Volume 2025, Issue 823

We prove the closure ordering conjecture on the local 𝐿-parameters of representations in local Arthur packets of G n = Sp 2 ⁢ n , SO 2 ⁢ n + 1 \mathrm{G}_{n}=\mathrm{Sp}_{2n},\mathrm{SO}_{2n+1} over a non-Archimedean local field of characteristic zero. Precisely, given any representation 𝜋 in a local Arthur packet Π ψ \Pi_{\psi} , the closure of the local 𝐿-parameter of 𝜋 in the Vogan variety must contain the local 𝐿-parameter corresponding to 𝜓. This conjecture reveals a geometric nature of local Arthur packets and is inspired by the work of Adams, Barbasch and Vogan, and the work of Cunningham, Fiori, Moussaoui, Mracek and Xu, on ABV-packets. As an application, for general quasi-split connected reductive groups, we show that the closure ordering conjecture implies the enhanced Shahidi conjecture, under certain reasonable assumptions. This provides a framework towards the enhanced Shahidi conjecture in general. We verify these assumptions for G n \mathrm{G}_{n} , hence give a new proof of the enhanced Shahidi conjecture. Finally, we show that local Arthur packets cannot be fully contained in other ones, which is in contrast to the situation over Archimedean local fields and is of independent interest.

We give formulas for 3-fold Massey products in the étale cohomology of the ring of integers of a number field and use these to find the first known examples of imaginary quadratic fields with class group of 𝑝-rank two possessing an infinite 𝑝-class field tower, where 𝑝 is an odd prime. Furthermore, we provide a necessary and sufficient condition, in terms of class groups of p-extensions, for the vanishing of 3-fold Massey products. As a consequence, we give an elementary and sufficient condition for the infinitude of class field towers of imaginary quadratic fields. We also disprove McLeman’s ( 3 , 3 ) (3,3) -conjecture. Lastly, we relate the vanishing of Massey products to the existence of Galois representations of G Q , S G_{\mathbb{Q},S} which realize an unexpectedly large class group for certain extensions of a quadratic imaginary number field.

Let 𝑀 be an open Riemann surface and 𝐴 the punctured cone in C n ∖ { 0 } \mathbb{C}^{n}\setminus\{0\} on a smooth projective variety 𝑌 in P n − 1 \mathbb{P}^{n-1} . Recently, Runge approximation theorems with interpolation for holomorphic immersions M → C n M\to\mathbb{C}^{n} , directed by 𝐴, have been proved under the assumption that 𝐴 is an Oka manifold. We prove analogous results in the algebraic setting, for regular immersions directed by 𝐴 from a smooth affine curve 𝑀 into C n \mathbb{C}^{n} . The Oka property is naturally replaced by the stronger assumption that 𝐴 is algebraically elliptic, which it is if 𝑌 is uniformly rational. Under this assumption, a homotopy-theoretic necessary and sufficient condition for approximation and interpolation emerges. We show that this condition is satisfied in many cases of interest.

After providing an explicit K-stability condition for a ℚ-Gorenstein log spherical cone, we prove the existence and uniqueness of an equivariant K-stable degeneration of the cone, and deduce uniqueness of the asymptotic cone of a given complete 𝐾-invariant Calabi–Yau metric in the trivial class of an affine 𝐺-spherical manifold, 𝐾 being the maximal compact subgroup of 𝐺. Next, we prove that the valuation induced by 𝐾-invariant Calabi–Yau metrics on affine 𝐺-spherical manifolds is in fact 𝐺-invariant. As an application, we point out an affine smoothing of a Calabi–Yau cone that does not admit any 𝐾-invariant Calabi–Yau metrics asymptotic to the cone. Another corollary is that, on C 3 \mathbb{C}^{3} , there are no complete Calabi–Yau metrics with maximal volume growth and spherical symmetry other than the standard flat metric and the Li–Conlon–Rochon–Székelyhidi metrics with horospherical asymptotic cone. This answers the question whether there is a nontrivial asymptotic cone with smooth cross section on C 3 \mathbb{C}^{3} raised by Conlon–Rochon when the symmetry is spherical.

Let 𝔛 be a smooth connected 𝑝-adic formal scheme. Based on the prismatic description of crystalline local systems, we prove an analogue of Fontaine’s conjecture for torsion crystalline local systems on the generic fiber of 𝔛. As an application, we show that the locus of crystalline local systems whose Hodge–Tate weights lie in a fixed interval cuts out a closed subscheme of the universal deformation ring.

The wavefront set is a fundamental invariant of an admissible representation arising from the Harish-Chandra–Howe local character expansion. In this paper, we give a precise formula for the wavefront set of an irreducible representation of real infinitesimal character in Lusztig’s category of unipotent representations in terms of the Deligne–Langlands–Lusztig correspondence. Our formula generalises the main result of [D. Ciubotaru, L. Mason-Brown and E. Okada, Wavefront sets of unipotent representations of reductive 𝑝-adic groups I, preprint (2021), https://arxiv.org/abs/2112.14354], where this formula was obtained in the Iwahori-spherical case. We deduce that, for any irreducible unipotent representation with real infinitesimal character, the algebraic wavefront set is a singleton. In the process, we establish new properties of the generalised Springer correspondence in relation to Lusztig’s families of unipotent representations of finite reductive groups.

In this paper, we study bi-Hermitian metrics on complex surfaces with split holomorphic tangent and construct 2 types of metric cones, which are natural analogues of the Kähler case. We introduce a new type of fully nonlinear geometric PDE on such surfaces and establish smooth solvability. As a geometric application, we solve the prescribed Bismut–Ricci problem. We also introduce a weighted version of this problem. With this, we obtain canonical metrics on two important classes of complex surfaces: primary Hopf surfaces and Inoue surfaces of type S M \mathcal{S}_{M} .

A version of the singular Yamabe problem in smooth domains in a closed manifold yields complete conformal metrics with negative constant scalar curvatures. In this paper, we study the blow-up phenomena of Ricci curvatures of these metrics on domains whose boundary is close to a certain limit set of a lower dimension. We will characterize the blow-up set according to the Yamabe invariant of the underlying manifold. In particular, we will prove that all points in the lower dimension part of the limit set belong to the blow-up set on manifolds not conformally equivalent to the standard sphere and that all but one point in the lower dimension part of the limit set belong to the blow-up set on manifolds conformally equivalent to the standard sphere. In certain cases, the blow-up set can be the entire manifold. We will demonstrate by examples that these results are optimal.