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1. Abelian varieties with isogenous reductions

   https://doi.org/https://doi.org/10.5802/crmath.129  

   Khare, Chandreshekhar; Larsen, Michael: (October 2020, Comptes rendus)

   null (Ed.)

   Let $$A_1$$ and $$A_2$$ be abelian varieties over a number field $$K$$. We prove that if there exists a non-trivial morphism of abelian varieties between reductions of $$A_1$$ and $$A_2$$ at a sufficiently high percentage of primes, then there exists a non-trivial morphism $$A_1\to A_2$$ over $$\bar K$$. Along the way, we give an upper bound for the number of components of a reductive subgroup of $$GL_n$$ whose intersection with the union of $$Q$$-rational conjugacy classes of $$GL_n$$ is Zariski-dense. This can be regarded as a generalization of the Minkowski-Schur theorem on faithful representations of finite groups with rational characters. 

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2. Nonvarying, affine and extremal geometry of strata of differentials

   https://doi.org/10.1017/S0305004123000567  

   CHEN, DAWEI (October 2023, Mathematical Proceedings of the Cambridge Philosophical Society)

   Abstract We prove that the nonvarying strata of abelian and quadratic differentials in low genus have trivial tautological rings and are affine varieties. We also prove that strata ofk-differentials of infinite area are affine varieties for allk. Vanishing of homology in degree higher than the complex dimension follows as a consequence for these affine strata. Moreover we prove that the stratification of the Hodge bundle for abelian and quadratic differentials of finite area is extremal in the sense that merging two zeros in each stratum leads to an extremal effective divisor in the boundary. A common feature throughout these results is a relation of divisor classes in strata of differentials as well as its incarnation in Teichmüller dynamics. 

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3. Fourier–Mukai transforms commuting with Frobenius

   https://doi.org/10.1112/blms.13145  

   Bragg, Daniel (November 2024, Bulletin of the London Mathematical Society)

   We show that a Fourier–Mukai equivalence between smooth projective varieties of characteristic that commutes with either pushforward or pullback along Frobenius is a composition of shifts, isomorphisms, and tensor products with invertible sheaves whose th tensor power is trivial. 

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4. Quantum K-theory of quiver varieties and many-body systems

   https://doi.org/10.1007/s00029-021-00698-3  

   Koroteev, Peter; Pushkar, Petr P.; Smirnov, Andrey V.; Zeitlin, Anton M. (November 2021, Selecta Mathematica)

   Abstract We define quantum equivariant K-theory of Nakajima quiver varieties. We discuss type A in detail as well as its connections with quantum XXZ spin chains and trigonometric Ruijsenaars-Schneider models. Finally we study a limit which produces a K-theoretic version of results of Givental and Kim, connecting quantum geometry of flag varieties and Toda lattice. 

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5. Quivers and curves in higher dimension

   https://doi.org/10.1090/tran/9232  

   Argüz, Hülya; Bousseau, Pierrick (September 2024, Transactions of the American Mathematical Society)

   We prove a correspondence between Donaldson–Thomas invariants of quivers with potential having trivial attractor invariants and genus zero punctured Gromov–Witten invariants of holomorphic symplectic cluster varieties. The proof relies on the comparison of the stability scattering diagram, describing the wall-crossing behavior of Donaldson–Thomas invariants, with a scattering diagram capturing punctured Gromov–Witten invariants via tropical geometry. 

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