We show that any Weyl group orbit of weights for the Tannakian group of semisimple holonomic 𝒟-modules on an abelian variety is realized by a Lagrangian cycle on the cotangent bundle. As applications we discuss a weak solution to the Schottky problem in genus five, an obstruction for the existence of summands of subvarieties on abelian varieties, and a criterion for the simplicity of the arising Lie

We compute a special case of base change of certain supercuspidal representations from a ramified unitary group to a general linear group, both defined over a p-adic field of odd residual characteristic. In this special case, we require the given supercuspidal representation to contain a skew maximal simple stratum, and the field datum of this stratum to be of maximal degree, tamely ramified over the

Journal Name: Journal für die reine und angewandte Mathematik Volume: 2021 Issue: 770 Pages: i-iv

Firstly, we confirm a conjecture asserting that any compact Kähler manifold N with Ric⊥>0 must be simply-connected by applying a new viscosity consideration to Whitney’s comass of (p,0)-forms. Secondly we prove the projectivity and the rational connectedness of a Kähler manifold of complex dimension n under the condition Rick>0 (for some k∈{1,…,n}, with Ricn being the Ricci curvature), generalizing

In this paper, we study a single-valued integration pairing between differential forms and dual differential forms which subsumes some classical constructions in mathematics and physics. It can be interpreted as a p-adic period pairing at the infinite prime. The single-valued integration pairing is defined by transporting the action of complex conjugation from singular to de Rham cohomology via the

We show that two properly embedded self-shrinkers in Euclidean space that are sufficiently separated at infinity must intersect at a finite point. The proof is based on a localized version of the Reilly formula applied to a suitable f-harmonic function with controlled gradient. In the immersed case, a new direct proof of the generalized half-space property is also presented.

The determinantal variety Σp⁢q is defined to be the set of all p×q real matrices with p≥q whose ranks are strictly smaller than q. It is proved that Σp⁢q is a minimal cone in ℝp⁢q and all its strata are regular minimal submanifolds.

A variant of Li–Tam theory, which associates to each end of a complete Riemannian manifold a positive solution of a given Schrödinger equation on the manifold, is developed. It is demonstrated that such positive solutions must be of polynomial growth of fixed order under a suitable scaling invariant Sobolev inequality. Consequently, a finiteness result for the number of ends follows. In the case when

We determine the stability/instability of the tangent bundles of the Fano varieties in a certain class of two orbit varieties, which are classified by Pasquier in 2009. As a consequence, we show that some of these varieties admit unstable tangent bundles, which disproves a conjecture on stability of tangent bundles of Fano manifolds.

We develop a theory of Frobenius functors for symmetric tensor categories (STC) 𝒞 over a field 𝒌 of characteristic p, and give its applications to classification of such categories. Namely, we define a twisted-linear symmetric monoidal functor F:𝒞→𝒞⊠Verp, where Verp is the Verlinde category (the semisimplification of Rep𝐤(ℤ/p)); a similar construction of the underlying additive functor appeared

Journal Name: Journal für die reine und angewandte Mathematik Volume: 2020 Issue: 769 Pages: i-iv

We study a cylindrical Lagrangian cobordism group for Lagrangian torus fibres in symplectic manifolds which are the total spaces of smooth Lagrangian torus fibrations. We use ideas from family Floer theory and tropical geometry to obtain both obstructions to and constructions of cobordisms; in particular, we give examples of symplectic tori in which the cobordism group has no non-trivial cobordism

We develop new techniques for studying concentration of Laplace eigenfunctions ϕλ as their frequency, λ, grows. The method consists of controlling ϕλ⁢(x) by decomposing ϕλ into a superposition of geodesic beams that run through the point x. Each beam is localized in phase-space on a tube centered around a geodesic whose radius shrinks slightly slower than λ-12. We control ϕλ⁢(x) by the L2-mass of ϕλ

Journal Name: Journal für die reine und angewandte Mathematik Volume: 2020 Issue: 768 Pages: 183-183

Journal Name: Journal für die reine und angewandte Mathematik Volume: 2020 Issue: 768 Pages: i-iv

