Showing papers in "Annales de l'Institut Fourier in 2021"

Sparse bounds for maximal rough singular integrals via the fourier transform

Abstract: Juan de la Cierva-Formaci\'on 2015 FJCI-2015-24547, Severo Ochoa Program SEV-2013-0323, Basque Government BERC Program 2014-2017

Whittaker supports for representations of reductive groups

Abstract: Let $F$ be either $\mathbb{R}$ or a finite extension of $\mathbb{Q}_p$, and let $G$ be a finite central extension of the group of $F$-points of a reductive group defined over $F$. Also let $\pi$ be a smooth representation of $G$ (Frechet of moderate growth if $F=\mathbb{R}$). For each nilpotent orbit $\mathcal{O}$ we consider a certain Whittaker quotient $\pi_{\mathcal{O}}$ of $\pi$. We define the Whittaker support WS$(\pi)$ to be the set of maximal $\mathcal{O}$ among those for which $\pi_{\mathcal{O}} eq 0$. In this paper we prove that all $\mathcal{O}\in\mathrm{WS}(\pi)$ are quasi-admissible nilpotent orbits, generalizing some of the results in [Moe96,JLS16]. If $F$ is $p$-adic and $\pi$ is quasi-cuspidal then we show that all $\mathcal{O}\in\mathrm{WS}(\pi)$ are $F$-distinguished, i.e. do not intersect the Lie algebra of any proper Levi subgroup of $G$ defined over $F$. We also give an adaptation of our argument to automorphic representations, generalizing some results from [GRS03,Shen16,JLS16,Cai] and confirming some conjectures from [Ginz06]. Our methods are a synergy of the methods of the above-mentioned papers, and of our preceding paper [GGS17].

Bilinear pseudo-differential operators with exotic symbols

Abstract: The boundedness from $H^p \times L^2$ to $L^r$, $1/p+1/2=1/r$, and from $H^p \times L^{\infty}$ to $L^p$ of bilinear pseudo-differential operators is proved under the assumption that their symbols are in the bilinear H\"ormander class $BS^m_{\rho,\rho}$, $0 \le \rho <1$, of critical order $m$, where $H^p$ is the Hardy space. This combined with the previous results of the same authors establishes the sharp boundedness from $H^p \times H^q$ to $L^r$, $1/p+1/q=1/r$, of those operators in the full range $0< p, q \le \infty$, where $L^r$ is replaced by $BMO$ if $r=\infty$.

Stratified spaces and synthetic Ricci curvature bounds

Abstract: We prove that a compact stratied space satises the Riemannian curvature-dimension condition RCD(K, N) if and only if its Ricci tensor is bounded below by K ∈ R on the regular set, the cone angle along the stratum of codimension two is smaller than or equal to 2π and its dimension is at most equal to N. This gives a new wide class of geometric examples of metric measure spaces satisfying the RCD(K, N) curvature-dimension condition, including for instance spherical suspensions, orbifolds, Kahler-Einstein manifolds with a divisor, Einstein manifolds with conical singularities along a curve. We also obtain new analytic and geometric results on stratied spaces, such as Bishop-Gromov volume inequality, Laplacian comparison, Levy-Gromov isoperimetric inequality.

A Ginzburg-Landau model with topologically induced free discontinuities

Abstract: We study a variational model which combines features of the Ginzburg-Landau model in 2D and of the Mumford-Shah functional. As in the classical Ginzburg-Landau theory, a prescribed number of point vortices appear in the small energy regime; the model allows for discontinuities, and the energy penalizes their length. The novel phenomenon here is that the vortices have a fractional degree 1/m with m 2 prescribed. Those vortices must be connected by line discontinuities to form clusters of total integer degrees. The vortices and line discontinuities are therefore coupled through a topological constraint. As in the Ginzburg-Landau model, the energy is parameterized by a small length scale e > 0. We perform a complete Γ-convergence analysis of the model as e ↓ 0 in the small energy regime. We then study the structure of minimizers of the limit problem. In particular, we show that the line discontinuities of a minimizer solve a variant of the Steiner problem. We finally prove that for small e > 0, the minimizers of the original problem have the same structure away from the limiting vortices.

Local metrics of the Gaussian free field

Abstract: We introduce the concept of a local metric of the Gaussian free field (GFF) $h$, which is a random metric coupled with $h$ in such a way that it depends locally on $h$ in a certain sense. This definition is a metric analog of the concept of a local set for $h$. We establish general criteria for two local metrics of the same GFF $h$ to be bi-Lipschitz equivalent to each other and for a local metric to be a.s. determined by $h$. Our results are used in subsequent works which prove the existence, uniqueness, and basic properties of the $\gamma$-Liouville quantum gravity (LQG) metric for all $\gamma \in (0,2)$, but no knowledge of LQG is needed to understand this paper.

