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1. Hodge classes and the Jacquet–Langlands correspondence

   https://doi.org/10.1017/fmp.2023.20  

   Ichino, Atsushi; Prasanna, Kartik (September 2023, Forum of Mathematics, Pi)

   We prove that the Jacquet–Langlands correspondence for cohomological automorphic forms on quaternionic Shimura varieties is realized by a Hodge class. Conditional on Kottwitz’s conjecture for Shimura varieties attached to unitary similitude groups, we also show that the image of this Hodge class in ℓ-adic cohomology is Galois invariant for all ℓ. 

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2. Cohomology of cluster varieties II: Acyclic case

   https://doi.org/10.1112/jlms.12809  

   Lam, Thomas; Speyer, David E. (August 2023, Journal of the London Mathematical Society)

   Abstract In the previous work, we initiated the study of the cohomology of locally acyclic cluster varieties. In the present work, we show that the mixed Hodge structure and point counts of acyclic cluster varieties are essentially determined by the combinatorics of the independent sets of the quiver. We use this to show that the mixed Hodge numbers of acyclic cluster varieties of really full rank satisfy a strong vanishing condition. 

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3. An Exploration of Degeneracy in Abelian Varieties of Fermat Type

   https://doi.org/10.1080/10586458.2024.2362953  

   Goodson, Heidi (June 2024, Experimental Mathematics)

   The term degenerate is used to describe abelian varieties whose Hodge rings contain exceptional cycles -- Hodge cycles that are not generated by divisor classes. We can see the effect of the exceptional cycles on the structure of an abelian variety through its Mumford-Tate group, Hodge group, and Sato-Tate group. In this article we examine degeneracy through these different but related lenses. We specialize to a family of abelian varieties of Fermat type, namely Jacobians of hyperelliptic curves of the form $y^2=x^m-1$. We prove that the Jacobian of the curve is degenerate whenever $$m$$ is an odd, composite integer. We explore the various forms of degeneracy for several examples, each illustrating different phenomena that can occur. 

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4. The integral Hodge conjecture for two-dimensional Calabi–Yau categories

   https://doi.org/10.1112/S0010437X22007266  

   Perry, Alexander (February 2022, Compositio Mathematica)

   We formulate a version of the integral Hodge conjecture for categories, prove the conjecture for two-dimensional Calabi–Yau categories which are suitably deformation equivalent to the derived category of a K3 or abelian surface, and use this to deduce cases of the usual integral Hodge conjecture for varieties. Along the way, we prove a version of the variational integral Hodge conjecture for families of two-dimensional Calabi–Yau categories, as well as a general smoothness result for relative moduli spaces of objects in such families. Our machinery also has applications to the structure of intermediate Jacobians, such as a criterion in terms of derived categories for when they split as a sum of Jacobians of curves. 

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5. 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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