Chern classes of automorphic vector bundles, II

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.


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 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 → S min which in dimension ≥ 2 is singular. This is minimal in the sense that each of the toroidal compactifications S tor Σ , subject to the extra choice of a fan Σ, admits a birational proper morphism S tor Σ → S 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 min . The automorphic bundle E admits a canonical extension E can on S tor Σ constructed by Mumford [Mum77,Theorem 3.1] and more generally in [Har89]; E can is locally free and defined over a finite extension of the reflex field. On S min , on the other hand, there is generally no locally free extension of E, except in dimension 1 when S tor Σ → S min is an isomorphism. In [EH17] we studied the continuous -adic Chern classes of E in case S is proper. In particular, we proved that the higher Chern classes in continuous -adic cohomology die if E is flat [EH17, Theorem 0.2]. A key ingredient in the proof is the study of the action of Hecke algebra on continuous -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 tor Σ . Thus we cannot apply the methods of loc.cit. to prove the analogous theorem for the canonical extensions of flat automorphic bundles on S tor . 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 4.19) but we leave the precise statement to the reader.
As the classes of E can are even defined in continuous -adic cohomology of S tor Q , where E →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 min Q , Q (n)) G E via the isomorphism H 2n (S min Q p , Q (n)) = H 2n (S min Q , Q (n)). We now describe the method of proof. The main tool used is the existence of a perfectoid space P min above 1 S 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 π HT : P min →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 ofQ p . This allows us to define vector bundles π * HT (E) on P min where E are equivariant vector bundles onX defining the automorphic bundles. So while the bundles E do not extend to S min , they do on Scholze's limit space P 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 tor above S tor . On P tor one has the pullback of the bundles π * HT (E) and the pullback via the tower defining the perfectoid space of the canonical extensions E can . Theorem 2.22 asserts that the two pullbacks on P tor are the same. This argument generalizes to Shimura varieties satisfying a certain technical Hypothesis 2.18. We expect Hypothesis 2.18 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 3.12), 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 A.
The classes c n (E) ∈ H 2n (S min Q p , Q (n)) map in particular to well-defined classes in -adic intersection cohomology IH 2n (S min Q p , Q (n)). On intersection cohomology of S min overQ, we identify the eigenspace under the volume character with the cohomology of the compact dualX of S (Proposition 4.16). If the classes in IH 2n (S min Q p , Q (n)), identified with IH 2n (S min Q , Q (n)), descended to continuous -adic intersection cohomology, we could try to apply the method developed in [EH17] to show that the classes of E can in continuous -adic cohomology on S tor over E die when E if flat. This would be in accordance with [EV02, Theorem 1.1] where it is shown that the higher Chern classes of E can in the rational Chow groups on the Siegel modular variety vanish when E is flat. However, we are not yet able to work with continuous -adic intersection cohomology. We prove in Lemma 4.18 the vanishing c n (E) = 0 for all n > 0 when 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.
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).

Generalities on automorphic bundles
If G is a reductive algebraic group over Q, by an admissible irreducible representation of G(A) we mean an irreducible admissible (g, K) × G(A f )-module, where g is the complexified Lie algebra of G, K ⊂ G(R) is a connected subgroup generated by the center of G(R) and a maximal compact connected subgroup, G(A) (G(A f )) is the group of (finite) adèles of G. If π is such a representation then we write π π ∞ ⊗ π f where π ∞ is an irreducible admissible (g, K)-module and π f is an irreducible admissible representation of G(A f ).
Let (G, X) be a Shimura datum: G is a reductive algebraic group over Q, and X is a (finite union of) hermitian symmetric spaces, endowed with a transitive action of G(R) satisfying a familiar list of axioms. The compact dualX of X is a (projective) flag variety for G. Thus one can speak of the category Vect G (X) of G-equivariant vector bundles onX; the choice of a base point h ∈ X ⊂X determines an equivalence of categories Rep(P h ) Vect G (X), where P h ⊂ G is the stabilizer of h, a maximal parabolic subgroup of G, and for any algebraic group H, Rep(H) denotes the tensor category of its representations. Let K h be the Levi quotient of P h ; the group K h,∞ := K h (C) ∩ G(R) can be identified with a maximal Zariski connected subgroup of G(R) that is compact modulo the center of G. If K ⊂ G(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 E ∈ Vect(X) the notation for the vector bundle on the compact dual, [E] K ∈ Vect( K S(G, X)) for the automorphic vector bundle associated to the underlying representation of the compact (mod center) group K h,∞ ⊂ G(R), the stabilizer of a chosen point h ∈ 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 h ; in particular, P h is also defined over E 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 H K = {T (g), g ∈ G(A f )} is recalled in Section 4. The Chern character and analogously in -adic cohomology. On the other hand, the construction of automorphic vector bundles gives rise to a homomorphism where v indicates the eigenvectors for the volume character.
then [E] is endowed with a natural flat connection, so its higher Chern classes in ⊕ i H 2i ( K S(G, X), Q(i)) are equal to 0. We obtain a morphism analogous to (1.1), where the subscript v denotes a certain eigenspace for the action of the unramified Hecke algebra.

