We prove that after inverting the Planck constant 
$h$, the Bezrukavnikov–Kaledin quantization 
$(X, {mathcal {O}}_h)$ of symplectic variety 
$X$ in characteristic 
$p$ with 
$H^2(X, {mathcal {O}}_X) =0$ is Morita equivalent to a certain central reduction of the algebra of differential operators on 
$X$.
We prove that the 
$p^infty$-torsion of the transcendental Brauer group of an abelian variety over a finitely generated field of characteristic 
$p>0$ is bounded. This answers a (variant of a) question asked by Skorobogatov and Zarhin for abelian varieties. To do this, we prove a ‘flat Tate conjecture’ for divisors. We also study other geometric Galois-invariant 
$p^infty$-torsion classes of the Brauer group which are not in the transcendental Brauer group. These classes, in contrast with our main theorem, can be infinitely 
$p$-divisible. We explain how the existence of these 
$p$-divisible towers is naturally related to the failure of surjectivity of specialisation morphisms of Néron–Severi groups in characteristic 
$p$.
We prove that any hyper-Kähler sixfold 
$K$ of generalized Kummer type has a naturally associated manifold 
$Y_K$ of 
$mathrm {K}3^{[3]}$ type. It is obtained as crepant resolution of the quotient of 
$K$ by a group of symplectic involutions acting trivially on its second cohomology. When 
$K$ is projective, the variety 
$Y_K$ is birational to a moduli space of stable sheaves on a uniquely determined projective 
$mathrm {K}3$ surface 
$S_K$. As an application of this construction we show that the Kuga–Satake correspondence is algebraic for the K3 surfaces 
We construct inhomogeneous isoparametric families of hypersurfaces with non-austere focal set on each symmetric space of non-compact type and rank 
${geq }3$. If the rank is 
${geq }4$, there are infinitely many such examples. Our construction yields the first examples of isoparametric families on any Riemannian manifold known to have a non-austere focal set. They can be obtained from a new general extension method of submanifolds from Euclidean spaces to symmetric spaces of non-compact type. This method preserves the mean curvature and isoparametricity, among other geometric properties.
DeMarco, Krieger, and Ye conjectured that there is a uniform bound B, depending only on the degree d, so that any pair of holomorphic maps 
$f, g :{mathbb {P}}^1to {mathbb {P}}^1$ with degree 
$d$ will either share all of their preperiodic points or have at most 
$B$ in common. Here we show that this uniform bound holds for a Zariski open and dense set in the space of all pairs, 
$mathrm {Rat}_d times mathrm {Rat}_d$, for each degree 
$dgeq 2$. The proof involves a combination of arithmetic intersection theory and complex-dynamical results, especially as developed recently by Gauthier and Vigny, Yuan and Zhang, and Mavraki and Schmidt. In addition, we present alternate proofs of the main results of DeMarco, Krieger, and Ye [Uniform Manin-Mumford for a family of genus 2 curves, Ann. of Math. (2) 191 (2020), 949–1001; Common preperiodic points for quadratic polynomials, J. Mod. Dyn. 18 (2022), 363–413] and of Poineau [Dynamique analytique sur 
$mathbb {Z}$ II : Écart uniforme entre Lattès et conjecture de Bogomolov-Fu-Tschinkel, Preprint (2022), arXiv:2207.01574 [math.NT]]. In fact, we prove a generalization of a conjecture of Bogomolov, Fu, and Tschinkel in a mixed setting of dynamical systems and elliptic
In this article, we establish the Grothendieck–Serre conjecture over valuation rings: for a reductive group scheme 
$G$ over a valuation ring 
$V$ with fraction field 
$K$, a 
$G$-torsor over 
$V$ is trivial if it is trivial over 
$K$. This result is predicted by the original Grothendieck–Serre conjecture and the resolution of singularities. The novelty of our proof lies in overcoming subtleties brought by general nondiscrete valuation rings. By using flasque resolutions and inducting with local cohomology, we prove a non-Noetherian counterpart of Colliot-Thélène–Sansuc's case of tori. Then, taking advantage of techniques in algebraization, we obtain the passage to the Henselian rank-one case. Finally, we induct on Levi subgroups and use the integrality of rational points of anisotropic groups to reduce to the semisimple anisotropic case, in which we appeal to properties of parahoric subgroups in Bruhat–Tits theory to conclude. In the last section, by using extension properties of reflexive sheaves on formal power series over valuation rings and patching of torsors, we prove a variant of Nisnevich's purity conjecture.
We study families of rational curves on irreducible holomorphic symplectic varieties. We give a necessary and sufficient condition for a sufficiently ample linear system on a holomorphic symplectic variety of 
$K3^{[n]}$-type to contain a uniruled divisor covered by rational curves of primitive class. In particular, for any fixed 
$n$, we show that there are only finitely many polarization types of holomorphic symplectic variety of 
$K3^{[n]}$-type that do not contain such a uniruled divisor. As an application, we provide a generalization of a result due to Beauville–Voisin on the Chow group of 
$0$-cycles on such varieties.
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