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1. Chern classes of automorphic vector bundles, II

   Esnault, H.; Harris, M. (October 2019, Épijournal de géométrie algébrique)

   We prove that the ℓ-adic Chern classes of canonical extensions of automorphic vector bundles, over toroidal compactifications of Shimura varieties of Hodge type over ℚ¯𝑝, descend to classes in the ℓ-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. 

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2. The Ax–Schanuel conjecture for variations of Hodge structures

   https://doi.org/10.1007/s00222-019-00863-8  

   Bakker, Benjamin and (January 2019, Inventiones mathematicae)

   We extend the Ax–Schanuel theorem recently proven for Shimura varieties by Mok–Pila–Tsimerman to all varieties supporting a pure polarizable integral variation of Hodge structures. In fact, Hodge theory provides a number of conceptual simplifications to the argument. The essential new ingredient is a volume bound for Griffiths transverse subvarieties of period domains. 

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3. Plectic Jacobians

   https://doi.org/10.1093/qmath/haae025  

   Fornea, Michele (June 2024, The Quarterly Journal of Mathematics)

   ABSTRACT Looking for a geometric framework to study plectic Heegner points, we define a collection of abelian varieties – called plectic Jacobians—using the middle-degree cohomology of quaternionic Shimura varieties (QSVs). The construction is inspired by the definition of Griffiths’ intermediate Jacobians and rests on Nekovář–Scholl’s notion of plectic Hodge structures. Moreover, we construct exotic Abel–Jacobi maps sending certain zero cycles on QSVs to plectic Jacobians. 

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4. Chern classes of automorphic vector bundles, II

   Esnault, Hélène; Harris, Michael (September 2019, Épijournal de géométrie algébrique)

   We prove that the l-adic Chern classes of canonical extensions of automorphic vector bundles, over toroidal compactifications of Shimura varieties of Hodge type over \bar{Q}_p, descend to classes in the l-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. 

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5. On integral models of Shimura varieties

   https://doi.org/10.1007/s00208-022-02387-8  

   Pappas, Georgios (April 2022, Mathematische Annalen)

   We show how to characterize integral models of Shimura varieties over places of the reflex field where the level subgroup is parahoric by formulating a definition of a ``canonical" integral model. We then prove that, in Hodge type cases and under a tameness hypothesis, the integral models constructed by the author and Kisin in previous work are canonical and, in particular, independent of choices. A main tool is a theory of displays with parahoric structure that we develop in this paper. 

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