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1. Quiver varieties and symmetric pairs

   https://doi.org/10.1090/ert/522  

   Li, Yiqiang (January 2019, Representation theory)

   We study fixed-point loci of Nakajima varieties under symplectomorphisms and their antisymplectic cousins, which are compositions of a diagram isomorphism, a reflection functor, and a transpose defined by certain bilinear forms. These subvarieties provide a natural home for geometric representation theory of symmetric pairs. In particular, the cohomology of a Steinberg-type variety of the symplectic fixed-point subvarieties is conjecturally related to the universal enveloping algebra of the subalgebra in a symmetric pair. The latter symplectic subvarieties are further used to geometrically construct an action of a twisted Yangian on a torus equivariant cohomology of Nakajima varieties. In the type A case, these subvarieties provide a quiver model for partial Springer resolutions of nilpotent Slodowy slices of classical groups and associated symmetric spaces, which leads to a rectangular symmetry and a refinement of Kraft-Procesi row/column removal reductions. 

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2. On the smooth locus of affine Schubert varieties

   https://doi.org/10.1007/s00208-025-03123-8  

   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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3. Cluster algebra structures on Poisson nilpotent algebras

   https://doi.org/10.1090/memo/1445  

   Goodearl, K; Yakimov, M (October 2023, Memoirs of the American Mathematical Society)

   Various coordinate rings of varieties appearing in the theory of Poisson Lie groups and Poisson homogeneous spaces belong to the large, axiomatically defined class of symmetric Poisson nilpotent algebras, e.g. coordinate rings of Schubert cells for symmetrizable Kac–Moody groups, affine charts of Bott-Samelson varieties, coordinate rings of double Bruhat cells (in the last case after a localization). We prove that every symmetric Poisson nilpotent algebra satisfying a mild condition on certain scalars is canonically isomorphic to a cluster algebra which coincides with the corresponding upper cluster algebra, without additional localizations by frozen variables. The constructed cluster structure is compatible with the Poisson structure in the sense of Gekhtman, Shapiro and Vainshtein. All Poisson nilpotent algebras are proved to be equivariant Poisson Unique Factorization Domains. Their seeds are constructed from sequences of Poisson-prime elements for chains of Poisson UFDs; mutation matrices are effectively determined from linear systems in terms of the underlying Poisson structure. Uniqueness, existence, mutation, and other properties are established for these sequences of Poisson-prime elements. 

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4. 𝔸-curves on log smooth varieties

   https://doi.org/10.1515/crelle-2017-0028  

   Chen, Qile; Zhu, Yi (November 2019, Journal für die reine und angewandte Mathematik (Crelles Journal))

   Abstract In this paper, we study {\mathbb{A}^{1}} -connected varieties from log geometry point of view, and prove a criterion for {\mathbb{A}^{1}} -connectedness. As applications, we provide many interesting examples of {\mathbb{A}^{1}} -connected varieties in the case of complements of ample divisors, and the case of homogeneous spaces. We also obtain a logarithmic version of Hartshorne conjecture characterizing projective spaces and affine spaces. 

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5. 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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