mathbb{Q}}_{p}},{\mathbb{Q}}_{\ell}(*))^{+}_{v}\right)\subset
H^{2*}\left({}_{K}S(G,X)_{\Sigma,\bar{\mathbb{Q}}_{p}},{\mathbb{Q}}_{\ell}(*)\right)$

is a $\mathbb{Q}_{\ell}$-sub-vectorspace which does not depend on the choice
of the relative polarization $\mathcal{L}$.

  * Proof.

By Theorem 2.16, the image of the Goresky-Pardon Chern classes
$c_{n}([{\mathcal{E}}]_{K})^{GP}$ in the algebra $IH^{2*}({}_{K}S(G,X)^{\rm
min},$ $\mathbb{Q}(*))$ lies in $IH^{2*}({}_{K}S(G,X)^{\rm
min},\mathbb{Q}(*))^{+}_{v}$ and spans this subalgebra over $\mathbb{Q}$. In
particular, the image of

$H^{2*}\left({}_{K}S(G,X)^{\rm min},\mathbb{Q}(*)\right)_{v}\xrightarrow{\rm
natural\ map}IH^{2*}\left({}_{K}S(G,X)^{\rm min},\mathbb{Q}(*)\right)_{v}$

contains $IH^{2*}({}_{K}S(G,X),\mathbb{Q}(*))^{+}_{v}$, and the map

$\varphi_{\mathcal{L}}^{*}:IH^{2*}\left({}_{K}S(G,X),\mathbb{Q}(*)\right)^{+}_{v}\to
H^{2*}_{\rm Chern}\left({}_{K}S(G,X)_{\Sigma},{\mathbb{Q}}(*)\right)^{+}$

is surjective. By comparison with $\ell$-adic intersection cohomology, the
image of

$H^{2*}\left({}_{K}S(G,X)_{\bar{\mathbb{Q}}_{p}},\mathbb{Q}_{\ell}(*)\right)_{v}\xrightarrow{\rm
natural\
map}IH^{2*}\left({}_{K}S(G,X)_{\bar{\mathbb{Q}}_{p}},\mathbb{Q}_{\ell}(*)\right)_{v}$

contains
$IH^{2*}({}_{K}S(G,X)_{\bar{\mathbb{Q}}_{p}},\mathbb{Q}_{\ell}(*))_{v}^{+}$,
and

$\varphi_{\mathcal{L}}^{*}:IH^{2*}\left({}_{K}S(G,X)_{\bar{\mathbb{Q}}_{p}},\mathbb{Q}(*)\right)^{+}_{v}\to
H^{2*}_{\rm
Chern}\left({}_{K}S(G,X)_{\Sigma,{\bar{\mathbb{Q}}_{p}}},{\mathbb{Q}}(*)\right)$

is surjective as well. We have no direct way to compare the Goresky-Pardon
classes to our classes from Theorem 4.13 on
${}_{K}S(G,X)_{\bar{\mathbb{Q}}_{p}}$. However, applying Proposition 4.29 for
$s=0$, i.e. for $K_{0}=K$, one has

$\displaystyle\varphi_{\mathcal{L}}^{*}\left(H^{2*}_{\rm
Chern}\left({}_{K}S(G,X)^{\rm
min}_{\bar{\mathbb{Q}}_{p}},{\mathbb{Q}}_{\ell}(*)\right)\right)=H^{2*}_{\rm
Chern}\left({}_{K}S(G,X)_{\Sigma,\bar{\mathbb{Q}}_{p}},{\mathbb{Q}}_{\ell}(*)\right).$

