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1. Which Hessenberg varieties are GKM?

   https://doi.org/10.4310/PAMQ.2023.v19.n4.a8  

   Goldin, Rebecca; Tymoczko, Julianna (September 2023, Pure and Applied Mathematics Quarterly)

   Karshon, Yael; Melrose, Richard; Uhlmann, Gunther; Uribe, Alejandro (Ed.)

   Hessenberg varieties H(X,H) form a class of subvarieties of the flag variety G/B, parameterized by an operator X and certain subspaces H of the Lie algebra of G. We identify several families of Hessenberg varieties in type A_{n−1} that are T -stable subvarieties of G/B, as well as families that are invariant under a subtorus K of T. In particular, these varieties are candidates for the use of equivariant methods to study their geometry. Indeed, we are able to show that some of these varieties are unions of Schubert varieties, while others cannot be such unions. Among the T-stable Hessenberg varieties, we identify several that are GKM spaces, meaning T acts with isolated fixed points and a finite number of one-dimensional orbits, though we also show that not all Hessenberg varieties with torus actions and finitely many fixed points are GKM. We conclude with a series of open questions about Hessenberg varieties, both in type A_{n−1} and in general Lie type. 

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2. On positivity for the Peterson variety

   Goldin, Rebecca (June 2024, Contemporary Mathematics, American Mathematical Society)

   Tu, Loring W (Ed.)

   We aim in this manuscript to describe a specific notion of geomet- ric positivity that manifests in cohomology rings associated to the flag variety G/B and, in some cases, to subvarieties of G/B. We offer an exposition on the the well-known geometric basis of the homology of G/B provided by Schubert varieties, whose dual basis in cohomology has nonnegative structure constants. In recent work [R. Goldin, L. Mihalcea, and R. Singh, Positivity of Peterson Schubert Calculus, arXiv2106.10372] we showed that the equivariant cohomology of Peterson varieties satisfies a positivity phenomenon similar to that for Schubert calculus for G/B. Here we explain how this positivity extends to this particular nilpotent Hessenberg variety, and offer some open questions about the ingredients for extending positivity results to other Hessenberg varieties. 

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3. Intersection complexes and unramified 𝐿-factors

   https://doi.org/10.1090/jams/990  

   Sakellaridis, Yiannis; Wang, Jonathan (July 2022, Journal of the American Mathematical Society)

   Let X X be an affine spherical variety, possibly singular, and L + X \mathsf L^+X its arc space. The intersection complex of L + X \mathsf L^+X , or rather of its finite-dimensional formal models, is conjectured to be related to special values of local unramified L L -functions. Such relationships were previously established in Braverman–Finkelberg–Gaitsgory–Mirković for the affine closure of the quotient of a reductive group by the unipotent radical of a parabolic, and in Bouthier–Ngô–Sakellaridis for toric varieties and L L -monoids. In this paper, we compute this intersection complex for the large class of those spherical G G -varieties whose dual group is equal to G ˇ \check G , and the stalks of its nearby cycles on the horospherical degeneration of X X . We formulate the answer in terms of a Kashiwara crystal, which conjecturally corresponds to a finite-dimensional G ˇ \check G -representation determined by the set of B B -invariant valuations on X X . We prove the latter conjecture in many cases. Under the sheaf–function dictionary, our calculations give a formula for the Plancherel density of the IC function of L + X \mathsf L^+X as a ratio of local L L -values for a large class of spherical varieties. 

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4. The essential dimension of congruence covers

   https://doi.org/10.1112/S0010437X21007594  

   Farb, Benson; Kisin, Mark; Wolfson, Jesse (November 2021, Compositio Mathematica)

   Abstract Consider the algebraic function $$\Phi _{g,n}$$ that assigns to a general $$g$$ -dimensional abelian variety an $$n$$ -torsion point. A question first posed by Klein asks: What is the minimal $$d$$ such that, after a rational change of variables, the function $$\Phi _{g,n}$$ can be written as an algebraic function of $$d$$ variables? Using techniques from the deformation theory of $$p$$ -divisible groups and finite flat group schemes, we answer this question by computing the essential dimension and $$p$$ -dimension of congruence covers of the moduli space of principally polarized abelian varieties. We apply this result to compute the essential $$p$$ -dimension of congruence covers of the moduli space of genus $$g$$ curves, as well as its hyperelliptic locus, and of certain locally symmetric varieties. These results include cases where the locally symmetric variety $$M$$ is proper . As far as we know, these are the first examples of nontrivial lower bounds on the essential dimension of an unramified, nonabelian covering of a proper algebraic variety. 

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5. Finiteness of reductions of Hecke orbits

   https://doi.org/10.4171/JEMS/1395  

   Kisin, Mark; Lam, Yeuk_Hay Joshua; Shankar, Ananth N; Srinivasan, Padmavathi (December 2023, Journal of the European Mathematical Society)

   We prove two finiteness results for reductions of Hecke orbits of abelian varieties over local fields: one in the case of supersingular reduction and one in the case of reductive monodromy. As an application, we show that only finitely many abelian varieties on a fixed isogeny leaf admit CM lifts, which in particular implies that in each fixed dimensiongonly finitely many supersingular abelian varieties admit CM lifts. Combining this with the Kuga–Satake construction, we also show that only finitely many supersingular K3surfaces admit CM lifts. Our tools includep-adic Hodge theory and group-theoretic techniques. 

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