Band 2022, Heft 783

Let F be a non-archimedean local field of characteristic 0, and 𝔬{{\mathfrak{o}}} the ring of integers in F . We give an explicit formula for the Siegel series of a half-integral matrix over 𝔬{{\mathfrak{o}}}. This formula expresses the Siegel series of a half-integral matrix B explicitly in terms of the Gross–Keating invariant of B and its related invariants.

We prove a local Gan–Gross–Prasad conjecture on predicting the branching law for the non-tempered representations of general linear groups in the case of non-Archimedean fields. We also generalize to Bessel and Fourier–Jacobi models and study a possible generalization to Ext-branching laws.

The prescribed Ricci curvature problem in the context of G -invariant metrics on a homogeneous space M=G/K{M=G/K} is studied. We focus on the metrics at which the map g↦Rc⁡(g){g\mapsto\operatorname{Rc}(g)} is, locally, as injective and surjective as it can be. Our main result is that such property is generic in the compact case. Our main tool is a formula for the Lichnerowicz Laplacian we prove in terms of the moment map for the variety of algebras.

In this paper, we investigate closed strictly convex hypersurfaces in ℝn+1{\mathbb{R}^{n+1}} which shrink self-similarly under a large family of fully nonlinear curvature flows by high powers of curvature. When the speed function is given by powers of a homogeneous of degree 1 and inverse concave function of the principal curvatures with power greater than 1, we prove that the only such hypersurfaces are round spheres. We also prove that slices are the only closed strictly convex self-similar solutions to such curvature flows in the hemisphere 𝕊+n+1{\mathbb{S}^{n+1}_{+}} with power greater than or equal to 1.

In this paper, we study the Ricci flow on manifolds with boundary. In the paper, we substantially improve Shen’s result [Y. Shen, On Ricci deformation of a Riemannian metric on manifold with boundary, Pacific J. Math. 173 1996, 1, 203–221] to manifolds with arbitrary initial metric. We prove short-time existence and uniqueness of the solution, in which the boundary becomes instantaneously totally geodesic for positive time. Moreover, we prove that the flow we constructed preserves natural boundary conditions. More specifically, if the initial metric has a convex boundary, then the flow preserves positive curvature operator and the PIC1, PIC2 conditions. Moreover, if the initial metric has a two-convex boundary, then the flow preserves the PIC condition.

We give explicit blowups of the projective plane in positive characteristic that contain smooth rational curves of arbitrarily negative self-intersection, showing that the Bounded Negativity Conjecture fails even for rational surfaces in positive characteristic. As a consequence, we show that any surface in positive characteristic admits a birational model failing the Bounded Negativity Conjecture.

We prove that torsors under parahoric group schemes on the punctured spectrum of Fontaine’s ring Ainf{A_{\mathrm{inf}}}, extend to the whole spectrum. Using descent we can extend a similar result for the ring 𝔖{\mathfrak{S}} of Kisin and Pappas to full generality. Moreover, we treat similarly the case of equal characteristic. As applications we extend results of Ivanov on exactness of the loop functor and present the construction of a canonical specialization map from the BdR+{B^{+}_{\mathrm{dR}}}-affine Grassmannian to the Witt vector affine flag variety.

We provide non-isomorphic finite 2-groups which have isomorphic group algebras over any field of characteristic 2, thus settling the Modular Isomorphism Problem.

We clarify the definition of the divisorial hull and recollect some basic facts. Then we correct Lemma 4.2 and Theorem 11.2 (1)–(2) in the original article.