We extend results on asymptotic invariants of line bundles on complex projective varieties to projective varieties over arbitrary fields. To do so over imperfect fields, we prove a scheme-theoretic version of the gamma construction of Hochster and Huneke to reduce to the setting where the ground field is $F$ -finite. Our main result uses the gamma construction to extend the ampleness criterion of de Fernex, Küronya, and Lazarsfeld using asymptotic cohomological functions to projective varieties over arbitrary fields, which was previously known only for complex projective varieties. We also extend Nakayama’s description of the restricted base locus to klt or strongly $F$ -regular varieties over arbitrary fields.

We describe the irreducible components of the jet schemes with origin in the singular locus of a two-dimensional quasi-ordinary hypersurface singularity. A weighted graph is associated with these components and with their embedding dimensions and their codimensions in the jet schemes of the ambient space. We prove that the data of this weighted graph is equivalent to the data of the topological type of the singularity. We also determine a component of the jet schemes (equivalent to a divisorial valuation on $\mathbb{A}^{3}$ ), that computes the log-canonical threshold of the singularity embedded in $\mathbb{A}^{3}$ . This provides us with pairs $X\subset \mathbb{A}^{3}$ whose log-canonical thresholds are not computed by monomial divisorial valuations. Note that for a pair $C\subset \mathbb{A}^{2}$ , where $C$ is a plane curve, the log-canonical threshold is always computed by a monomial divisorial valuation (in suitable coordinates of $\mathbb{A}^{2}$ ).

In order to work with non-Nagata rings which are Nagata “up-to-completely-decomposed-universal-homeomorphism,” specifically finite rank Hensel valuation rings, we introduce the notions of pseudo-integral closure, pseudo-normalization, and pseudo-Hensel valuation ring. We use this notion to give a shorter and more direct proof that $H_{\operatorname{cdh}}^{n}(X,F_{\operatorname{cdh}})=H_{l\operatorname{dh}}^{n}(X,F_{l\operatorname{dh}})$ for homotopy sheaves $F$ of modules over the $\mathbb{Z}_{(l)}$ -linear motivic Eilenberg–Maclane spectrum. This comparison is an alternative to the first half of the author’s volume Astérisque 391 whose main theorem is a cdh-descent result for Voevodsky motives. The motivating new insight is really accepting that Voevodsky’s motivic cohomology (with $\mathbb{Z}[\frac{1}{p}]$ -coefficients) is invariant not just for nilpotent thickenings, but for all universal homeomorphisms.

We will establish a nearby and vanishing cycle formalism for the arithmetic $\mathscr{D}$ -module theory following Beilinson’s philosophy. As an application, we define smooth objects in the framework of arithmetic $\mathscr{D}$ -modules whose category is equivalent to the category of overconvergent isocrystals.

We prove that the integral closure of a strongly Golod ideal in a polynomial ring over a field of characteristic zero is strongly Golod, positively answering a question of Huneke. More generally, the rational power $I_{\unicode[STIX]{x1D6FC}}$ of an arbitrary homogeneous ideal is strongly Golod for $\unicode[STIX]{x1D6FC}\geqslant 2$ and, if $I$ is strongly Golod, then $I_{\unicode[STIX]{x1D6FC}}$ is strongly Golod for $\unicode[STIX]{x1D6FC}\geqslant 1$ . We also show that all the coefficient ideals of a strongly Golod ideal are strongly Golod.

In this article, I give a crystalline characterization of abelian varieties amongst the class of smooth projective varieties with trivial tangent bundles in characteristic $p>0$ . Using my characterization, I show that a smooth, projective, ordinary variety with trivial tangent bundle is an abelian variety if and only if its second crystalline cohomology is torsion-free. I also show that a conjecture of KeZheng Li about smooth projective varieties with trivial tangent bundles in characteristic $p>0$ is true for smooth projective surfaces. I give a new proof of a result by Li and prove a refinement of it. Based on my characterization of abelian varieties, I propose modifications of Li’s conjecture, which I expect to be true.

We prove the existence of the global attractor in ${\dot{H}}^{s}$ , $s>11/12$ for the weakly damped and forced mKdV on the one-dimensional torus. The existence of global attractor below the energy space has not been known, though the global well-posedness below the energy space has been established. We directly apply the $I$ -method to the damped and forced mKdV, because the Miura transformation does not work for the mKdV with damping and forcing terms. We need to make a close investigation into the trilinear estimates involving resonant frequencies, which are different from the bilinear estimates corresponding to the KdV.

