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1. Feigin–Odesskii brackets associated with Kodaira cycles and positroid varieties

   https://doi.org/10.1112/plms.70054  

   Hua, Zheng; Polishchuk, Alexander (May 2025, Proceedings of the London Mathematical Society)

   Abstract We establish a link between open positroid varieties in the Grassmannians and certain moduli spaces of complexes of vector bundles over Kodaira cycle , using the shifted Poisson structure on the latter moduli spaces and relating them to the standard Poisson structure on . This link allows us to solve a classification problem for extensions of vector bundles over . Based on this solution we further classify the symplectic leaves of all positroid varieties in with respect to the standard Poisson structure. Moreover, we get an explicit description of the moduli stack of symplectic leaves of with the standard Poisson structure as an open substack of the stack of vector bundles on . 

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2. Schubert defects in Lagrangian Grassmannians

   https://doi.org/10.1007/JHEP06(2025)148  

   Gu, Wei; Mihalcea, Leonardo; Sharpe, Eric; Xu, Weihong; Zhang, Hao; Zou, Hao (June 2025, Journal of High Energy Physics)

   A<sc>bstract</sc> In this paper, we propose a construction of GLSM defects corresponding to Schubert cycles in Lagrangian Grassmannians, following recent work of Closset-Khlaif on Schubert cycles in ordinary Grassmannians. In the case of Lagrangian Grassmannians, there are superpotential terms in both the bulk GLSM as well as on the defect itself, enforcing isotropy constraints. We check our construction by comparing the locus on which the GLSM defect is supported to mathematical descriptions, checking dimensions, and perhaps most importantly, comparing defect indices to known and expected polynomial invariants of the Schubert cycles in quantum cohomology and quantum K theory. 

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3. Block perturbation of symplectic matrices in Williamson’s theorem

   Babu, Gajendra; Mishra, Hemant K (August 2023, Canadian Mathematical Bulletin)

   Williamson's theorem states that for any 2n×2n real positive definite matrix A, there exists a 2n×2n real symplectic matrix S such that STAS=D⊕D, where D is an n×n diagonal matrix with positive diagonal entries which are known as the symplectic eigenvalues of A. Let H be any 2n×2n real symmetric matrix such that the perturbed matrix A+H is also positive definite. In this paper, we show that any symplectic matrix S̃ diagonalizing A+H in Williamson's theorem is of the form S̃ =SQ+(‖H‖), where Q is a 2n×2n real symplectic as well as orthogonal matrix. Moreover, Q is in symplectic block diagonal form with the block sizes given by twice the multiplicities of the symplectic eigenvalues of A. Consequently, we show that S̃ and S can be chosen so that ‖S̃ −S‖=(‖H‖). Our results hold even if A has repeated symplectic eigenvalues. This generalizes the stability result of symplectic matrices for non-repeated symplectic eigenvalues given by Idel, Gaona, and Wolf [Linear Algebra Appl., 525:45-58, 2017]. 

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4. An elementary alternative to ECH capacities

   https://doi.org/10.1073/pnas.2203090119  

   Hutchings, Michael (August 2022, Proceedings of the National Academy of Sciences)

   The embedded contact homology (ECH) capacities are a sequence of numerical invariants of symplectic four-manifolds that give (sometimes sharp) obstructions to symplectic embeddings. These capacities are defined using embedded contact homology, and establishing their basic properties currently requires Seiberg–Witten theory. In this paper we define a sequence of symplectic capacities in four dimensions using only basic notions of holomorphic curves. The capacities satisfy the same basic properties as ECH capacities and agree with the ECH capacities for the main examples for which the latter have been computed, namely convex and concave toric domains. The capacities are also useful for obstructing symplectic embeddings into closed symplectic four-manifolds. This work is inspired by a recent preprint of McDuff and Siegel [D. McDuff, K. Siegel, arXiv [Preprint] (2021)], giving a similar elementary alternative to symplectic capacities from rational symplectic field theory (SFT). 

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5. Rational cuspidal curves and symplectic fillings

   https://doi.org/10.4310/jsg.241101004405  

   Golla, Marco; Starkston, Laura (October 2024, Journal of Symplectic Geometry)

   A symplectic rational cuspidal curve with positive self-intersection number admits a concave neighborhood, and thus a corresponding contact manifold on the boundary. In this article, we study symplectic fillings of such contact manifolds, providing a complementary perspective to our earlier article on symplectic isotopy classes of rational cuspidal curves. We explore aspects of these symplectic fillings through Stein handlebodies and rational blow-downs. We give examples of such contact manifolds which are identifiable as links of normal surface singularities, other examples which admit no symplectic fillings, and further examples where the fillings can be fully classified. 

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