As $\varphi_{\mathcal{L}}^{*}$ is injective, we deduce that the natural map
sends the $\mathbb{Q}_{\ell}$-subalgebra $H^{2*}_{\rm Chern}({}_{K}S(G,X)^{\rm
min}_{\bar{\mathbb{Q}}_{p}},{\mathbb{Q}}_{\ell}(*))$ onto
$IH^{2*}({}_{K}S(G,X)^{\rm
min}_{\bar{\mathbb{Q}}_{p}},{\mathbb{Q}}_{\ell}(*))^{+}_{v}$. This proves that
the natural map in the proposition and $\varphi^{*}$ are surjective. It
remains to prove that they are injective as well. The Chern character has by
definition values in $H^{2*}_{\rm Chern}({}_{K}S(G,X)^{\rm
min}_{\bar{\mathbb{Q}}_{p}},{\mathbb{Q}}_{\ell}(*))$ and the whole diagram
commutes. This proves that $\varphi^{*}$ is injective. It follows that the
homomorphisms ${\rm ch}_{\ell},\ \varphi^{*}$ and the natural map are
isomorphisms. Finally, the image of $\varphi^{*}$ being equal to the image of
$\varphi^{*}_{\mathcal{L}}$, the latter does not depend on $\mathcal{L}$. This
finishes the proof. $\Box$

###### Lemma 4.46

The classes

$\displaystyle c_{n,\ell}([{\mathcal{E}}]_{p}){}_{K_{r}}\in
H^{2n}\left({}_{K_{r}}S(G,X)^{\rm
min}_{\bar{{\mathbb{Q}}}_{p}},{\mathbb{Q}}_{\ell}(n)\right)_{v}^{G_{E_{w}}}$

constructed in Theorem 4.13 vanish for all $n>0$ when $\mathcal{E}$ is flat,
that is when $\mathcal{E}$ comes from a $G$-representation.

  * Proof.

Under the assumption of the lemma, $[\mathcal{E}]_{p}$ is trivial (see
Notation 4.14). The construction of Theorem 4.13 thus implies the lemma.
$\Box$

###### Remarks 4.47

(i) If we ignore the action of the Galois group, the Chern classes constructed
above depend only on the connected components of ${}_{K}S(G,X)^{\rm min}$.
Thus the results above extend to general Shimura varieties of abelian type. We
leave the precise formulation to the reader. (ii) We naturally expect that
Proposition 4.36 remains true without the superscripts +, which in any case
change nothing when $G$ satisfies condition (iv) of Theorem 2.11. In the
excluded case, it is easy to see that the classes denoted $e_{r}$ in §2.A.a
have non-trivial images in $IH^{n}({}_{K}S(G,X)^{\rm
min}_{\bar{\mathbb{Q}}_{p}},{\mathbb{Q}}_{\ell}(*))$; indeed, this follows
from the results of [GP02] and the equality $e_{r}^{2}=c_{1,r}^{n}$ for each
$r$. However, we also have

$(c_{1,r}^{k-1})^{2}=c_{1,r}^{n},$

so it is possible that $c_{1,r}$ and $\pm e_{r}$ have the same image in
$IH^{n}$. We have not attempted to prove that this is not the case.

##  5\. Comparison of torsors

Let $(G,X)=(\mathrm{GSp}(2g),X_{2g})$, write ${\hat{X}}_{2g}$ for the compact
dual of $X_{2g}$, fix a base point $h\in X_{2g}\subset{\hat{X}}_{2g}$, and let
$\Omega_{h}$ denote the fiber at $h$ of the cotangent bundle to
${\hat{X}}_{2g}$. Let $V_{g}$ denote the vector group $\mathbb{G}_{a}^{g}$.
Let $K_{h,g}\subset G$ denote the stabilizer of $h$. Then $K_{h,g}$ can be
identified with $\mathrm{GL}(g)\times\mathbb{G}_{m}={\rm
Aut}(V_{g})\times\mathbb{G}_{m}$ in such a way that the restriction to
$\mathrm{GL}(g)$ of the isotropy action of $K_{h,g}$ on $\Omega_{h}$ is
equivalent to ${\rm Sym}^{2}(V_{g})$. We take $\mathrm{St}$ to be the standard
representation $K_{h,g}\mapsto{\rm Aut}(V_{g})$, and let
${\mathcal{E}}_{\mathrm{St}}$ be the corresponding equivariant vector bundle
on ${\hat{X}}_{2g}$. Fix $K_{g}=K_{g,p}\cdot K^{p}_{g}$ a neat open compact
subgroup of $\mathrm{GSp}(2g)({{\mathbf{A}}_{f}})$, and fix a toroidal datum
$\Sigma$ for $K$. As in §2, we write