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 der of G is isotropic over 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 veigenspace most naturally appears in intersection cohomology of the minimal compactification. This has been worked out in detail in the C ∞ and the L 2 theory by Goresky and Pardon in [GP02]. In what follows, we let j K : K S(G, X) → K S(G, X) min denote the minimal (Baily-Borel-Satake) compactification. The minimal compactification is functorial, thus if K ⊂ K, there is a unique morphism K S(G, X) min → K S(G, X) min extending the natural map of open Shimura varieties, and if g ∈ G(A f ), then the Hecke correspondence T (g) extends canonically to a correspondence on K S(G, X) min × K S(G, X) min where the product is taken over the reflex field. In particular, for any cohomology theory H as above, we have Hecke operators The minimal compactification is always singular, except when G ad is a product of copies of PGL(2), and the automorphic vector bundles do not in general extend as bundles to K S(G, X) 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) min , * ). On the other hand, there is a large collection of toroidal compactifications K S(G, X) → K S(G, X) Σ indexed by combinatorial data Σ (see [AMRT75, III, §6, Main Theorem] for details; the adelic construction is in [Har89,Pin90]). The set of Σ is adapted to the level subgroup K. It is partially ordered by refinement: if Σ is a refinement of Σ, then there is a natural proper morphism extending the identity map on the open Shimura variety. Any two Σ and Σ can be simultaneously refined by some Σ . Further, the open embedding where ϕ Σ 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(A f ), in the sense of [Har89] or [Pin90,0.6]. In that case, we can choose Σ so that K S(G, X) Σ is smooth and projective, and the boundary divisor has normal crossings. We do so unless we specify otherwise. With these conventions, ϕ Σ is always a desingularization of the minimal compactification, which is constructed as in [AMRT75], loc.cit.; moreover, for any refinement Σ of Σ, K S(G, X) Σ is again smooth and projective.
Mumford proved in [Mum77, Theorem 3.1] that, if E ∈ Vect ss G (X) (recall from [EH17] that the upper index ss stands for semi-simple) the automorphic vector bundle The adelic construction is carried out in Section 4 of [Har89], where Mumford's result was generalized to arbitrary E ∈ Vect G (X). In particular it was shown there that if E = (G×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 withX, then [E] K is a vector bundle on K S(G, X) with a flat connection, and its canonical extension [E] can K on K S(G, X) Σ is exactly Deligne's canonical extension. In particular, the connection has logarithmic poles along Finally, for any fixed Σ, is a monoidal functor from Vect G (X) to the category of vector bundles on K S(G, X) Σ . Unfortunately, the g ∈ G(A f ) generally permute the set of Σ and thus the Hecke correspondences T (g) in general do not extend to correspondences on a given K S(G, X) Σ . 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 [E] can K in continuous -adic cohomology of K S(G, X) Σ over its reflex field, when [E] K is a flat automorphic vector bundle.
One can obtain information about the classes of [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, Q), resp. IH * (Z, Q) denotes Betti resp. intersection cohomology of a complex variety Z, while H * (Z, Q ), resp. IH * (Z, Q ) denote Q -étale cohomology.
Remark 2.2. The action of Hecke correspondences on Betti intersection cohomology IH( K S(G, X), 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), Q) and thus on IH( K S(G, X), Q ) 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 2.3. There is a canonical isomorphism of algebras
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 C. By Zucker's Conjecture [Loo88,SS90], IH * ( K S(G, X) min , C) is computed using square-integrable automorphic forms and Matsushima's formula [Bor83, 3.6, formula (1)]. Say the space A (2) (G) of squareintegrable automorphic forms on G(Q)\G(A) decomposes as the direct sum where π runs over irreducible admissible representations of G(A) and m(π) is a non-negative integer, which is positive for a countable set of π 2 . Then This implies where q runs over rational primes. Now if q is unramified for K then π K f = 0 unless π q is spherical. Now the trivial representation of G(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(Q q ) with this property. Thus (π K f ) v 0 implies that π q is the trivial representation for all q that are unramified for K. It then follows from weak approximation that π is in fact the trivial representation. Thus for all i, 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(Q)\G(A) that are invariant under the action of G(A). In other words, the only π that contributes is the space of constant functions on G(Q)\G(A), which are square integrable modulo the center of G(R). Thus IH 2 * ( K S(G, X) min , Q( * )) v can be viewed as L 2 -Chern classes of automorphic vector bundles. In addition IH 2 * +1 ( K S(G, X) min , Q( * )) v = 0. Although most automorphic vector bundles do not extend as bundles to the minimal compactification K S(G, X) min , it was proved by Goresky and Pardon in [GP02, Main Theorem] that under the natural homomorphism (2.6) the classes in IH * ( K S(G, X) min , C) v lift canonically, as differential forms, to ordinary singular cohomology We use the notation of the diagram (2.1). Let K be a neat open compact subgroup of G(A f ), as above. Let [E] K be the automorphic vector bundle on K S(G, X) attached to the homogeneous vector bundle be the Chern classes in Betti cohomology. The following theorem summarizes the main results of the article [GP02].
such that for any n the diagram

