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.
Automorphic forms of $øverline\partial$-cohomology type as coherent cohomology classes
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Van Der Geer, Gerard; Looijenga, Eduard, 2021, Lifting Chern Classes By Means Of EkedahlâOortstrata, Tunisian Journal Of Mathematics, 3, 3, pp. 469-480, 10.2140/tunis.2021.3.469.