$\mathfrak{q}_{g}:_{K^{p}_{g}}\mathcal{A}_{g}^{\rm tor}{\leavevmode\nobreak\
\rightarrow\leavevmode\nobreak\ }_{K^{p}_{g}}\mathcal{A}_{g}^{\rm min}$

for the morphism denoted $q$ above.

######  Proposition 5.1 ([PS16])

There is a canonical isomorphism

$\theta_{g}:\mathfrak{q}_{g}^{*}[{\mathcal{E}}_{\mathrm{St}}]_{p}{\leavevmode\nobreak\
\overset{\sim}{\longrightarrow}\leavevmode\nobreak\
}[{\mathcal{E}}_{\mathrm{St}}]_{dR,\Sigma}$

of vector bundles over ${}_{K^{p}_{g}}\mathcal{A}_{g}^{\rm tor}$.

  * Proof.

This is essentially equivalent to Corollaire 1.16 of [PS16]. More precisely,
the generic fiber of the bundle denoted $\omega^{\rm mod}_{A}$ in [PS16] is
exactly $[{\mathcal{E}}_{\mathrm{St}}]_{dR,\Sigma}$. Indeed, with our
notation, the restriction of $[{\mathcal{E}}_{\mathrm{St}}]^{\rm can}$ to the
open Siegel modular variety in (neat) finite level $K$ is the sheaf of
relative sections of the relative $1$-forms of the universal polarized abelian
scheme, and its canonical extension to the toroidal compactification is
isomorphic to the sheaf of relative sections of the relative invariant
$1$-forms on the corresponding semi-abelian scheme; see [Lan12, Proposition
6.9] or [Lan17, (1.3.3.15)]; the comparison between the algebraic and analytic
constructions is Theorem 5.2.12 of [Lan12a]. $\Box$

We now restrict the information to Shimura varieties of Hodge type. Let
$(G,X)$ be a Shimura datum of Hodge type, with compact dual ${\hat{X}}$. We
fix a symplectic embedding

$\displaystyle\iota:(G,X){\leavevmode\nobreak\ \rightarrow\leavevmode\nobreak\
}(\mathrm{GSp}(2g),X_{2g}).$

Let $h\in X$ be a base point, let $K_{h}\subset G$ be its stabilizer, let
$K_{h,g}\subset\mathrm{GSp}(2g)$ be the stabilizer of $\iota(h)$, and denote
by $\iota_{h}$ the inclusion of $K_{h}$ in $K_{h,g}$. Let
$\mathrm{St}_{h}=\mathrm{St}\circ\iota_{h}:K_{h}{\leavevmode\nobreak\
\rightarrow\leavevmode\nobreak\ }{\rm Aut}(V_{g})$, with $\mathrm{St}$ the
standard faithful representation of $K_{h,g}$. Finally, let
${\mathcal{E}}_{\mathrm{St}_{h}}$ be the equivariant vector bundle on
${\hat{X}}$ with isotropy representation $\mathrm{St}_{h}$. Let $K=K_{p}\cdot
K^{p}$ be a neat open compact subgroup of $G({{\mathbf{A}}_{f}})$.

###### Corollary 5.2

We fix a natural number $s$. The morphism $\theta_{g}$ induces by pullback for
any $s$ a canonical isomorphism

$\theta(s):Q^{*}[{\mathcal{E}}_{\mathrm{St}_{h}}]_{p}{\leavevmode\nobreak\
\overset{\sim}{\longrightarrow}\leavevmode\nobreak\
}[{\mathcal{E}}_{\mathrm{St}_{h}}]_{dR,\Sigma,K_{s}}$

of vector bundles over $Z_{s}$.

