Volume 2024, Issue 812

F-signature is an important numeric invariant of singularities in positive characteristic that can be used to detect strong F-regularity. One would like to have a variant that rather detects F-rationality, and there are two theories that aim to fill this gap: F-rational signature of Hochster and Yao and dual F-signature of Sannai. Unfortunately, several important properties of the original F-signature are unknown for these invariants. We find a modification of the Hochster–Yao definition that agrees with Sannai’s dual F-signature and push further the united theory to achieve a complete generalization of F-signature.

We introduce a distributional Jacobian determinant det D ⁢ V β ⁢ ( D ⁢ v ) \det DV_{\beta}(Dv) in dimension two for the nonlinear complex gradient V β ⁢ ( D ⁢ v ) = | D ⁢ v | β ⁢ ( v x 1 , − v x 2 ) V_{\beta}(Dv)=\lvert Dv\rvert^{\beta}(v_{x_{1}},-v_{x_{2}}) for any β > − 1 \beta>-1 , whenever v ∈ W loc 1 , 2 v\in W^{1\smash{,}2}_{\mathrm{loc}} and β ⁢ | D ⁢ v | 1 + β ∈ W loc 1 , 2 \beta\lvert Dv\rvert^{1+\beta}\in W^{1\smash{,}2}_{\mathrm{loc}} . This is new when β ≠ 0 \beta\neq 0 . Given any planar ∞-harmonic function 𝑢, we show that such distributional Jacobian determinant det D ⁢ V β ⁢ ( D ⁢ u ) \det DV_{\beta}(Du) is a nonnegative Radon measure with some quantitative local lower and upper bounds. We also give the following two applications. Applying this result with β = 0 \beta=0 , we develop an approach to build up a Liouville theorem, which improves that of Savin. Precisely, if 𝑢 is an ∞-harmonic function in the whole R 2 \mathbb{R}^{2} with lim inf R → ∞ inf c ∈ R 1 R ⁢ ⨍ B ⁢ ( 0 , R ) | u ⁢ ( x ) − c | ⁢ d x < ∞ , \liminf_{R\to\infty}\inf_{c\in\mathbb{R}}\frac{1}{R}\barint_{B(0,R)}\lvert u(x)-c\rvert\,dx<\infty, then u = b + a ⋅ x u=b+a\cdot x for some b ∈ R b\in\mathbb{R} and a ∈ R 2 a\in\mathbb{R}^{2} . Denoting by u p u_{p} the 𝑝-harmonic function having the same nonconstant boundary condition as 𝑢, we show that det D ⁢ V β ⁢ ( D ⁢ u p ) → det D ⁢ V β ⁢ ( D ⁢ u ) \det DV_{\beta}(Du_{p})\to\det DV_{\beta}(Du) as p → ∞ p\to\infty in the weak-⋆ sense in the space of Radon measure. Recall that V β ⁢ ( D ⁢ u p ) V_{\beta}(Du_{p}) is always quasiregular mappings, but V β ⁢ ( D ⁢ u ) V_{\beta}(Du) is not in general.

We construct first examples of Fano varieties with torsion in their third cohomology group. The examples are constructed as double covers of linear sections of rank loci of symmetric matrices, and can be seen as higher-dimensional analogues of the Artin–Mumford threefold. As an application, we answer a question of Voisin on the coniveau and strong coniveau filtrations of rationally connected varieties.

We determine the asymptotic distribution of Manin’s iterated integrals of length at most 2. For all lengths, we compute all the asymptotic moments. We show that if the length is at least 3, these moments do in general not determine a unique distribution.

Let 𝐷 be the nonsplit quaternion algebra over Q p \mathbb{Q}_{p} . We prove that a class of admissible unitary Banach space representations of D × D^{\times} of global origin are topologically of finite length.

For every d ≥ 3 d\geq 3 , we construct a noncompact smooth 𝑑-dimensional Riemannian manifold with strictly positive sectional curvature without isoperimetric sets for any volume below 1. We construct a similar example also for the relative isoperimetric problem in (unbounded) convex sets in R d \mathbb{R}^{d} . The examples we construct have nondegenerate asymptotic cone. The dimensional constraint d ≥ 3 d\geq 3 is sharp. Our examples exhibit nonexistence of isoperimetric sets only for small volumes; indeed, in nonnegatively curved spaces with nondegenerate asymptotic cones, isoperimetric sets with large volumes always exist. This is the first instance of noncollapsed nonnegatively curved space without isoperimetric sets.

For each prime 𝑝 and each positive integer 𝑑, we construct the first examples of second countable, topologically simple 𝑝-adic Lie groups of dimension 𝑑 whose Lie algebras are abelian. This answers several questions of Glöckner and Caprace–Monod. The proof relies on a generalization of small cancellation methods that applies to central extensions of acylindrically hyperbolic groups.

We use spectral analysis to give an asymptotic formula for the number of matrices in SL ⁢ ( n , Z ) \mathrm{SL}(n,\mathbb{Z}) of height at most 𝑇 with strong error terms, far beyond the previous known, both for small and large rank.

This paper is the fourth in a series of four papers aiming to describe the (almost integral) Chow ring of M ̄ 3 \overline{\mathcal{M}}_{3} , the moduli stack of stable curves of genus 3. In this paper, we finally compute the Chow ring of M ̄ 3 \overline{\mathcal{M}}_{3} with Z ⁢ [ 1 / 6 ] \mathbb{Z}[1/6] -coefficients.