This is an investigation into the possible existence and consequences of a Birch–Swinnerton-Dyer-type formula for L-functions of elliptic curves twisted by Artin representations. We translate expected properties of L-functions into purely arithmetic predictions for elliptic curves, and show that these force some peculiar properties of the Tate–Shafarevich group, which do not appear to be tractable

Anderson generating functions have received a growing attention in function field arithmetic in the last years. Despite their introduction by Anderson in the 1980s where they were at the heart of comparison isomorphisms, further important applications, e.g., to transcendence theory have only been discovered recently. The Anderson–Thakur special function interpolates L-values via Pellarin-type identities

Elementary abelian groups are finite groups in the form of A=(ℤ/p⁢ℤ)r for a prime number p. For every integer ℓ>1 and r>1, we prove a non-trivial upper bound on the ℓ-torsion in class groups of every A-extension. Our results are pointwise and unconditional. This establishes the first case where for some Galois group G, the ℓ-torsion in class groups are bounded non-trivially for every G-extension and

This work concerns with the existence and detailed asymptotic analysis of type II singularities for solutions to complete non-compact conformally flat Yamabe flow with cylindrical behavior at infinity. We provide the specific blow-up rate of the maximum curvature and show that the solution converges, after blowing-up around the curvature maximum points, to a rotationally symmetric steady soliton. It

In this paper we investigate the Hausdorff dimension of limit sets of Anosov representations. In this context we revisit and extend the framework of hyperconvex representations and establish a convergence property for them, analogue to a differentiability property. As an application of this convergence, we prove that the Hausdorff dimension of the limit set of a hyperconvex representation is equal

Let M be a simply connected homogeneous three-manifold with isometry group of dimension 4, and let Σ be any compact surface of genus zero immersed in M whose mean, extrinsic and Gauss curvatures satisfy a smooth elliptic relation Φ⁢(H,Ke,K)=0. In this paper we prove that Σ is a sphere of revolution, provided that the unique inextendible rotational surface S in M that satisfies this equation and touches

In their study of local models of Shimura varieties for totally ramified extensions, Pappas and Rapoport posed a conjecture about the reducedness of a certain subscheme of n×n matrices. We give a positive answer to their conjecture in full generality. Our main ideas follow naturally from two of our previous works. The first is our proof of a conjecture of Kreiman, Lakshmibai, Magyar, and Weyman on

Journal Name: Journal für die reine und angewandte Mathematik Volume: 2020 Issue: 767 Pages: i-iv

We give a formula of an asymptotic expansion of a function, provided in the form of a fiber integral around a critical value of a holomorphic map of toric type.

In this paper, we show an energy convexity and thus uniqueness for weakly intrinsic bi-harmonic maps from the unit 4-ball B1⊂ℝ4 into the sphere 𝕊n. In particular, this yields a version of uniqueness of weakly harmonic maps on the unit 4-ball which is new. We also show a version of energy convexity along the intrinsic bi-harmonic map heat flow into 𝕊n, which in particular yields the long-time existence

Journal Name: Journal für die reine und angewandte Mathematik Volume: 2020 Issue: 766 Pages: i-iv

Journal Name: Journal für die reine und angewandte Mathematik Volume: 2020 Issue: 765 Pages: i-iv

We obtain a Bonnet–Myers theorem under a spectral condition: a closed Riemannian (Mn,g) manifold for which the lowest eigenvalue of the Ricci tensor ρ is such that the Schrödinger operator Δ+(n-2)⁢ρ is positive has finite fundamental group. Further, as a continuation of our earlier results, we obtain isoperimetric inequalities from Kato-type conditions on the Ricci curvature. We also obtain the Kato

In 1980, Oniščik [A. L. Oniščik, Totally geodesic submanifolds of symmetric spaces, Geometric methods in problems of algebra and analysis. Vol. 2, Yaroslav. Gos. Univ., Yaroslavl’ 1980, 64–85, 161] introduced the index of a Riemannian symmetric space as the minimal codimension of a (proper) totally geodesic submanifold. He calculated the index for symmetric spaces of rank ≤2, but for higher rank it

We determine the image and the fibers for solvable base change.