Isoperimetric inequalities, shapes of Følner sets and groups with Shalom’s property ${H_{\protect \mathrm{FD}}}$

Abstract: — We prove an isoperimetric inequality for groups. As an application we show that any Grigrochuk group of intermediate growth has at least exponential Følner function. As another application, we obtain lower bounds on Følner functions in various nilpotent-by-cyclic groups. Under a regularity assumption, we obtain a characterization of Følner functions of these groups. As a further application, we evaluate the asymptotics of the Følner function of Sym(Z) o Z. We study examples of groups with Shalom’s property HFD among nilpotent-bycyclic groups. We show that there exist lacunary hyperbolic groups with property HFD. We find groups with property HFD, which are direct products of lacunary hyperbolic groups and have arbitrarily large Følner functions. Résumé. — Nous prouvons une inégalité isopérimétrique pour les groupes. En application, nous montrons que la fonction de Følner de tout groupe de Grigorchuk à croissance intermédiaire est au moins exponentielle. En tant qu’autre application, nous obtenons des bornes inférieures sur les fonctions de Følner dans divers groupes nilpotents par cycliques. Sous une hypothèse de régularité, nous obtenons une caractérisation des fonctions de Følner de ces groupes. Comme autre application, nous évaluons le comportement asymptotique de la fonction de Følner de Sym(Z) o Z. Nous étudions des exemples de groupes avec la propriété de Shalom HFD parmi les extensions d’un groupe nilpotent par un groupe cyclique. Nous montrons qu’il existe des groupes hyperboliques lacunaires avec la propriété HFD. Nous trouvons des groupes avec la propriété HFD, qui sont des produits directs de groupes hyperbolique lacunaires et ont des fonctions Følner arbitrairement grandes.

On the Effective Freeness of the Direct Images of Pluricanonical Bundles

Abstract: We give effective bounds on the generation of pushforwards of log-pluricanonical bundles twisted by ample line bundles. This gives a partial answer to a conjecture proposed by Popa and Schnell. We prove two types of statements: first, more in the spirit of the general conjecture, we show generic global generation with predicted bound when the dimesnion of the variety if less than 4 and more generally, with a quadratic Angehrn-Siu type bound. Secondly, assuming that the relative canonical bundle is relatively semi-ample, we make a very precise statement. In particular, when the morphism is smooth, it solves the conjecture with the same bounds, for certain pluricanonical bundles.

Effective operators for Robin eigenvalues in domains with corners

Abstract: We study the eigenvalues of the Laplacian with a strong attractive Robin boundary condition in curvilinear polygons. It was known from previous works that the asymptotics of several first eigenvalues is essentially determined by the corner openings, while only rough estimates were available for the next eigenvalues. Under some geometric assumptions, we go beyond the critical eigenvalue number and give a precise asymptotics of any individual eigenvalue by establishing a link with an effective Schr\"odinger-type operator on the boundary of the domain with boundary conditions at the corners.

On the non-vanishing of $p$ -adic heights on CM abelian varieties, and the arithmetic of Katz $p$ -adic $L$ -functions

Abstract: Let $B$ be a simple CM abelian variety over a CM field $E$, $p$ a rational prime. Suppose that $B$ has potentially ordinary reduction above $p$ and is self-dual with root number $-1$. Under some further conditions, we prove the generic non-vanishing of (cyclotomic) $p$-adic heights on $B$ along anticyclotomic $\mathbb{Z}_{p}$-extensions of $E$. This provides evidence towards Schneider's conjecture on the non-vanishing of $p$-adic heights. For CM elliptic curves over $\mathbb{Q}$, the result was previously known as a consequence of work of Bertrand, Gross--Zagier and Rohrlich in the 1980s. Our proof combines non-vanishing results for Katz $p$-adic $L$-functions and a Gross--Zagier formula relating the latter to families of rational points on $B$.

Fixed-point spectrum for group actions by affine isometries on $L_{p}$ -spaces

Abstract: The fixed-point spectrum of a locally compact second countable group G on lp is defined to be the set of real numbers p such that every action by affine isometries of G on lp admits a fixed-point. We show that this set is either empty, or is equal to a set of one of the following forms : [1,\\pc[, [1,\\pc[\\{2} for some \\pc<\\infty or \\pc=\\infty, or [1,\\pc], [1,\\pc]\\{2} for some pc

On the joint spectral radius for isometries of non-positively curved spaces and uniform growth

Abstract: We recast the notion of joint spectral radius in the setting of groups acting by isometries on non-positively curved spaces and give geometric versions of results of Berger-Wang and Bochi valid for $\delta$-hyperbolic spaces and for symmetric spaces of non-compact type. This method produces nice hyperbolic elements in many classical geometric settings. Applications to uniform growth are given, in particular a new proof and a generalization of a theorem of Besson-Courtois-Gallot.

