Chern classes of automorphic vector bundles

Abstract

We prove that the -adic Chern classes of canonical extensions of automorphic vector bundles, over toroidal compactifications of Shimura varieties of Hodge type overQ p , descend to classes in the -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. [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 surQ p , les classes de Chern -adiques des extensions canoniques des fibrés vectoriels automorphes descendent en des classes de cohomologie -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.

Chern classes of automorphic vector bundles, II

“…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 .…”

“…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 [E] K on K S(G, X) admits a canonical extension [E] can K to K S(G, X) Σ ; we write [E] Σ K 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 E ∈ Vect G (X).…”

“…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.…”

“…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].…”

“…, 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.…”