We study the behaviour of the Arakelov metric on a smooth curve under semistable degeneration. The final result is a complicated formula involving the local discriminants of the singularities, and the graph governing the degeneration.

We calculate the direct sum of the mod-two cohomology of all alternating groups, with both cup and transfer product structures, which in particular determines the additive structure and ring structure of the cohomology of individual groups. We show that there are no nilpotent elements in the cohomology rings of individual alternating groups. We calculate the action of the Steenrod algebra and discuss

We construct the ancient solutions of the hypersurface flows in Euclidean spaces studied by B. Andrews in 1994.

We construct smooth rational real algebraic varieties of every dimension ≥4 which admit infinitely many pairwise non-isomorphic real forms.

We formulate and prove a profinite rigidity theorem for the twisted Alexander polynomials up to several types of finite ambiguity. We also establish torsion growth formulas of the twisted homology groups in a ℤ-cover of a 3-manifold with use of Mahler measures. We examine several examples associated to Riley’s parabolic representations of two-bridge knot groups and give a remark on hyperbolic volumes

In this paper, we construct virtual cycles on moduli spaces of solutions to the perturbed gauged Witten equation over a fixed smooth r-spin curve, under the framework of [G. Tian and G. Xu, Analysis of gauged Witten equation, J. reine angew. Math. 740 (2018), 187–274]. Together with the wall-crossing formula proved in the companion paper [G. Tian and G. Xu, A wall-crossing formula for the correlation

Let f:ℂ→X be a transcendental holomorphic curve into a complex projective manifold X. Let L be a very ample line bundle on X. Let s be a very generic holomorphic section of L and D the zero divisor given by s. We prove that the geometric defect of D (defect of truncation 1) with respect to f is zero. We also prove that f almost misses general enough analytic subsets on X of codimension 2.

In this paper we study the structure of the pointed-Gromov–Hausdorff limits of sequences of Ricci shrinkers. We define a regular-singular decomposition following the work of Cheeger–Colding for manifolds with a uniform Ricci curvature lower bound, and prove that the regular part of any non-collapsing Ricci shrinker limit space is strongly convex, inspired by Colding–Naber’s original idea of parabolic

Let X be a compact Kähler manifold. Given a big cohomology class {θ}, there is a natural equivalence relation on the space of θ-psh functions giving rise to 𝒮⁢(X,θ), the space of singularity types of potentials. We introduce a natural pseudo-metric d𝒮 on 𝒮⁢(X,θ) that is non-degenerate on the space of model singularity types and whose atoms are exactly the relative full mass classes. In the presence

Journal Name: Journal für die reine und angewandte Mathematik Volume: 2020 Issue: 764 Pages: i-iv

We apply the local removable singularity theorem for minimal laminations [W. H. Meeks III, J. Pérez and A. Ros, Local removable singularity theorems for minimal laminations, J. Differential Geom. 103 (2016), no. 2, 319–362] and the local picture theorem on the scale of topology [W. H. Meeks III, J. Pérez and A. Ros, The local picture theorem on the scale of topology, J. Differential Geom. 109 (2018)

We construct constant mean curvature surfaces in euclidean space with genus zero and n ends asymptotic to Delaunay surfaces using the DPW method.

We give asymptotics for the level set equation for mean curvature flow on a convex domain near the point where it attains a maximum. It is known that solutions are not necessarily C3, and we recover this result and construct non-smooth solutions which are C3. We also construct solutions having prescribed behavior near the maximum. We do this by analyzing the asymptotics for rescaled mean curvature

We introduce a class of almost homogeneous varieties contained in the class of spherical varieties and containing horospherical varieties as well as complete symmetric varieties. We develop Kähler geometry on these varieties, with applications to canonical metrics in mind, as a generalization of the Guillemin–Abreu–Donaldson geometry of toric varieties. Namely we associate convex functions with Hermitian

In this paper we prove a gap theorem for Kähler manifolds with nonnegative orthogonal bisectional curvature and nonnegative Ricci curvature, which generalizes an earlier result of the first author [L. Ni, An optimal gap theorem, Invent. Math. 189 2012, 3, 737–761]. We also prove a Liouville theorem for plurisubharmonic functions on such a manifold, which generalizes a previous result of L.-F. Tam and