Let ${\mathcal{V}}$ be a complete discrete valuation ring of unequal characteristic with perfect residue field, $u:{\mathcal{Z}}{\hookrightarrow}\mathfrak{X}$ be a closed immersion of smooth, quasi-compact, separated formal schemes over ${\mathcal{V}}$ , $T$ be a divisor of $X$ such that $U:=T\cap Z$ is a divisor of $Z$ , and $\mathfrak{D}$ a strict normal crossing divisor of $\mathfrak{X}$ such that $u^{-1}(\mathfrak{D})$ is a strict normal crossing divisor of ${\mathcal{Z}}$ . We pose $\mathfrak{X}^{\sharp }:=(\mathfrak{X},\mathfrak{D})$ , ${\mathcal{Z}}^{\sharp }:=({\mathcal{Z}},u^{-1}\mathfrak{D})$ and $u^{\sharp }:{\mathcal{Z}}^{\sharp }{\hookrightarrow}\mathfrak{X}^{\sharp }$ the exact closed immersion of smooth logarithmic formal schemes over ${\mathcal{V}}$ . In Berthelot’s theory of arithmetic ${\mathcal{D}}$ -modules, we work with the inductive system of sheaves of rings $\widehat{{\mathcal{D}}}_{\mathfrak{X}^{\sharp }}^{(\bullet )}(T):=(\widehat{{\mathcal{D}}}_{\mathfrak{X}^{\sharp }}^{(m)}(T))_{m\in \mathbb{N}}$ , where $\widehat{{\mathcal{D}}}_{\mathfrak{X}^{\sharp }}^{(m)}$ is the $p$ -adic completion of the ring of differential operators of level $m$ over $\mathfrak{X}^{\sharp }$ and where $T$ means that we add overconvergent singularities along the divisor $T$ . Moreover, Berthelot introduced the sheaf ${\mathcal{D}}_{\mathfrak{X}^{\sharp }}^{\dagger }(\text{}^{\dagger }T)_{\mathbb{Q}}:=\underset{\underset{m}{\longrightarrow }}{\lim }\,\widehat{{\mathcal{D}}}_{\mathfrak{X}^{\sharp }}^{(m)}(T)\otimes _{\mathbb{Z}}\mathbb{Q}$ of differential operators over $\mathfrak{X}^{\sharp }$ of finite level with overconvergent singularities along $T$ . Let ${\mathcal{E}}^{(\bullet )}\in \underset{\displaystyle \longrightarrow }{LD}\text{}_{\mathbb{Q},\text{coh}}^{\text{b}}(\widehat{{\mathcal{D}}}_{\mathfrak{X}^{\sharp }}^{(\bullet )}(T))$ and ${\mathcal{E}}:=\varinjlim ~({\mathcal{E}}^{(\bullet )})$ be the corresponding object of $D_{\text{coh}}^{\text{b}}({\mathcal{D}}_{\mathfrak{X}^{\sharp }}^{\dagger }(\text{}^{\dagger }T)_{\mathbb{Q}})$ . In this paper, we study sufficient conditions on ${\mathcal{E}}$ so that if $u^{\sharp !}({\mathcal{E}})\in D_{\text{coh}}^{\text{b}}({\mathcal{D}}_{{\mathcal{Z}}^{\sharp }}^{\dagger }(\text{}^{\dagger }U)_{\mathbb{Q}})$ then $u^{\sharp (\bullet )!}({\mathcal{E}}^{(\bullet )})\in \underset{\displaystyle \longrightarrow }{LD}\text{}_{\mathbb{Q},\text{coh}}^{\text{b}}(\widehat{{\mathcal{D}}}_{{\mathcal{Z}}^{\sharp }}^{(\bullet )}(U))$ . For instance, we check that this is the case when ${\mathcal{E}}$ is a coherent ${\mathcal{D}}_{\mathfrak{X}^{\sharp }}^{\dagger }(\text{}^{\dagger }T)_{\mathbb{Q}}$ -module such that the cohomological spaces of $u^{\sharp !}({\mathcal{E}})$ are isocrystals on ${\mathcal{Z}}^{\sharp }$ overconvergent along $U$ .

Our aim in this paper is to deal with integrability of maximal functions for Herz–Morrey spaces on the unit ball with variable exponent $p_{1}(\cdot )$ approaching $1$ and for double phase functionals $\unicode[STIX]{x1D6F7}_{d}(x,t)=t^{p_{1}(x)}+a(x)t^{p_{2}}$ , where $a(x)^{1/p_{2}}$ is nonnegative, bounded and Hölder continuous of order $\unicode[STIX]{x1D703}\in (0,1]$ and $1/p_{2}=1-\unicode[STIX]{x1D703}/N>0$ . We also establish Sobolev type inequality for Riesz potentials on the unit ball.

Soit $\unicode[STIX]{x1D70B}$ un module de plus haut poids unitaire du groupe $G=\mathbf{Sp}(2n,\mathbb{R})$ . On s’intéresse aux paquets d’Arthur contenant  $\unicode[STIX]{x1D70B}$ . Lorsque le plus haut poids est scalaire, on détermine les paramètres de ces paquets, on établit la propriété de multiplicité $1$ de $\unicode[STIX]{x1D70B}$ dans le paquet, et l’on calcule le caractère $\unicode[STIX]{x1D70C}_{\unicode[STIX]{x1D70B}}$ (du groupe des composantes connexes du centralisateur du paramètre dans le groupe dual) associé à $\unicode[STIX]{x1D70B}$ et qui joue un grand rôle dans la théorie d’Arthur. On fait de même pour certains modules de plus haut poids unitaires unipotents  $\unicode[STIX]{x1D70E}_{n,k}$ , ou bien lorsque le caractère infinitésimal est régulier.