On resonances generated by conic diffraction

Abstract: We describe the resonances closest to the real axis generated by diffraction of waves among cone points on a manifold with Euclidean ends These resonances lie asymptotically evenly spaced along a curve of the form $$\frac{\Im \lambda}{\log \left |\Re \lambda\right |}= - u;$$ here $ u=(n-1)/2 L_0$ where $n$ is the dimension and $L_0$ is the length of the longest geodesic connecting two cone points Moreover there are asymptotically no resonances below this curve and above the curve $$ \frac{\Im \lambda}{\log \left |\Re \lambda\right |}= -\Lambda $$ for a fixed $\Lambda> u$

Analytic properties of approximate lattices

Abstract: We introduce a notion of cocycle-induction for strong uniform approximate lattices in locally compact second countable groups and use it to relate (relative) Kazhdan- and Haagerup-type of approximate lattices to the corresponding properties of the ambient locally compact groups. Our approach applies to large classes of uniform approximate lattices (though not all of them) and is flexible enough to cover the $L^p$-versions of Property (FH) and a-(FH)-menability as well as quasified versions thereof a la Burger--Monod and Ozawa.

Seshadri constants and Grassmann bundles over curves

Abstract: Let $X$ be a smooth complex projective curve, and let $E$ be a vector bundle on $X$ which is not semistable. For a suitably chosen integer $r$, let $\text{Gr}(E)$ be the Grassmann bundle over $X$ that parametrizes the quotients of the fibers of $E$ of dimension $r$. Assuming some numerical conditions on the Harder-Narasimhan filtration of $E$, we study Seshadri constants of ample line bundles on $\text{Gr}(E)$. In many cases, we give the precise value of Seshadri constant. Our results generalize various known results for ${\rm rank}(E)=2$.

Local foliation of manifolds by surfaces of willmore type

Abstract: We show the existence of a local foliation of a three dimensional Riemannian manifold by critical points of the Willmore functional subject to a small area constraint around non-degenerate critical points of the scalar curvature. This adapts a method developed by Rugang Ye to construct foliations by surfaces of constant mean curvature.

Inversion of Rankin–Cohen operators via Holographic Transform

Abstract: The analysis of branching problems for restriction of representations brings the concept of symmetry breaking transform and holographic transform. Symmetry breaking operators decrease the number of variables in geometric models, whereas holographic operators increase it. Various expansions in classical analysis can be interpreted as particular occurrences of these transforms. From this perspective, we investigate two remarkable families of differential operators: the Rankin-Cohen operators and the holomorphic Juhl conformally covariant operators. Then we establish for the corresponding symmetry breaking transforms the Parseval-Plancherel type theorems and find explicit inversion formulae with integral expression of holographic operators. The proof uses the F-method which provides a duality between symmetry breaking operators in the holomorphic model and holographic operators in the $L^2$-model, leading us to deep links between special orthogonal polynomials and branching laws for infinite-dimensional representations of real reductive Lie groups.

Embeddings of finite groups in $B_n/\Gamma_k(P_n)$ for $k=2, 3$

Abstract: Let $n \geq 3$. In this paper, we study the problem of whether a given finite group $G$ embeds in a quotient of the form $B_n/\Gamma_k(P_n)$, where $B_n$ is the $n$-string Artin braid group, $k \in \{2, 3\}$, and $\{\Gamma_l(P_n)\}_{l\in N}$ is the lower central series of the $n$-string pure braid group $P_n$. Previous results show that a necessary condition for such an embedding to exist is that $|G|$ is odd (resp. is relatively prime with $6$) if $k=2$ (resp. $k=3$). We show that any finite group $G$ of odd order (resp. of order relatively prime with $6$) embeds in $B_{|G|}/\Gamma_2(P_{|G|})$ (resp. in $B_{|G|}/\Gamma_3(P_{|G|})$), where $|G|$ denotes the order of $G$. The result in the case of $B_{|G|}/\Gamma_2(P_{|G|})$ has been proved independently by Beck and Marin. One may then ask whether $G$ embeds in a quotient of the form $B_n/\Gamma_k(P_n)$, where $n < |G|$ and $k \in \{2, 3\}$. If $G$ is of the form $Z_{p^r} \rtimes_{\theta} Z_d$, where the action $\theta$ is injective, $p$ is an odd prime (resp. $p \geq 5$ is prime) $d$ is odd (resp. $d$ is relatively prime with $6$) and divides $p-1$, we show that $G$ embeds in $B_{p^r}/\Gamma_2(P_{p^r})$ (resp. in $B_{p^r}/\Gamma_3(P_{p^r})$). In the case $k=2$, this extends a result of Marin concerning the embedding of the Frobenius groups in $B_n/\Gamma_2(P_n)$, and is a special case of another result of Beck and Marin. Finally, we construct an explicit embedding in $B_9/\Gamma_2(P_9)$ of the two non-Abelian groups of order $27$, namely the semi-direct product $Z_9 \rtimes Z_3$, where the action is given by multiplication by $4$, and the Heisenberg group mod $3$.

