Volume 2022, Issue 789

We prove a general form of the regularity theorem for uniformity norms, and deduce an inverse theorem for these norms which holds for a class of compact nilspaces including all compact abelian groups, and also nilmanifolds; in particular we thus obtain the first non-abelian versions of such theorems. We derive these results from a general structure theorem for cubic couplings, thereby unifying these results with the Host–Kra Ergodic Structure Theorem. A unification of this kind had been propounded as a conceptual prospect by Host and Kra. Our work also provides new results on nilspaces. In particular, we obtain a new stability result for nilspace morphisms. We also strengthen a result of Gutman, Manners and Varjú, by proving that a k -step compact nilspace of finite rank is a toral nilspace (in particular, a connected nilmanifold) if and only if its k -dimensional cube set is connected. We also prove that if a morphism from a cyclic group of prime order into a compact finite-rank nilspace is sufficiently balanced (i.e. equidistributed in a certain quantitative and multidimensional sense), then the nilspace is toral. As an application of this, we obtain a new proof of a refinement of the Green–Tao–Ziegler inverse theorem.

We study dominant rational maps preserving singular holomorphic codimension one foliations on projective manifolds and that exhibit non-trivial transverse dynamics.

While there is much work and many conjectures surrounding the intersection theory of the moduli space of curves, relatively little is known about the intersection theory of the Hurwitz space ℋk,g{\mathcal{H}_{k,g}} parametrizing smooth degree k , genus g covers of ℙ1{\mathbb{P}^{1}}. Let k=3,4,5{k=3,4,5}. We prove that the rational Chow rings of ℋk,g{\mathcal{H}_{k,g}} stabilize in a suitable sense as g tends to infinity. In the case k=3{k=3}, we completely determine the Chow rings for all g . We also prove that the rational Chow groups of the simply branched Hurwitz space ℋk,gs⊂ℋk,g{\mathcal{H}^{s}_{k,g}\subset\mathcal{H}_{k,g}} are zero in codimension up to roughly gk{\frac{g}{k}}. In [ S. Canning and H. Larson , The Chow rings of the moduli spaces of curves of genus 7,8{7,8} and 9, preprint 2021, arXiv:2104.05820], results developed in this paper are used to prove that the Chow rings of ℳ7,ℳ8,{\mathcal{M}_{7},\mathcal{M}_{8},} and ℳ9{\mathcal{M}_{9}} are tautological.

Given a number field K and an integer m≥0{m\geq 0}, let Km{K_{m}} denote the maximal m -step solvable Galois extension of K and write GKm{G_{K}^{m}} for the maximal m -step solvable Galois group Gal⁡(Km/K){\operatorname{Gal}(K_{m}/K)} of K . In this paper, we prove that the isomorphy type of K is determined by the isomorphy type of GK3{G_{K}^{3}}. Further, we prove that Km/K{K_{m}/K} is determined functorially by GKm+3{G_{K}^{m+3}} (resp. GKm+4{G_{K}^{m+4}}) for m≥2{m\geq 2} (resp. m≤1{m\leq 1}). This is a substantial sharpening of a famous theorem of Neukirch and Uchida. A key step in our proof is the establishment of the so-called local theory, which in our context characterises group-theoretically the set of decomposition groups (at nonarchimedean primes) in GKm{G_{K}^{m}}, starting from GKm+2{G_{K}^{m+2}}.

The first part of the paper studies the boundary behavior of holomorphic isometric mappings F=(F1,…,Fm){F=(F_{1},\dots,F_{m})} from the complex unit ball 𝔹n{\mathbb{B}^{n}}, n≥2{n\geq 2}, to a bounded symmetric domain Ω=Ω1×⋯×Ωm{\Omega=\Omega_{1}\times\cdots\times\Omega_{m}} up to constant conformal factors, where Ωi′{\Omega_{i}^{\prime}}s are irreducible factors of Ω. We prove every non-constant component Fi{F_{i}} must map generic boundary points of 𝔹n{\mathbb{B}^{n}} to the boundary of Ωi{\Omega_{i}}. In the second part of the paper, we establish a rigidity result for local holomorphic isometric maps from the unit ball to a product of unit balls and Lie balls.

We consider the case with boundary of the classical Kazdan–Warner problem in dimension greater or equal than three, i.e. the prescription of scalar and boundary mean curvatures via conformal deformations of the metric. We deal in particular with negative scalar curvature and boundary mean curvature of arbitrary sign, which to our knowledge has not been treated in the literature. We employ a variational approach to prove new existence results, especially in three dimensions. One of the principal issues for this problem is to obtain compactness properties, due to the fact that bubbling may occur with profiles of hyperbolic balls or horospheres, and hence one may lose either pointwise estimates on the conformal factor or the total conformal volume. We can sometimes prevent them using integral estimates, Pohozaev identities and domain-variations of different types.

In this paper, we prove the abundance theorem for numerically trivial canonical divisors on strongly F -regular varieties, assuming that the geometric generic fibers of the Albanese morphisms are strongly F -regular.

We express the order of the pole and the leading coefficient of the L -function of a (large class of) ℓ{\ell}-adic coefficients (ℓ{\ell} any prime) over a quasi-projective variety over a finite field of characteristic p . We use the technique of [13] with coefficients with, as new ingredient, the use of F -gauges and their equivalence, in the derived category, with Raynaud modules proved by Ekedahl.