  * Proof of Theorem 3.22.

We fix $s$ as in Section 3. As $\mathrm{St}_{h}$ is faithful, any irreducible
representation $V$ of $K_{h}$ defined over $\bar{\mathbb{Q}}$ is a direct
factor of the representation $\mathrm{St}_{h}^{\otimes
m}\otimes\mathrm{St}_{h}^{\vee\otimes n}$ for some pair of natural numbers
$(m,n)$ [DMOS82, I, Proposition 3.1 (a), II, Proposition 2.23]. Write

$V\xrightarrow{\sigma}\mathrm{St}_{h}^{\otimes
m}\otimes\mathrm{St}_{h}^{\vee\otimes n}\xrightarrow{\tau}V$

for the splitting. The isomorphism of vector bundles $\theta(s)$ induces an
isomorphism of vector bundles

$\theta(s)^{\otimes m}\otimes\theta(s)^{\vee\otimes
n}:\mathfrak{q}_{g}^{*}\left([{\mathcal{E}}_{\mathrm{St}_{h}}]^{\otimes
m}_{p}\otimes[{\mathcal{E}}_{\mathrm{St}_{h}}]^{\vee\otimes
n}_{p}\right){\leavevmode\nobreak\
\overset{\sim}{\longrightarrow}\leavevmode\nobreak\
}[{\mathcal{E}}_{\mathrm{St}_{h}}]_{dR,\Sigma,K_{s}}^{\otimes
m}\otimes[{\mathcal{E}}_{\mathrm{St}_{h}}]_{dR,\Sigma,K_{s}}^{\vee\otimes n}$

while $\sigma$ and $\tau$ induce the splittings of vector bundles over
${}_{K^{p}}{\mathcal{S}}(G,X)^{\rm min}$ and over $Z_{s}$

$\displaystyle[\mathcal{V}]_{p}\xrightarrow{\sigma_{p}}[{\mathcal{E}}_{\mathrm{St}_{h}}]^{\otimes
m}_{p}\otimes[{\mathcal{E}}_{\mathrm{St}_{h}}]^{\vee\otimes
n}_{p}\xrightarrow{\tau_{p}}[\mathcal{V}]_{p}$
$\displaystyle[\mathcal{V}]_{dR,\Sigma,K_{s}}\xrightarrow{\sigma_{dR,\Sigma,r}}[{\mathcal{E}}_{\mathrm{St}_{h}}]^{\otimes
m}_{dR,\Sigma,K_{s}}\otimes[{\mathcal{E}}_{\mathrm{St}_{h}}]^{\vee\otimes
n}_{p}\xrightarrow{\tau_{dR,\Sigma,K_{s}}}[\mathcal{V}]_{dR,\Sigma,K_{s}}.$

This yields the following morphisms of coherent sheaves over $Z_{s}$

$\displaystyle\alpha:=\tau_{dR,\Sigma,K_{s}}\circ\theta(s)^{\otimes
m}\otimes\theta(s)^{\vee\otimes n}\circ
Q^{*}\sigma_{p}:Q^{*}[\mathcal{V}]_{p}\to[\mathcal{V}]_{dR,\Sigma,K_{s}}$
$\displaystyle\beta:=Q^{*}\tau_{p}\circ(\theta(s)^{\otimes
m}\otimes\theta(s)^{\vee\otimes
n})^{-1}\circ\sigma_{dR,\Sigma,K_{s}}:[\mathcal{V}]_{dR,\Sigma,s}\to
Q^{*}[\mathcal{V}]_{p}$

which pull back to inverse isomorphisms on $Z^{o}_{s}$ under
$\mathfrak{j}_{s}$ (see (4.8)). Thus