Duality of random planar maps via percolation

Abstract: We discuss duality properties of critical Boltzmann planar maps such that the degree of a typical face is in the domain of attraction of a stable distribution with parameter $\alpha\in(1,2]$. We consider the critical Bernoulli bond percolation model on a Boltzmann map in the dilute and generic regimes $\alpha \in (3/2,2]$, and show that the open percolation cluster of the origin is itself a Boltzmann map in the dense regime $\alpha \in (1,3/2)$, with parameter \[\alpha':= \frac{2\alpha+3}{4\alpha-2}.\] This is the counterpart in random planar maps of the duality property $\kappa \leftrightarrow 16/\kappa$ of Schramm--Loewner Evolutions and Conformal Loop Ensembles, recently established by Miller, Sheffield and Werner. As a byproduct, we identify the scaling limit of the boundary of the percolation cluster conditioned to have a large perimeter. The cases of subcritical and supercritical percolation are also discussed. In particular, we establish the sharpness of the phase transition through the tail distribution of the size of the percolation cluster.

Super-expanders and warped cones

Abstract: For a Banach space $X$, we show that any family of graphs quasi-isometric to levels of a warped cone $\mathcal O_\Gamma Y$ is an expander with respect to $X$ if and only if the induced $\Gamma$-representation on $L^2(Y;X)$ has a spectral gap. This provides examples of graphs that are an expander with respect to all Banach spaces of non-trivial type.

Uniform rectifiability and $\varepsilon $ -approximability of harmonic functions in $L^p$

Abstract: Suppose that $E \subset \mathbb{R}^{n+1}$ is a uniformly rectifiable set of codimension $1$. We show that every harmonic function is $\varepsilon$-approximable in $L^p(\Omega)$ for every $p \in (1,\infty)$, where $\Omega := \mathbb{R}^{n+1} \setminus E$. Together with results of many authors this shows that pointwise, $L^\infty$ and $L^p$ type $\varepsilon$-approximability properties of harmonic functions are all equivalent and they characterize uniform rectifiability for codimension $1$ Ahlfors-David regular sets. Our results and techniques are generalizations of recent works of T. Hytonen and A. Rosen and the first author, J. M. Martell and S. Mayboroda.

$C^{*}$ -simplicity of HNN extensions and groups acting on trees

Abstract: The first author is supported by a PhD stipend from the Danish National Research Foundation (DNRF) through the Centre for Symmetry and Deformation at University of Copenhagen.

On the anti-canonical geometry of weak $\protect \mathbb{Q}$ -Fano threefolds II

Abstract: By a canonical (resp. terminal) weak Q-Fano 3-fold we mean a normal projective one with at worst canonical (resp. terminal) singularities on which the anti-canonical divisor is QCartier, nef and big. For a canonical weak Q-Fano 3-fold V , we show that there exists a terminal weak Q-Fano 3-fold X , being birational to V , such that the m-th anti-canonical map defined by | −mKX | is birational for all m ≥ 52. As an intermediate result, we show that for any K-Mori fiber space Y of a canonical weak Q-Fano 3-fold, the m-th anti-canonical map defined by | −mKY | is birational for all m ≥ 52.

Structure of extensions of free Araki–Woods factors

Abstract: We investigate the structure of crossed product von Neumann algebras arising from Bogoljubov actions of countable groups on Shlyakhtenko's free Araki-Woods factors. Among other results, we settle the questions of factoriality and Connes' type classification. We moreover provide general criteria regarding fullness and strong solidity. As an application of our main results, we obtain examples of type ${\rm III_0}$ factors that are prime, have no Cartan subalgebra and possess a maximal amenable abelian subalgebra. We also obtain a new class of strongly solid type ${\rm III}$ factors with prescribed Connes' invariants that are not isomorphic to any free Araki-Woods factors.