commutes. In other words, the isomorphism of Proposition 2.3 tensor C factors through
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 -adic cohomology of Shimura varieties of abelian type.

2.A.a. Even-dimensional quadrics
Theorem 2.7 excludes the case where G der (R) contains a factor isomorphic to SO(2k − 2, 2) with k > 2. Assuming G der is Q-simple, there is then a totally real field F, with [F : Q] = d, say, such thatX C is isomorphic to a product Q d n 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 = SO(n + 2)/SO(n) × SO(2) of dimension n, with C = C or Q . Then A contains a subalgebra A + isomorphic to C[c 1 ]/(c n+1 1 ), with c 1 ∈ H 2 (Q n , C) given by the Chern class of the line bundle corresponding to the standard representation of 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 with e the Euler class of the the vector bundle arising from the standard representation of SO(n). In [BH58], the class c 1 is denoted x 1 and the class e is denoted n i=2 x i ; the equation e 2 = c n 1 then follows immediately for formula (6) of [BH58].
More generally, ifX C is isomorphic to Q d n as above, we denote by c 1,r ∈ H 2 (X C , C(1)) the class of the line bundle defined above corresponding to the r-th factor of Q d n , r = 1, . . . , d, and let e r correspond to the Euler class in the r-th factor. The isomorphism of Proposition 2.3 is valid in all cases, and Goresky and Pardon showed that the image in IH 2j ( K S(G, X) min , Q(j)) v of the classes c j 1,r ∈ H 2j (X C , C(j)) lift canonically to the cohomology of the minimal compactification (the twist j here is unnecessary for C = C, we write it for the case C = Q ). However, when n = 2k − 2 > 2, they were unable to show that the e r lift. We can extend the statement of Theorem 2.7 with the following definition.
Definition 2.9. (i) Suppose G der is Q-simple and G der (R) SO(2k − 2, 2) d for some integer d, with k > 2.
to be the subalgebras generated respectively by the classes c 1,r , r = 1, . . . , d, and their images under the isomorphism of Proposition 2.3. We similarly define to be the inverse image of H * (X, C) + under the isomorphism (1.1).
We continue to use the notation E to denote (virtual) bundles in V ect G (X) + .
Theorem 2.12. Suppose G satisfies either (i) or (ii) of Definition 2.9. Then the conclusions of Theorem 2.7 hold with H * (X, C), Vect G (X), and IH * ( K S(G, X) min , Q(n)) v replaced by the versions with superscript + , and for E ∈ V ect G (X) + .
The theorem, and its application in Proposition 4.16 below, naturally extend to groups G such that G der is a product of groups of type (i) and (ii) in Definition 2.9. We omit the details.
, k → k p be the projection. Denote by K p its projection to G(Q p ). For r ≥ 0 we let K p,r ⊂ K p be a decreasing family of subgroups of finite index, with K p,r ⊃ K p,r+1 for all r, and such that r K p,r = {1}. Let K r = {k ∈ K | k p ∈ K p,r }, and let K p = ∩ r K r . We identify K p with its projection to the prime-to-p adèles G(A p f ); then K p is an open compact subgroup of G(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 (GSp(2g), X 2g ) for some g, where X 2g is the union of the Siegel upper and lower half-spaces. We letX 2g denote the compact dual flag variety of X 2g . Let C denote the completion of an algebraic closure Q p of Q p . Denote by K r S(G, X), resp. K r S(G, X) min the adic space over C attached to K r S(G, X), resp. K r S(G, X) min . We assume that K p is contained in the principal congruence subgroup of level N for some N ≥ 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 r for (GSp(2g), X 2g ) such that K r = K r ∩ G(A f ), and such that the scheme theoretic image of K r S(G, X) min in K r S(GSp(2g), X 2g ) min does not depend on the choice of K r . We denote it by K r S(G, X) min , and by K r S(G, X) min its associated C-adic space.

