Volume 2025, Issue 827

We prove that the Tate conjecture for divisors is “generically true” for mod ⁡ p \operatorname{mod}p reductions of complex projective varieties with h 2 , 0 = 1 h^{2,0}=1 , under a mild assumption on moduli. By refining this general result, we establish a new case of the BSD conjecture over global function fields, and the Tate conjecture for a class of general type surfaces of geometric genus 1.

Rudin’s version of the classical Julia–Wolff–Carathéodory theorem is a cornerstone of holomorphic function theory in the unit ball of C d \mathbb{C}^{d} . In this paper, we obtain a complete generalization of Rudin’s theorem for a holomorphic map f : D → D ′ f\colon D\to D^{\prime} between convex domains of finite type. In particular, given a point ξ ∈ ∂ D \xi\in\partial D with finite dilation, we show that the 𝐾-limit of 𝑓 at 𝜉 exists and is a point η ∈ ∂ D ′ \eta\in\partial D^{\prime} , and we obtain asymptotic estimates for all entries of the Jacobian matrix of the differential d ⁢ f z df_{z} in terms of the multitypes at the points 𝜉 and at 𝜂. We introduce a generalization of Bracci–Patrizio–Trapani’s pluricomplex Poisson kernel which, together with the dilation at 𝜉, gives a formula for the restricted 𝐾-limit of the normal component of the normal derivative ⟨ d ⁢ f z ⁢ ( n ξ ) , n η ⟩ \langle df_{z}(n_{\xi}),n_{\eta}\rangle . Our principal tools are methods from Gromov hyperbolicity theory, a scaling in the normal direction, and the strong asymptoticity of complex geodesics. To obtain our main result, we prove a conjecture by Abate on the Kobayashi type of a vector 𝑣, proving that it is equal to the reciprocal of the line type of 𝑣, and we give new extrinsic characterizations of both 𝐾-convergence and restricted convergence to a point ξ ∈ ∂ D \xi\in\partial D in terms of the multitype at 𝜉.

Using a mixed-characteristic incarnation of fusion, we prove an analog of Nekovář–Scholl’s plectic conjecture for local Shimura varieties. We apply this to obtain results on the plectic conjecture for (global) Shimura varieties after restricting to a decomposition group. Along the way, we prove a 𝑝-adic uniformization theorem for the basic locus of abelian type Shimura varieties at hyperspecial level, which is of independent interest.

In this paper, we complete the framework of Arveson’s version of the Gauss–Bonnet–Chern formula by proving that Arveson’s version of the Gauss–Bonnet–Chern formula holding true for the quotient module in the Drury–Arveson module is equivalent to the associated submodule being locally algebraic. Moreover, we establish the asymptotic Arveson curvature invariant and the asymptotic Euler characteristic for contractive Hilbert modules over the polynomial ring in infinitely many variables, and obtain the infinitely-many-variables analogue of Arveson’s version of the Gauss–Bonnet–Chern formula. Finally, we solve the finite defect problem for submodules of the Drury–Arveson module H 2 H^{2} in infinitely many variables by proving that H 2 H^{2} has no nontrivial submodules of finite rank.

In his “Four Lectures”, Gromov conjectured a scalar curvature extremality property of convex polytopes. Moreover, Gromov outlined a proof of the conjecture in the special case when the dihedral angles are acute. Gromov’s argument relies on Dirac operator techniques together with a smoothing construction. In this paper, we give the details of such a smoothing construction, thereby providing a detailed proof of Gromov’s theorem.

We study the geometry of hyperconvex representations of hyperbolic groups in PSL ⁢ ( d , C ) \mathrm{PSL}(d,\mathbb{C}) and establish two structural results: a group admitting a hyperconvex representation is virtually isomorphic to a Kleinian group, and its hyperconvex limit set in the appropriate flag manifold has Hausdorff dimension strictly smaller than 2.

We construct the universal central extension of the Lie algebra of exact divergence-free vector fields, proving a conjecture by Claude Roger from 1995. The proof relies on the analysis of a Leibniz algebra that underlies these vector fields. As an application, we construct the universal central extension of the (infinite-dimensional) Lie group of exact divergence-free diffeomorphisms of a compact 3-dimensional manifold.

Fix an 𝐼-adically complete Noetherian ring 𝐴 and suppose 𝑋 is a proper 𝐴-scheme. This article concerns the relationship between the Brauer group of 𝑋 and that of the various X n X_{n} where X n X_{n} is the fiber over A / I n + 1 A/I^{n+1} . In particular, we answer a question of Grothendieck by showing that, in positive and mixed characteristic, there are examples of 𝑋 with nontrivial Brauer classes that restrict to zero on all the X n X_{n} . We characterize such behavior, prove this cannot happen in characteristic zero, and deduce a formal GAGA statement for Brauer classes.

In this work, we investigate compact Kähler manifolds with non-negative or quasi-positive mixed curvature coming from a linear combination of the Ricci and holomorphic sectional curvature, which covers various notions of curvature considered in the literature. Specifically, we prove a splitting theorem, analogous to the Cheeger–Gromoll splitting theorem, for complete Kähler manifolds with non-negative mixed curvature containing a line, and then establish a structure theorem for compact Kähler manifolds with non-negative mixed curvature. We also show that the Hodge numbers of compact Kähler manifolds with quasi-positive mixed curvature must vanish. Both results are based on the conformal perturbation method.