$\beta\circ\alpha-{\rm Id}_{Q^{*}[\mathcal{V}]_{p}}:Q^{*}[\mathcal{V}]_{p}\to
Q^{*}[\mathcal{V}]_{p},\quad\alpha\circ\beta-{\rm
Id}_{[\mathcal{V}]_{dR,\Sigma,K_{s}}}:[\mathcal{V}]_{dR,\Sigma,K_{s}}\to[\mathcal{V}]_{dR,\Sigma,K_{s}}$

are homomorphisms of coherent sheaves which pull back to zero on $Z^{o}_{s}$.
As $Q^{*}[\mathcal{V}]_{p}$ and $[\mathcal{V}]_{dR,\Sigma,K_{s}}$ are vector
bundles, thus are both torsion free, we conclude that both maps are $0$. This
finishes the proof of Theorem 3.22. As for Theorem 2.38, the proof is
identical, except that we do not have to fix $s$ in the beginning and we
replace $Q$ by $q$ everywhere. $\Box$

## Appendix A. Geometry

We collect here the statements of some standard results in the theory of
schemes, in versions adapted to perfectoid spaces. The proofs are due to Peter
Scholze.

######  Proposition A.1 (Scholze)

Let ${\mathcal{W}}$ be a vector bundle of rank $r$ over a perfectoid space
${\mathcal{X}}$over $C$. Let ${\mathbb{P}}({\mathcal{W}})$ denote the
corresponding projective bundle over ${\mathcal{X}}$, viewed as an adic space
over $C$ _(_ see Remark A.2 below _)_. Then
$\mathcal{O}_{{\mathbb{P}}({\mathcal{W}})}(1)$ has a first Chern class

$(z_{m})\in\varprojlim_{m}H^{2}\left({\mathbb{P}}({\mathcal{W}}),\mathbb{Z}/\ell^{m}(1)\right)$

such that the homomorphism

$\displaystyle\varprojlim_{m}H^{b}({\mathbb{P}}({\mathcal{W}}),\mathbb{Z}/\ell^{m}(a))\leftarrow\oplus_{i=0}^{r-1}\varprojlim_{m}H^{b-2i}({\mathcal{X}},\mathbb{Z}/\ell^{m}(a-i))\cdot(z_{n})^{i}$

is an isomorphism.

###### Remark A.2

One of the referees asked how ${\mathbb{P}}({\mathcal{W}})$ can be defined as
an adic space. We forwarded the question to Laurent Fargues and David Hansen,
who sent essentially identical replies. Here is an (almost literal) paraphrase
of Hansen’s argument. First, without loss of generality, we may localize on
${\mathcal{X}}$ and assume that ${\mathcal{X}}$ is affinoid perfectoid, say
$\mathrm{Spa}(R,R^{+})$, and that ${\mathcal{W}}$ is the trivial bundle. Thus
we need to define $\mathrm{Spa}(R,R^{+})\times{\mathbb{P}}^{r-1}$ as an adic
space. Next, by the usual covering of ${\mathbb{P}}^{r-1}$ by $r$ affinoid
balls, we are reduced to showing that the ring $R\langle
T_{1},\dots,T_{r-1}\rangle$ (power series with coefficients in $R$ that tend
to $0$) is sheafy [SW17, Definition 3.1.11]. In fact, $R\langle
T_{1},\dots,T_{r-1}\rangle$ is sousperfectoid, in the terminology of Hansen
and Kedlaya, and in particular is sheafy; see §6.3 of [SW17] for the proofs.
Moreover, $R\langle T_{1},\dots,T_{r-1}\rangle$ is analytic [SW17, Definition
4.4.2]. We are warned, however, that there is as yet no reference for the
étale cohomology in the generality of sousperfectoid spaces. We have seen that
${\mathbb{P}}({\mathcal{W}})$ is an analytic adic space; thus, in order to
define the étale cohomology spaces used in Proposition A.1, one can pass to
the associate locally spatial diamond (see [Sch17, §15]), and apply the
cohomological formalism developed in [Sch17].

