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   Pappas, Georgios; Zhou, Rong (March 2025, Mathematische Annalen)

   Abstract We give a simple and uniform proof of a conjecture of Haines–Richarz characterizing the smooth locus of Schubert varieties in twisted affine Grassmannians. Our method is elementary and avoids any representation theoretic techniques, instead relying on a combinatorial analysis of tangent spaces of Schubert varieties. 

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2. Singer conjecture for varieties with semismall Albanese map and residually finite fundamental group

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   Di Cerbo, Luca F.; Lombardi, Luigi (April 2024, Proceedings of the Royal Society of Edinburgh: Section A Mathematics)

   We prove the Singer conjecture for varieties with semismall Albanese map and residually finite fundamental group. 

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3. Arithmetic diagonal cycles on unitary Shimura varieties

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   Rapoport, M.; Smithling, B.; Zhang, W. (September 2020, Compositio Mathematica)

   null (Ed.)

   We define variants of PEL type of the Shimura varieties that appear in the context of the arithmetic Gan–Gross–Prasad (AGGP) conjecture. We formulate for them a version of the AGGP conjecture. We also construct (global and semi-global) integral models of these Shimura varieties and formulate for them conjectures on arithmetic intersection numbers. We prove some of these conjectures in low dimension. 

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4. Singular fibers of very general Lagrangian fibrations

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   Sawon, Justin (November 2022, Communications in Contemporary Mathematics)

   Let [Formula: see text] be a (holomorphic) Lagrangian fibration that is very general in the moduli space of Lagrangian fibrations. We conjecture that the singular fibers in codimension one must be semistable degenerations of abelian varieties. We prove a partial result towards this conjecture, and describe an example that provides further evidence. 

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5. Heights on stacks and a generalized Batyrev–Manin–Malle conjecture

   https://doi.org/10.1017/fms.2023.5  

   Ellenberg, Jordan S.; Satriano, Matthew; Zureick-Brown, David (January 2023, Forum of Mathematics, Sigma)

   Abstract We define a notion of height for rational points with respect to a vector bundle on a proper algebraic stack with finite diagonal over a global field, which generalizes the usual notion for rational points on projective varieties. We explain how to compute this height for various stacks of interest (for instance: classifying stacks of finite groups, symmetric products of varieties, moduli stacks of abelian varieties, weighted projective spaces). In many cases, our uniform definition reproduces ways already in use for measuring the complexity of rational points, while in others it is something new. Finally, we formulate a conjecture about the number of rational points of bounded height (in our sense) on a stack $$\mathcal {X}$$ , which specializes to the Batyrev–Manin conjecture when $$\mathcal {X}$$ is a scheme and to Malle’s conjecture when $$\mathcal {X}$$ is the classifying stack of a finite group. 

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