Here, on the left side, j is an open embedding of perfectoid Shimura varieties over C, on the right side, lim ← − −r j r denotes the formal inverse system of open embeddings of adic spaces over C, and the notation ∼ is defined precisely in [SW13, Definition 2.4.1]. Moreover, there is a G(Q p )-equivariant Hodge-Tate morphism
which is compatible with change of tame level subgroup K p . Discussion 2.15. Here we use the notationX 2g for the adic space over C attached to the flag variety X 2g . In loc. cit. the notation X * K p is used for the perfectoid Shimura variety K p S(G, X) min . The notation ∼ indicates that the right hand side of (2.14) 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 (2.16) 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]. LetX ⊂X 2g be the closed embedding of C-adic spaces corresponding to the closed embeddingX ⊂X 2g of the compact duals of X and X 2g . Theorem 2.1.3 in [CS17]). The Hodge-Tate morphism π HT in Theorem 2.13 factors through the inclusionX ⊂X 2g , yielding a G(Q p )-equivariant Hodge-Tate morphism

Theorem 2.17 (
Proof. The existence of π HT is stated for the open perfectoid Shimura variety K p S(G, X) with values inX . SinceX is closed inX 2g , the extension to the boundary then follows from Theorem 2.13 by continuity, as in [C+6, Theorem 3.3.4].
Assume for the moment that (G, X) = (GSp(2g), X 2g ). Fix K g ⊂ GSp(2g, A f ) a neat compact open subgroup, with K p = GSp(2g, Z p ) and write K g A g instead of K g S(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 ⊂ K g be the principal congruence subgroup of K g of level p r . Let K g A tor g = K g A g,Σ g be a smooth projective toroidal compactification of K g A g for some combinatorial datum Σ g as above. Following Pilloni-Stroh in [PS16], §1.3, but with a change of notation, we let K g,r A tor g denote the corresponding toroidal compactification of K g,r A g for each r, with the same Σ g . The K g,r A tor g form a projective system of adic spaces over C. The authors construct a projective system of normal models K g,r A tor g,O C over Spec(O C ) for each r, and define a perfectoid space K p g A tor g which is the generic fiber of the projective limit of the K g,r A 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 X (p ∞ ) tor−mod , is indeed perfectoid. There is a natural map Let (G, X) be any Shimura datum of Hodge type, with ι : (G, X) → (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 (GSp(2g), X 2g ): Hypothesis 2.18. The projective system of K r S(G, X) has a compatible projective system of toroidal compactifications K r S(G, X) Σ such that, if K r S(G, X) Σ is the associated adic space over C, then there is a perfectoid space It is likely the Hypothesis 2.18 is valid, but its proof is probably as elaborate as the construction of integral toroidal compactifications in [Lan13] 5 . 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 2.18 is considerably simpler, and it is valid, by [PS16], in the Siegel modular case.
Under Hypothesis 2.18, there is automatically a map q : K p S(G, X) Σ → K p S(G, X) min as well as the maps q r : K r S(G, X) Σ → K r S(G, X) min at finite level (the existence of which does not depend on the hypothesis). By [Har85, Section 3.4], there is a correspondence where b is a family of G-torsors, functorial with respect to inclusions K ⊂ K and translation by elements g ∈ G(A f ), and a is a G-equivariant  Choose a base point h ∈ X ⊂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 µ ). Any faithful representation α : K h → GL(W h ) defines by pullback to P h a G-equivariant vector bundle W h over X, and thus an automorphic vector bundle [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) → K S(G, X) with a K h -equivariant morphism a : K T h (G, X) →X as the moduli space of trivializations of [W h ] as above. More precisely, letting R u P h denote the unipotent radical of P h , the natural morphismX h := G/R u P h →X is canonically a G-equivariant K h -torsor, whose pullback a −1X 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 morphismX →X is denoted by T . The K h -torsor b has a Mumford extension T can → K S(G, X) Σ as a G-torsor. See [HZ94], specifically Lemma 4.4.2 and pages 320-321, where the Mumford extension of the G-torsor b in (2.19) is constructed; the K h -torsor is constructed in the same way. One denotes by M can = K M can the pullback of T can via the ringed space morphism We define the K h -torsor