The Dehn function measures the area of minimal discs that fill closed curves in a space; it is an important invariant in analysis, geometry, and geometric group theory. There are several equivalent ways to define the Dehn function, varying according to the type of disc used. In this paper, we introduce a new definition of the Dehn function and use it to prove several theorems. First, we generalize

We construct solutions for the fractional Yamabe problem that are singular at a prescribed number of isolated points. This seems to be the first time that a gluing method is successfully applied to a non-local problem in order to construct singular solutions. There are two main steps in the proof: to construct an approximate solution by gluing half bubble towers at each singular point, and then an

Journal Name: Journal für die reine und angewandte Mathematik Volume: 2020 Issue: 763 Pages: i-iv

We give an affirmative solution to a conjecture of Cheng proposed in 1979 which asserts that the Bergman metric of a smoothly bounded strongly pseudoconvex domain in ℂn,n≥2, is Kähler–Einstein if and only if the domain is biholomorphic to the ball. We establish a version of the classical Kerner theorem for Stein spaces with isolated singularities which has an immediate application to construct a hyperbolic

In this paper we prove a result on the effective generation of pluri-canonical linear systems on foliated surfaces of general type. Fix a function P:ℤ≥0→ℤ, then there exists an integer N>0 such that if (X,ℱ) is a canonical or nef model of a foliation of general type with Hilbert polynomial χ⁢(X,𝒪X⁢(m⁢Kℱ))=P⁢(m) for all m∈ℤ≥0, then |m⁢Kℱ| defines a birational map for all m≥N.

Given a stability condition on a smooth projective variety X, we construct a family of stability conditions on X×C, where C is a smooth projective curve. In particular, this gives the existence of stability conditions on arbitrary products of curves. The proof uses, by following an idea of Toda, the positivity lemma established by Bayer and Macrì and weak stability conditions on the Abramovich-Polishchuk

The Cremona group is the group of birational transformations of the complex projective plane. In this paper we classify its subgroups that consist only of elliptic elements using elementary model theory. This yields in particular a description of the structure of torsion subgroups. As an application, we prove the Tits alternative for arbitrary subgroups of the Cremona group, generalizing a result of

In recent years, a number of papers have been devoted to the study of zeros of period polynomials of modular forms. In the present paper, we study cohomological analogues of the Eichler–Shimura period polynomials corresponding to higher L-derivatives. We state a general conjecture about the locations of the zeros of the full and odd parts of the polynomials, in analogy with the existing literature

We correct a heavy misprint in our paper.

We show that any proper Lie groupoid admits a compatible real analytic structure. Our proof hinges on a Weyl unitary trick of sorts for appropriate local holomorphic groupoids.

In this paper we obtain rigidity results for a non-constant entire solution u of the Allen–Cahn equation in ℝn, whose level set {u=0} is contained in a half-space. If n≤3, we prove that the solution must be one-dimensional. In dimension n≥4, we prove that either the solution is one-dimensional or stays below a one-dimensional solution and converges to it after suitable translations. Some generalizations

We study deformations of rational curves and their singularities in positive characteristic. We use this to prove that if a smooth and proper surface in positive characteristic p is dominated by a family of rational curves such that one member has all δ-invariants (resp. Jacobian numbers) strictly less than 12⁢(p-1) (resp. p), then the surface has negative Kodaira dimension. We also prove similar,

The Plateau–Douglas problem asks to find an area minimizing surface of fixed or bounded genus spanning a given finite collection of Jordan curves in Euclidean space. In the present paper we solve this problem in the setting of proper metric spaces admitting a local quadratic isoperimetric inequality for curves. We moreover obtain continuity up to the boundary and interior Hölder regularity of solutions

The main goal of this article is to understand the trace properties of nonlocal minimal graphs in ℝ3, i.e. nonlocal minimal surfaces with a graphical structure.

Unlike Hochschild (co)homology and K-theory, global and dominant dimensions of algebras are far from being invariant under derived equivalences in general. We show that, however, global dimension and dominant dimension are derived invariant when restricting to a class of algebras with anti-automorphisms preserving simples. Such anti-automorphisms exist for all cellular algebras and in particular for