We define the Chern class $c_{i}({\mathcal{W}})$ by Grothendieck’s standard
equation

$\varprojlim_{m}H^{2r}({\mathbb{P}}({\mathcal{W}}),\mathbb{Z}/\ell^{m}(r))\ni\sum_{i=0}^{r}(-1)^{i}c_{i}({\mathcal{W}})\cdot(z_{m})^{r-i}=0,\quad(c_{i}({\mathcal{W}})_{m})\in\varprojlim_{m}H^{2i}({\mathbb{P}}({\mathcal{W}}),\mathbb{Z}/\ell^{m}(i)).$

  * Proof.

One defines $z$ as usual, as the projective limit over $m$ of the images
$z_{m}\in H^{2}({\mathbb{P}}({\mathcal{W}}),\mathbb{Z}/\ell^{m}(1))$ of the
class of $\mathcal{O}_{{\mathbb{P}}({\mathcal{W}})}(1)$ in
$H^{1}({\mathbb{P}}({\mathcal{W}}),\mathbb{G}_{m})$ via the connecting
homomorphism of the étale Kummer exact sequence
$1\to\mu_{\ell^{m}}\to\mathbb{G}_{m}\xrightarrow{\ell^{m}}\mathbb{G}_{m}\to
1$. To prove the statement, it is enough to prove that the map

$\displaystyle
H^{b}({\mathbb{P}}({\mathcal{W}}),\mathbb{Z}/\ell^{m}(a))\leftarrow\oplus_{i=0}^{n-1}H^{b-2i}({\mathcal{X}},\mathbb{Z}/\ell^{m}(a-i))\cdot
z_{m}^{i}$

is an isomorphism. This is a local property on ${\mathcal{X}}$, reduced by
[CS17, Lemma 4.4.1] to the computation of the étale cohomology of
${\mathbb{P}}(\mathcal{W)}$ over a geometric point
$\bar{x}={\rm\mathrm{Spa}}(C(\bar{x}),C(\bar{x})^{+})$, which then is the
standard computation. $\Box$

###### Remark A.3

We avoid here the delicate question whether the surjection

$H^{j}(\mathcal{X},\mathbb{Z}_{\ell}(i))\to\varprojlim_{m}H^{j}(\mathcal{X},\mathbb{Z}/\ell^{m}(i))$

is an isomorphism as it is irrelevant for our purpose.

Recall that we have level subgroups $K_{p,r}\subset G({{\mathbb{Q}}_{p}})$. We
denote by $\Sigma^{H}$ the invariants under a group $H$ acting on the set
$\Sigma$. We use this notation for cohomology invariants.

######  Proposition A.4 (Scholze)

The morphisms of ringed spaces

${}_{K^{p}}\overline{{\mathcal{S}}(G,X)}^{\rm min}\rightarrow_{K^{p}\cdot
K_{p,r}}\overline{\mathcal{S}(G,X)}^{\rm min}$

induce for all pairs of integers $(i,j)$ and all $r\geq 1$ homomorphisms

$\displaystyle H^{j}\left({}_{K^{p}}\overline{{\mathcal{S}}(G,X)}^{\rm
min},\mathbb{Z}/\ell^{m}(i)\right)^{K_{p,r}}\to H^{j}\left({}_{K^{p}\cdot
K_{p,r}}S(G,X)^{\rm
min}_{\overline{\mathbb{Q}}_{p}},\mathbb{Z}/\ell^{m}(i)\right)$

and

$\left[\varprojlim_{m}H^{j}\left({}_{K^{p}}\overline{{\mathcal{S}}(G,X)}^{\rm
min},\mathbb{Z}/\ell^{m}(i)\right)\right]^{K_{p,r}}=\varprojlim_{m}\left[H^{j}\left({}_{K^{p}}\overline{{\mathcal{S}}(G,X)}^{\rm
min},\mathbb{Z}/\ell^{m}(i)\right)\right]^{K_{p,r}}\\!\\!\\!\to
H^{j}\left({}_{K^{p}\cdot K_{p,r}}S(G,X)^{\rm
min}_{\overline{\mathbb{Q}}_{p}},\mathbb{Z}_{\ell}(i)\right).$

  * Proof.