Proposition 2.21 (Proposition 2.3.9 in [CS17]). For any neat level subgroup K, there is a canonical isomorphism
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 M p and M dR,Σ,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. 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 2.22 for general Shimura varieties of Hodge type, under Hypothesis 2.18. In the following section, we show how to dispense with Hypothesis 2.18 and prove Theorem 3.12 as an alternative to Theorem 2.22. In Section 4 we construct -adic Chern classes using either Theorem 2.22 (when it is available) or Theorem 3.12 for general Shimura varieties of Hodge type.

Construction of perfectoid covers
Let (G, X) be any Shimura datum of Hodge type, with ι : (G, X) → (GSp(2g), X 2g ) a fixed symplectic embedding. Choose a combinatorial datum Σ for K S(G, X) that is compatible with the fixed Σ g chosen above, so that there is a morphism (3.1) We denote the corresponding C-adic spaces by K S(G, X) Σ and K S(G, X) Σ . If we admitted Hypothesis 2.18 then we would be able to replace K S(G, X) Σ by K S(G, X) Σ in what follows; however, we will not assume Hypothesis 2.18 in the remainder of this section. By [Sch12,Prop. 6.18] the fibre product exists in the category of perfectoid spaces. We denote bȳ the projection on the second factor, and bȳ q r : K p S(G, X) Σ → K r S(G, X) min the composition with the canonical map of K p S(G, X) min to K r S(G, X) min . By definition, in the sense of [SW13,Definition 2.4.1], one has One deduces In particular, by [Sch12, Corollary 7.1.8], one has the relation On the other hand, Scholze's Hodge-Tate map yields the composite map which by definition is the same as the composite map This defines a map in the category of C-adic spaces. It is G(Q p )-equivariant.
In what follows the idea to use the space denoted P N ,perf to define a perfectoid space above K s S(G, X) Σ is due to Bhargav Bhatt. We fix a natural number s. Associated to it we have the normalization morphism ν s : K s S(G, X) Σ → K s S(G, X) Σ . It is a finite morphism, thus it factors as for some natural number N , where η s is a closed embedding. We set Y s = Image(η s ).
By definition η s induces an isomorphism between K s S(G, X) Σ and Y s . For any ρ ≥ 0 we also define We fix coordinates (x 0 : . . . : x N ) of P N and define for ρ ∈ N the finite flat morphisms φ ρ : Then the fiber product K p S(G, X) Σ,N := K p S(G, X) Σ × C P N ,perf exists as a perfectoid space, by [Sch12, Proposition 6.18], and we have where the map K s+ρ S(G, X) Σ × C P N → K s S(G, X) Σ × C P N is given by the natural projection in the first factor and φ ρ in the second factor. Proposition 3.9. There is a closed perfectoid subspace Proof. We define Z s ⊂ K p S(G, X) Σ,N by the pullback of the ideal of Y s in K s S(G, X) Σ × C P N via (3.8) It follows from [Bha17,Proposition 9.4.1] that Z s is perfectoid.
By construction, Z s maps to K p S(G, X) Σ which itself maps to K p S(G, X) min , and to K s S(G, X) Σ , but not necessarily to K r S(G, X) Σ for r > s. On the other hand, K s S(G, X) Σ naturally maps to K s S(G, X) min , and thus we have a composite diagram We let Q denote the composite of the three maps, and let q s = u s • 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 The homomorphism on cohomology induced from (i), is injective, where is a prime different from p. The injectivity holds also true with Q -coefficients when s = 0.
Proof. In what follows we assume G ad contains no Q-simple factor isomorphic to 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 On the other hand, the proper maps Y s+ρ → Y s+ρ−1 for ρ ≥ 1 are finite, of degree a p-power for s ≥ 1. Indeed, the finiteness of this map is reduced by Definition (3.7) to the corresponding assertion for the maps 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 A g,Σ g and the compactifications K S(G, X) min are finite. Indeed, let ω g denote the determinant of the relative cotangent bundle of the universal abelian scheme over K g A g ; in other words, ω 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 ω g extends to an ample line bundle over K g A min g , and that the pullback of this extension to K S(G, X) 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 S(G, X) min to its image in K g A min g is finite, and the morphisms from K S(G, X) min to K S(G, X) min are finite, and from the commutative square