The category of Cohen–Macaulay modules of an algebra $B_{k,n}$ is used in Jensen et al. (A categorification of Grassmannian cluster algebras, Proc. Lond. Math. Soc. (3) 113(2) (2016), 185–212) to give an additive categorification of the cluster algebra structure on the homogeneous coordinate ring of the Grassmannian of $k$ -planes in $n$ -space. In this paper, we find canonical Auslander–Reiten sequences and study the Auslander–Reiten translation periodicity for this category. Furthermore, we give an explicit construction of Cohen–Macaulay modules of arbitrary rank. We then use our results to establish a correspondence between rigid indecomposable modules of rank 2 and real roots of degree 2 for the associated Kac–Moody algebra in the tame cases.

For a character of the absolute Galois group of a complete discrete valuation field, we define a lifting of the refined Swan conductor, using higher dimensional class field theory.

We prove that for $n\geqslant 4$ , every knot has infinitely many conjugacy classes of $n$ -braid representatives if and only if it has one admitting an exchange move.

The relationship between interpolation and separation properties of hypersurfaces in Bargmann–Fock spaces over $\mathbb{C}^{n}$ is not well understood except for $n=1$ . We present four examples of smooth affine algebraic hypersurfaces that are not uniformly flat, and show that exactly two of them are interpolating.

This paper examines Euler characteristics and characteristic classes in the motivic setting. We establish a motivic version of the Becker–Gottlieb transfer, generalizing a construction of Hoyois. Making calculations of the Euler characteristic of the scheme of maximal tori in a reductive group, we prove a generalized splitting principle for the reduction from $\operatorname{GL}_{n}$ or $\operatorname{SL}_{n}$ to the normalizer of a maximal torus (in characteristic zero). Ananyevskiy’s splitting principle reduces questions about characteristic classes of vector bundles in $\operatorname{SL}$ -oriented, $\unicode[STIX]{x1D702}$ -invertible theories to the case of rank two bundles. We refine the torus-normalizer splitting principle for $\operatorname{SL}_{2}$ to help compute the characteristic classes in Witt cohomology of symmetric powers of a rank two bundle, and then generalize this to develop a general calculus of characteristic classes with values in Witt cohomology.

In this article, we establish a new estimate for the Gaussian curvature of open Riemann surfaces in Euclidean three-space with a specified conformal metric regarding the uniqueness of the holomorphic maps of these surfaces. As its applications, we give new proofs on the unicity problems for the Gauss maps of various classes of surfaces, in particular, minimal surfaces in Euclidean three-space, constant mean curvature one surfaces in the hyperbolic three-space, maximal surfaces in the Lorentz–Minkowski three-space, improper affine spheres in the affine three-space and flat surfaces in the hyperbolic three-space.

For a complex function $F$ on $\mathbb{C}$ , we study the associated composition operator $T_{F}(f):=F\circ f=F(f)$ on Wiener amalgam $W^{p,q}(\mathbb{R}^{d})\;(1\leqslant p<\infty ,1\leqslant q<2)$ . We have shown $T_{F}$ maps $W^{p,1}(\mathbb{R}^{d})$ to $W^{p,q}(\mathbb{R}^{d})$ if and only if $F$ is real analytic on $\mathbb{R}^{2}$ and $F(0)=0$ . Similar result is proved in the case of modulation spaces $M^{p,q}(\mathbb{R}^{d})$ . In particular, this gives an affirmative answer to the open question proposed in Bhimani and Ratnakumar (J. Funct. Anal. 270(2) (2016), 621–648).

We show that every Fricke-invariant meromorphic modular form for $\unicode[STIX]{x1D6E4}_{0}(N)$ whose divisor on $X_{0}(N)$ is defined over $\mathbb{Q}$ and supported on Heegner divisors and the cusps is a generalized Borcherds product associated to a harmonic Maass form of weight $1/2$ . Further, we derive a criterion for the finiteness of the multiplier systems of generalized Borcherds products in terms of the vanishing of the central derivatives of $L$ -functions of certain weight  $2$ newforms. We also prove similar results for twisted Borcherds products.

We provide a complete classification of all algebras of generalized dihedral type, which are natural generalizations of algebras which occurred in the study of blocks of group algebras with dihedral defect groups. This gives a description by quivers and relations coming from surface triangulations.

In this paper, we construct Chern classes from the relative $K$ -theory of modulus pairs to the relative motivic cohomology defined by Binda–Saito. An application to relative motivic cohomology of henselian dvr is given.