As before, we write $K_{r}=K^{p}\cdot K_{p,r}$. By Theorem 2.17 and [Sch12,
Corollary 7.18], one has

$H^{j}\left({}_{K^{p}}\overline{{\mathcal{S}}(G,X)}^{\rm
min},\mathbb{Z}/\ell^{m}(i)\right)=\varinjlim_{r}H^{j}\left({}_{K_{r}}\overline{{\mathcal{S}}(G,X)}^{\rm
min},\mathbb{Z}/\ell^{m}(i)\right)=\varinjlim_{r}H^{j}\left({}_{K_{r}}\overline{S(G,X)}^{\rm
min}_{\bar{\mathbb{Q}}_{p}},\mathbb{Z}/\ell^{m}(i)\right).$

The second equality comes from the comparison isomorphism [Hub98, Theorem
4.2]. On the other hand, for $r^{\prime}>r\geq 1$ one has
$K_{r}/K_{r^{\prime}}=K_{p,r}/K_{p,r^{\prime}}$, which is a $p$-group. It
follows that the transition maps in the inductive system are injective as
${}_{K_{r+1}}\overline{{\mathcal{S}}(G,X)}^{\rm
min}\to{}_{K_{r}}\overline{{\mathcal{S}}(G,X)}^{\rm min}$ is finite surjective
of degree prime to $\ell$. This implies

$\displaystyle H^{j}\left({}_{K^{p}}\overline{{\mathcal{S}}(G,X)}^{\rm
min},\mathbb{Z}/\ell^{m}(i)\right)^{K_{p,r}}$
$\displaystyle=\varinjlim_{r^{\prime}}H^{j}\left({}_{K_{r^{\prime}}}\overline{{\mathcal{S}}(G,X)}^{\rm
min},\mathbb{Z}/\ell^{m}(i)\right)^{K_{p,r}/K_{p,r^{\prime}}}$
$\displaystyle=\varinjlim_{r^{\prime}}H^{j}\left([_{K_{r^{\prime}}}\overline{{\mathcal{S}}(G,X)}^{\rm
min}]/[K_{p,r}/K_{p,r^{\prime}}],\mathbb{Z}/\ell^{m}(i)\right)$ (A.5)

if $r\geq 1$, because $K_{p,r}/K_{p,r^{\prime}}$ is a $p$-group. Here
$[_{K_{r^{\prime}}}\overline{{\mathcal{S}}(G,X)}^{\rm
min}]/[K_{p,r}/K_{p,r^{\prime}}]$ denotes the quotient of the (projective)
scheme ${}_{K_{r^{\prime}}}\overline{{\mathcal{S}}(G,X)}^{\rm min}$ by the
finite $p$-group $K_{p,r}/K_{p,r^{\prime}}$. If $r=0$, the last two groups are
equal up to finite error that is bounded independently of $r^{\prime}$ and
$m$; we return to this case below. For any $r^{\prime}\geq r$, the quotient
${}_{K_{r^{\prime}}}\overline{{\mathcal{S}}(G,X)}^{\rm
min}/[K_{p,r}/K_{p,r^{\prime}}]$ lies in a sequence of finite morphisms

${}_{K_{r}}{\mathcal{S}}(G,X)^{\rm min}{\leavevmode\nobreak\
\rightarrow\leavevmode\nobreak\
}_{K_{r^{\prime}}}\overline{{\mathcal{S}}(G,X)}^{\rm
min}/[K_{p,r}/K_{p,r^{\prime}}]{\leavevmode\nobreak\
\rightarrow\leavevmode\nobreak\ }_{K_{r}}\overline{{\mathcal{S}}(G,X)}^{\rm
min}.$

Indeed, the minimal compactification is normal, which explains the left arrow,
and the right one is just the image of the middle term in the minimal
compactification of the Siegel space at level $K_{r}$. Define