Construction of -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 3.12or on Theorem 2.22, if we assume Hypothesis 2.18. Proofs are given without assuming Hypothesis 2.18; to pass from the situation with Hypothesis 2.18, we just formally set Q = q.

4.A. Construction using π HT
As in [CS17, Theorem 2.1.3 (2)], the existence of the K h -torsor M p over K p S(G, X) min implies that there is a functor (4.1) Similarly, the existence of the K h torsor M dR,Σ,K over K S(G, X) Σ gives rise to a functor (4.2) With the notation of Theorem 3.12, we define 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 ⊂ G(A f ). Let h ∈ G(A f ) and consider K h = K ∩ hKh −1 ⊂ K. Right multiplication by h defines an isomorphism r h : hKh −1 S(G, X) ∼ −→ K S(G, X). One defines (4.6) The projection formula implies As one has pullbacks and push-downs for K 0,Q and for -adic cohomology, one has on them an action of the Hecke algebra H K , spanned by the T (h) for h ∈ G(A f ), with h trivial at p and at ramified places for K. The finite morphism π K,K h : K h S(G, X) → K S(G, X) extends to a finite morphism π min K,K h : K h S(G, X) min → K S(G, X) min and the action of H K on H 2n ( K S(G, X), Z (n)) extends to an action on H 2n ( K S(G, X) min , Z (n)).
Proof. When G = GL(2) Q the Shimura varieties are finite unions of modular curves, and the result in that case is standard. We may thus assume G ad contains no factor isomorphic to PGL(2) Q . Then we know by Theorem 10.14 of [BB66] that H 0 ( K S(G, X), ω n ) is finite dimensional for any n ≥ 0, where ω is the dualizing sheaf. The isomorphism r h induces an algebra isomorphism Similarly the finite cover π K,K h induces an injective algebra morphism ⊕ n∈N H 0 ( K S(G, X), ω n ) → ⊕ n∈N H 0 ( K h S(G, X), ω n ).
Since K S(G, X) min (resp. K h S(G, X) min ) is Proj of the left-hand side (resp. right-hand side) of the last diagram [BB66, Theorem 10.11], the induced map on the Proj defines the extensions r min h and π min K,K h . On the other hand, pullback on cohomology is defined, while the trace map Tr : π K,K h * Z → Z extends to j K * π K,K h * Z = π min K,K h * j K h * Z = π min K,K h * Z → j K * Z = Z . This proves the second part.
Let E(G, X) =: E be the reflex field. Fix an embedding E →Q. We choose a place w of E dividing the in which all morphisms are compatible with the polarized Hodge structure, and ϕ * L is injective (see e.g. [dCM05, Corollary 2.8.2]). For a prime p and any s ≥ 0, the diagram (4.11) has an -adic version This is a scheme of finite type as it is dominated by K r S(G, X) min . Again we have a sequence of finite morphisms K r S(G, X) min → K r S(G, X) min,# → K r S(G, X) min .