Extension of the curvature form of the relative canonical line bundle on families of Calabi–Yau manifolds and applications

Abstract: Given a proper, open, holomorphic map of Kahler manifolds, whose general fibers are Calabi-Yau manifolds, the volume forms for the Ricci-flat metrics induce a hermitian metric on the relative canonical bundle over the regular locus of the family. We show that the curvature form extends as a closed positive current. Consequently the Weil-Petersson metric extends as a positive current. In the projective case, the Weil-Petersson form is known to be the curvature of a certain determinant line bundle, equipped with a Quillen metric. As an application we get that after blowing up the singular locus, the determinant line bundle extends, and the Quillen metric extends as singular hermitian metric, whose curvature is a positive current.

The dual actions, equivariant autoequivalences and stable tilting objects

Abstract: For a finite abelian group action on a linear category, we study the dual action given by the character group acting on the category of equivariant objects. We prove that the groups of equivariant autoequivalences on these two categories are isomorphic. In the triangulated situation, this isomorphism implies that the classifications of stable tilting objects for these two categories are in a natural bijection. We apply these results to stable tilting complexes on weighted projective lines of tubular type.

The lemniscate tree of a random polynomial

Abstract: To each generic complex polynomial $p(z)$ there is associated a labeled binary tree (here referred to as a "lemniscate tree") that encodes the topological type of the graph of $|p(z)|$. The branching structure of the lemniscate tree is determined by the configuration (i.e., arrangement in the plane) of the singular components of those level sets $|p(z)|=t$ passing through a critical point. In this paper, we address the question "How many branches appear in a typical lemniscate tree?" We answer this question first for a lemniscate tree sampled uniformly from the combinatorial class and second for the lemniscate tree arising from a random polynomial generated by i.i.d. zeros. From a more general perspective, these results take a first step toward a probabilistic treatment (within a specialized setting) of Arnold's program of enumerating algebraic Morse functions.

Surface singularities and planar contact structures

Abstract: We prove that if a contact 3-manifold admits an open book decomposition of genus 0, a certain intersection pattern cannot appear in the homology of any of its minimal symplectic fillings, and moreover, fillings cannot contain symplectic surfaces of positive genus. Applying these obstructions to canonical contact structures on links of normal surface singularities, we show that links of isolated singularities of surfaces in the complex 3-space are planar only in the case of $A_n$-singularities. In general, we characterize completely planar links of normal surface singularities (in terms of their resolution graphs); these singularities are precisely rational singularities with reduced fundamental cycle (also known as minimal singularities). We also establish non-planarity of tight contact structures on certain small Seifert fibered L-spaces and of contact structures arising from the Boothby--Wang construction applied to surfaces of positive genus. Additionally, we prove that every finitely presented group is the fundamental group of a Lefschetz fibration with planar fibers.

Slant products on the Higson-Roe exact sequence

Abstract: We construct a slant product $/ \colon \mathrm{S}_p(X \times Y) \times \mathrm{K}_{1-q}(\mathfrak{c}^{\mathrm{red}}Y) \to \mathrm{S}_{p-q}(X)$ on the analytic structure group of Higson and Roe and the K-theory of the stable Higson corona of Emerson and Meyer. The latter is the domain of the co-assembly map $\mu^\ast \colon \mathrm{K}_{1-\ast}(\mathfrak{c}^{\mathrm{red}}Y) \to \mathrm{K}^\ast(Y)$. We obtain such products on the entire Higson--Roe sequence. They imply injectivity results for external product maps. Our results apply to products with aspherical manifolds whose fundamental groups admit coarse embeddings into Hilbert space. To conceptualize the class of manifolds where this method applies, we say that a complete $\mathrm{spin}^{\mathrm{c}}$-manifold is Higson-essential if its fundamental class is detected by the co-assembly map. We prove that coarsely hypereuclidean manifolds are Higson-essential. We draw conclusions for positive scalar curvature metrics on product spaces, particularly on non-compact manifolds. We also obtain equivariant versions of our constructions and discuss related problems of exactness and amenability of the stable Higson corona.

Formal classification of two-dimensional neighborhoods of genus $g\ge 2$ curves with trivial normal bundle

Abstract: In this paper we study the formal classication of two-dimensional neighborhoods of genus g ≥ 2 curves with trivial normal bundle. We first construct formal foliations on such neighborhoods with holonomy vanishing along many loops, then give the formal/analytic classication of neighborhoods equipped with two foliations, and finally put this together to obtain a description of the space of neighborhoods up to formal equivalence.