We prove that an algebraic stack with affine stabilizers over an arbitrary base is & eacute;talelocally a quotient stack around any point with a linearly reductive stabilizer. This generalizes earlier work by the authors (stacks over algebraically closed fields) and by Abramovich, Olsson and Vistoli (stacks with finite inertia). In addition, we prove a number of foundational results, which are new even over a field. These include various coherent completeness and effectivity results for adic sequences of algebraic stacks. Finally, we give several applications of our results and methods, such as structure theorems for linearly reductive group schemes and generalizations to the relative setting of Sumihiro's theorem on torus actions and Luna's & eacute;tale slice theorem.
We characterize the simplicity of Pimsner algebras for non-proper C*-correspondences. With the aid of this criterion, we give a systematic strategy to produce outer actions of unitary tensor categories on Kirchberg algebras. In particular, every countable unitary tensor category admits an outer action on the Cuntz algebra O2. We also study the realizability of modules over fusion rings as K-groups of Kirchberg algebras acted on by unitary tensor categories, which turns out to be generically true for every unitary fusion category. Several new examples are provided, among which actions on Cuntz algebras of 3-cocycle twists of cyclic groups are constructed for all possible 3-cohomological classes, thereby answering a question asked by Izumi.
We realize the quantum loop groups and shifted quantum loop groups of arbitrary types, possibly non symmetric, using critical K-theory. This generalizes the Nakajima construction of symmetric quantum loop groups via quiver varieties to non symmetric types. We also give a new geometric construction of some simple modules of both quantum loop groups and shifted quantum loop groups.
We prove that the topological entropy of any dominant rational self-map of a projective variety defined over a complete non-Archimedean field is bounded from above by the maximum of its dynamical degrees, thereby extending a theorem of Gromov and Dinh-Sibony from the complex to the non-Archimedean setting. We proceed by proving that any regular self-map which admits a regular extension to a projective model defined over the valuation ring has necessarily zero entropy. To this end we introduce the e-reduction of a Berkovich analytic space, a notion of independent interest.
We study topological properties of moduli spaces of p-adic shtukas and local Shimura varieties. On one hand, we construct and study the specialization map for moduli spaces of p-adic shtukas at parahoric level whose target is an affine Deligne-Lusztig variety. On the other hand, given a p-adic shtuka datum (G, b, & micro;), with G unramified over Qp and such that (b, & micro;) is HN-irreducible, we determine the set of geometric connected components of infinite level moduli spaces of p-adic shtukas. In other words, we understand 0(ShtG,b,& micro;,1 x SpdCp) with its right G(Qp) x Gb(Qp) x WE-action. As a corollary, we prove new cases of a conjecture of Rapoport and Viehmann.
We prove a variant of the Beauville-Bogomolov decomposition for weakly ordinary, or generally globally F-split, varieties X with K-X similar to 0, in characteristic p > 0. We also show that the weakly ordinary assumption in our statement cannot be dropped. Additionally, if the assumption K-X similar to 0 is replaced by-K-X being semi-ample, we show the weaker statement that all closed fibers of the Albanese morphism are isomorphic. Finally, we apply our main theorem to draw consequences to the behavior of rational points and fundamental groups of weakly ordinary K-trivial varieties in positive characteristic.
We prove an abstract result giving a (t)(epsilon) upper bound on the growth of the Sobolev norms of a time-dependent Schr & ouml;dinger equation of the form i Psi = H-0 Psi +V(t)Psi. Here H-0 is assumed to be the Hamiltonian of a steep quantum integrable system and to be a pseudodifferential operator of order d > 1; V(t) is a time-dependent family of pseudodifferential operators, unbounded, but of order b < d. The abstract theorem is then applied to perturbations of the quantum anharmonic oscillators in dimension 2 and to perturbations of the Laplacian on a manifold with integrable geodesic flow, and in particular Zoll manifolds, rotation-invariant surfaces and Lie groups. The proof is based on a quantum version of the proof of the classical Nekhoroshev theorem.
A rigid cohomology class on a complex manifold is a class that is represented by a unique closed positive current. The positive current representing a rigid class is also called rigid. For a compact Kahler manifold $X$ all eigenvectors of hyperbolic automorphisms acting on $H^{1,1}(X)$ that have non-unit eigenvalues are rigid classes. Such classes are always parabolic, namely, they belong to the boundary of the Kahler cone and have vanishing volume. We study parabolic $(1,1)$-classes on compact hyperkahler manifolds with $b_2 \geq 7$. We show that a parabolic class is rigid if it is not orthogonal to a rational vector with respect to the BBF form. This implies that a general parabolic class on a hyperkahler manifold is rigid.
The present paper is the first in a series devoted to the study of asymptotic geometry of Riemann surfaces and their moduli spaces. We introduce the moduli space of hybrid curves as a new compactification of the moduli space of curves, refining the one obtained by Deligne and Mumford. This is the moduli space for multiscale geometric objects which mix complex and higher rank tropical and non-Archimedean geometries, reflecting both discrete and continuous features. We define canonical measures on hybrid curves which combine and generalize Arakelov-Bergman measures on Riemann surfaces and Zhang measures on metric graphs. We then show that the universal family of canonically measured hybrid curves over this moduli space varies continuously. This provides a precise link between the non-Archimedean Zhang measure and variations of Arakelov-Bergman measures in families of Riemann surfaces, answering a question which has been open since the pioneering work of Zhang on admissible pairing in the nineties.
We prove the Massey vanishing conjecture, due to Min & aacute;c and T & acirc;n, for n = 4 and p = 2. That is, we show that for all fields F, if a fourfold Massey product modulo 2 is defined over F, then it vanishes over F.
Given a non-increasing function W N ! R+ such that s n .s/ tends to zero ass goes to infinity, we show that the set of points in Rn that are exactly-approximable is non-empty, and we compute its Hausdorff dimension. This answers questions of Jarnik and of Beresnevich, Dickinson and Velani.
Roughly speaking, an ALF metric of real dimension 4n should be a metric such that it has a (4n-1)-dimensional asymptotic cone, the volume growth of this metric is of order 4n-1 and its sectional curvature tends to 0 at infinity. In this paper, we first show that the Taub-NUT deformation of a hyperk & auml;hler cone with respect to a locally free S1-symmetry is ALF hyperk & auml;hler. Using this metric at infinity, we establish the existence of ALF Calabi-Yau metric on certain crepant resolutions. In particular, we prove that there exist ALF Calabi-Yau metrics on canonical bundles of classical homogeneous Fano contact manifolds.
The circle method has been successfully used over the last century to study rational points on hypersurfaces. More recently, a version of the method over function fields, combined with spreading out techniques, has led to a range of results about moduli spaces of rational curves on hypersurfaces. In this paper a version of the circle method is implemented in the setting of the Grothendieck ring of varieties. This allows us to approximate the classes of these moduli spaces directly, without relying on point counting, and leads to a deeper understanding of their geometry.
We are investigating the lifting problem for local actions involving semidirect products of a cyclic p-group with a cyclic group prime to p, where p represents the characteristic of the special fiber. We establish a criterion based on the Harbater-Katz-Gabber compactification of local actions, enabling us to determine whether a given local action can be lifted or not. Specifically, in the case of the dihedral group, we present an example of a local dihedral action that cannot be lifted. This instance provides a more potent obstruction than the KGB obstruction.