${}_{K_{r}}\overline{{\mathcal{S}}(G,X)}^{\rm
min,\\#}=\varprojlim_{r^{\prime}}{}_{K_{r^{\prime}}}\overline{{\mathcal{S}}(G,X)}^{\rm
min}/[K_{p,r}/K_{p,r^{\prime}}].$

This is a scheme of finite type as it is dominated by
${}_{K_{r}}{\mathcal{S}}(G,X)^{\rm min}$. Again we have a sequence of finite
morphisms

${}_{K_{r}}{\mathcal{S}}(G,X)^{\rm min}{\leavevmode\nobreak\
\rightarrow\leavevmode\nobreak\ }_{K_{r}}\overline{{\mathcal{S}}(G,X)}^{\rm
min,\\#}{\leavevmode\nobreak\ \rightarrow\leavevmode\nobreak\
}_{K_{r}}\overline{{\mathcal{S}}(G,X)}^{\rm min}.$

It follows from the above considerations that, for any $m$, $j$, and $i$, we
have

$H^{j}\left({}_{K^{p}}\overline{{\mathcal{S}}(G,X)}^{\rm
min},\mathbb{Z}/\ell^{m}(i)\right)^{K_{p,r}}=\varinjlim_{r}H^{j}\left([_{K_{r}}\overline{{\mathcal{S}}(G,X)}^{\rm
min,\\#}],\mathbb{Z}/\ell^{m}(i)\right)$

if $r\geq 1$, and with finite error independent of $m$ if $r=0$. So composing
with the natural map

$H^{*}\left({}_{K_{r}}{\mathcal{S}}(G,X)^{\rm
min,\\#},\mathbb{Z}/\ell^{m}(i)\right){\leavevmode\nobreak\
\rightarrow\leavevmode\nobreak\ }H^{*}\left({}_{K_{r}}{\mathcal{S}}(G,X)^{\rm
min},\mathbb{Z}/\ell^{m}(i)\right)$

one obtains the homomorphism

$H^{j}\left({}_{K^{p}}\overline{{\mathcal{S}}(G,X)}^{\rm
min},\mathbb{Z}/\ell^{m}(i)\right)^{K_{p,r}}\to
H^{j}\left({}_{K_{r}}{\mathcal{S}}(G,X)^{\rm
min},\mathbb{Z}/\ell^{m}(i)\right)=H^{j}\left({}_{K_{r}}S(G,X)^{\rm
min}_{\overline{\mathbb{Q}}_{p}},\mathbb{Z}/\ell^{m}(i)\right)$ (A.6)

(see again [Hub98], loc.cit., for the comparison isomorphism), if $r\geq 1$.
Finally, by definition, one has

$\displaystyle\varprojlim_{m}H^{j}\left({}_{K^{p}}{\mathcal{S}}(G,X)^{\rm
min},\mathbb{Z}/\ell^{m}(i)\right)^{K_{p,r}}=\left[\varprojlim_{m}H^{j}\left({}_{K^{p}}{\mathcal{S}}(G,X)^{\rm
min},\mathbb{Z}/\ell^{m}(i)\right)\right]^{K_{p,r}},$

which thus maps to

$\displaystyle\varprojlim_{m}H^{j}\left({}_{K_{r}}S(G,X)^{\rm
min}_{\overline{\mathbb{Q}}_{p}},\mathbb{Z}/\ell^{m}(i)\right)=H^{j}\left({}_{K_{r}}S(G,X)^{\rm
min}_{\overline{\mathbb{Q}}_{p}},\mathbb{Z}_{\ell}(i)\right).$

$\Box$

If $r=0$ this remains true with $\mathbb{Z}/\ell^{m}(i)$ replaced by
$\mathbb{Q}_{\ell}(i)$: the maps in (A.5) are not necessarily isomorphisms but
the kernels and cokernels are of order bounded independently of $m$ and $i$.

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