Inventiones mathematicae最新文献

We prove that the quotient of a klt type singularity by a reductive group is of klt type in characteristic 0. In particular, given a klt variety (X) endowed with the action of a reductive group (G) and admitting a quasi-projective good quotient (Xrightarrow X/!/G), we can find a boundary (B) on (X/!/G) so that the pair ((X/!/G,B)) is klt. This applies for example to GIT-quotients of klt varieties. Our main result has consequences for complex spaces obtained as quotients of Hamiltonian Kähler (G)-manifolds, for collapsings of homogeneous vector bundles as introduced by Kempf, and for good moduli spaces of smooth Artin stacks. In particular, it implies that the good moduli space parametrizing (n)-dimensional K-polystable smooth Fano varieties of volume (v) has klt type singularities. As a corresponding result regarding global geometry, we show that quotients of Mori Dream Spaces with klt Cox rings are Mori Dream Spaces with klt Cox ring. This in turn applies to show that projective GIT-quotients of varieties of Fano type are of Fano type; in particular, projective moduli spaces of semistable quiver representations are of Fano type.

We show that direct summands (or more generally, pure images) of klt type singularities are of klt type. As a consequence, we give a different proof of a recent result of Braun, Greb, Langlois and Moraga that reductive quotients of klt type singularities are of klt type.

We prove that the stationary measures for the free-energy increment process for the geometric last passage percolation (LPP) and log-gamma polymer model on a diagonal strip is given by a marginal of a two-layer Gibbs measure with a simple and explicit description. This result is shown subject to certain restrictions on the parameters controlling the weights on the boundary of the strip. However, from this description and an analytic continuation argument we are able to access the stationary measure for all boundary parameters. Taking an intermediate disorder limit of the log-gamma polymer stationary measure in a strip we readily recover (modulo convergence of the polymer to the open KPZ equation, Conjecture 4.2) the conjectural description from (Barraquand, Le Doussal in Europhys. Lett. 137(6):61003, 2022) of the open KPZ stationary measure for all choices of boundary parameters (u,vin mathbb{R}) (thus going beyond the restriction (u+vgeq 0) from (Corwin, Knizel in Stationary measure for the open KPZ equation, 2021, arXiv:2103.12253)).

In this paper, we explore the version of Hairer’s regularity structures based on a greedier index set than trees, as introduced in (Otto et al. in A priori bounds for quasi-linear SPDEs in the full sub-critical regime, 2021, arXiv:2103.11039) and algebraically characterized in (Linares et al. in Comm. Am. Math. Soc. 3:1–64, 2023). More precisely, we construct and stochastically estimate the renormalized model postulated in (Otto et al. in A priori bounds for quasi-linear SPDEs in the full sub-critical regime, 2021, arXiv:2103.11039), avoiding the use of Feynman diagrams but still in a fully automated, i. e. inductive way. This is carried out for a class of quasi-linear parabolic PDEs driven by noise in the full singular but renormalizable range. We assume a spectral gap inequality on the (not necessarily Gaussian) noise ensemble. The resulting control on the variance of the model naturally complements its vanishing expectation arising from the BPHZ-choice of renormalization. We capture the gain in regularity on the level of the Malliavin derivative of the model by describing it as a modelled distribution. Symmetry is an important guiding principle and built-in on the level of the renormalization Ansatz. Our approach is analytic and top-down rather than combinatorial and bottom-up.

We construct a fundamental piece of the boundary of the maximal globally hyperbolic development (MGHD) of Cauchy data for the multi-dimensional compressible Euler equations, which is necessary for the local shock development problem. For an open set of compressive and generic (H^{7}) initial data, we construct unique (H^{7}) solutions to the Euler equations in the maximal spacetime region below a given time-slice, beyond the time of the first singularity; at any point in this spacetime, the solution can be smoothly and uniquely computed by tracing both the fast and slow acoustic characteristic surfaces backward-in-time, until reaching the Cauchy data prescribed along the initial time-slice. The future temporal boundary of this spacetime region is a singular hypersurface, containing the union of three sets: first, a co-dimension-2 surface of “first singularities” called the pre-shock; second, a downstream hypersurface called the singular set emanating from the pre-shock, on which the Euler solution experiences a continuum of gradient catastrophes; third, an upstream hypersurface consisting of a Cauchy horizon emanating from the pre-shock, which the Euler solution cannot reach. We develop a new geometric framework for the description of the acoustic characteristic surfaces which is based on the Arbitrary Lagrangian Eulerian (ALE) framework, and combine this with a new type of differentiated Riemann variables which are linear combinations of gradients of velocity, sound speed, and the curvature of the fast acoustic characteristic surfaces. With these new variables, we establish uniform (H^{7}) Sobolev bounds for solutions to the Euler equations without derivative loss and with optimal regularity.

We prove an asymptotic for the second moment of quadratic twists of a modular (L)-function. This was previously known conditionally on GRH by the work of Soundararajan and Young (J. Eur. Math. Soc. 12(5):1097–1116, 2010).

A subset (A) of a group (G) is called product-free if there is no solution to (a=bc) with (a,b,c) all in (A). It is easy to see that the largest product-free subset of the symmetric group (S_{n}) is obtained by taking the set of all odd permutations, i.e. (S_{n} backslash A_{n}), where (A_{n}) is the alternating group. In 1985 Babai and Sós (Eur. J. Comb. 6(2):101–114, 1985) conjectured that the group (A_{n}) also contains a product-free set of constant density. This conjecture was refuted by Gowers (whose result was subsequently improved by Eberhard), still leaving the long-standing problem of determining the largest product-free subset of (A_{n}) wide open. We solve this problem for large (n), showing that the maximum size is achieved by the previously conjectured extremal examples, namely families of the form (left { pi :,pi (x)in I, pi (I)cap I=varnothing right } ) and their inverses. Moreover, we show that the maximum size is only achieved by these extremal examples, and we have stability: any product-free subset of (A_{n}) of nearly maximum size is structurally close to an extremal example. Our proof uses a combination of tools from Combinatorics and Non-abelian Fourier Analysis, including a crucial new ingredient exploiting some recent theory developed by Filmus, Kindler, Lifshitz and Minzer for global hypercontractivity on the symmetric group.

For the Schrödinger equation with a cubic-quintic, focusing-focusing nonlinearity in one space dimension, this article proves the local asymptotic completeness of the family of small standing solitary waves under even perturbations in the energy space. For this model, perturbative of the integrable cubic Schrödinger equation for small solutions, the linearized equation around a small solitary wave has an internal mode, whose contribution to the dynamics is handled by the Fermi golden rule.

In this paper we prove the conjecture claiming that, over a flexible field, isotropic Chow groups coincide with numerical Chow groups (with ({Bbb {F}}_{p})-coefficients). This shows that Isotropic Chow motives coincide with Numerical Chow motives. In particular, homs between such objects are finite groups and ⊗ has no zero-divisors. It provides a large supply of new points for the Balmer spectrum of the Voevodsky motivic category. We also prove the Morava K-theory version of the above result, which permits to construct plenty of new points for the Balmer spectrum of the Morel-Voevodsky ({mathbb{A}}^{1})-stable homotopy category. This substantially improves our understanding of the mentioned spectra whose description is a major open problem.