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Creators/Authors contains: "Polishchuk, Alexander"

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  1. ; ; ; (, International Mathematics Research Notices)
    Abstract We study Hecke operators associated with curves over a non-archimedean local field $$K$$ and over the rings $$O/\mathfrak{m}^{N}$$, where $$O\subset K$$ is the ring of integers. Our main result is commutativity of a certain “small” local Hecke algebra over $$O/\mathfrak{m}^{N}$$, associated with a connected split reductive group $$G$$ such that $[G,G]$ is simply connected. The proof uses a Hecke algebra associated with $$G(K(\!(t)\!))$$ and a global argument involving $$G$$-bundles on curves. 
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    Free, publicly-accessible full text available April 1, 2026
  2. ; (, 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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  3. ; (, manuscripta mathematica)
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  4. ; (, Mathematische Annalen)
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  5. ; (, Journal of Geometry and Physics)
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  6. ; ; (, Ukrainian Mathematical Journal)
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  7. ; (, Journal of Topology)
    Abstract In this paper, generalizing our previous construction, we equip the relative moduli stack of complexes over a Calabi–Yau fibration (possibly with singular fibers) with a shifted Poisson structure. Applying this construction to the anticanonical linear systems on surfaces, we get examples of compatible Poisson brackets on projective spaces extending Feigin–Odesskii Poisson brackets. Computing explicitly the corresponding compatible brackets coming from Hirzebruch surfaces, we recover the brackets defined by Odesskii–Wolf. 
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  8. ; ; (, American Journal of Mathematics)
    Abstract: We prove that the moduli of stable supercurves with punctures is a smooth proper DM stack and study an analog of the Mumford's isomorphism for its canonical line bundle. 
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  9. (, Communications in Mathematical Physics)
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  10. ; (, Symmetry, Integrability and Geometry: Methods and Applications)
    We give explicit formulas for ten compatible Poisson brackets on $$\mathbb P^5$$ found in arXiv:2007.12351. 
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