Showing papers in "Annales de l'Institut Fourier in 2017"

Uniform k-stability, duistermaat-heckman measures and singularities of pairs

Abstract: The purpose of the present paper is to set up a formalism inspired from non-Archimedean geometry to study K-stability. We rst provide a detailed analysis of Duistermaat-Heckman measures in the context of test congurations, characterizing in par- ticular the trivial case. For any normal polarized variety (or, more generally, polarized pair in the sense of the Minimal Model Program), we introduce and study the non-Archimedean analogues of certain classical functionals in Kahler geometry. These functionals are dened on the space of test congurations, and the Donaldson-Futaki invariant is in particular interpreted as the non-Archimedean version of the Mabuchi functional, up to an explicit error term. Finally, we study in detail the relation between uniform K-stability and sin- gularities of pairs, reproving and strengthening Y. Odaka's results in our formalism. This provides various examples of uniformly K-stable varieties.

Sobolev spaces on graded Lie groups

Abstract: — In this article, we study the Lp-properties of powers of positive Rockland operators and define Sobolev spaces on general graded Lie groups. We establish that the defined Sobolev spaces are independent of the choice of a positive Rockland operator, and that they are interpolation spaces. Although this generalises the case of sub-Laplacians on stratified groups studied by G. Folland in [12], many arguments have to be different since Rockland operators are usually of higher degree than two. We also prove results regarding duality and Sobolev embeddings, together with inequalities of Hardy–Littlewood–Sobolev type and of Gagliardo–Nirenberg type. Résumé. — Dans cet article, nous étudions les propriétés Lp des puissances des opérateurs de Rockland positifs et nous définissons les espaces de Sobolev sur tous les groupes de Lie nilpotents gradués. Nous montrons que les espaces de Sobolev ainsi définis sont indépendants du choix de l’opérateur de Rockland positif et qu’ils sont des espaces d’interpolation. Quoique cela généralise le cas des sous-laplaciens sur les groupes stratifiés étudiés par G. Folland dans [12], plusieurs arguments sont différents car les opérateurs de Rockland sont souvent de degrée plus haut que deux. Nous montrons aussi des résultats concernant la dualité et les injections de Sobolev, ainsi que des inégalités de type Littlewood–Sobolev et de type Gagliardo– Nirenberg.

On the mean curvature flow of grain boundaries

Abstract: Suppose that $\Gamma_0\subset\mathbb R^{n+1}$ is a closed countably $n$-rectifiable set whose complement $\mathbb R^{n+1}\setminus \Gamma_0$ consists of more than one connected component. Assume that the $n$-dimensional Hausdorff measure of $\Gamma_0$ is finite or grows at most exponentially near infinity. Under these assumptions, we prove a global-in-time existence of mean curvature flow in the sense of Brakke starting from $\Gamma_0$. There exists a finite family of open sets which move continuously with respect to the Lebesgue measure, and whose boundaries coincide with the space-time support of the mean curvature flow.

SRB measures for higher dimensional singular partially hyperbolic attractors

Abstract: — We prove the existence and the uniqueness of the SRB measure for any singular hyperbolic attractor in dimension d > 3. The proof does not use Poincaré sectional maps, but uses basic properties of thermodynamical formalism. Résumé. — Nous prouvons que tout attracteur partiellement hyperbolique de dimension finie et avec singularité(s) admet une unique mesure SRB. La preuve utilise des outils simples et généraux du formalisme thermodynamique et ne nécessite pas de recourir à une section de Poincaré.

Highest weight categories and recollements

Abstract: We provide several equivalent descriptions of a highest weight category using recollements of abelian categories. Also, we explain the connection between sequences of standard and exceptional objects.

On short sums of trace functions

Abstract: We consider sums of oscillating functions on intervals in cyclic groups of size close to the square root of the size of the group. We first prove non-trivial estimates for intervals of length slightly larger than this square root (bridging the "Poly\'a-Vinogradov gap" in some sense) for bounded functions with bounded Fourier transforms. We then prove that the existence of non-trivial estimates for ranges slightly below the square-root bound is stable under the discrete Fourier transform, and we give applications related to trace functions over finite fields.

On the convergence of arithmetic orbifolds

Abstract: — We discuss the geometry of some arithmetic orbifolds locally isometric to a product X of real hyperbolic spaces H m of dimension m = 2, 3, and prove that certain sequences of non-compact orbifolds are convergent to X in a geometric (" Benjamini–Schramm ") sense for low-dimensional cases (when X is equal to H 2 × H 2 or H 3). We also deal with sequences of maximal arithmetic three–dimensional hyperbolic lattices defined over a quadratic or cubic field. A motivating application is the study of Betti numbers of Bianchi groups. Resume. — Cet article est consacre a l'etude de la geometrie globale de certaines orbi-varietes localement isometriques a un produit d'espaces tridimen-sionnels et de plans hyperboliques. On demontre que pour les peties dimensions (pour l'espace ou le plan hyperbolique, ou un produit de plans hyperboliques) cer-taines suites de telles orbi-varietes non-compactes de volume fini convergent vers l'espace symetrique en un sens geometrique precis (« convergence de Benjamini– Schramm »). On traite aussi le cas des reseaux arithmetiques maximaux en dimension trois dont les corps de traces sont quadratiques ou cubiques. Une des princi-pales motivations est d'etudier l'asymptotique des nombres de Betti des groupes de Bianchi.

Strong scarring of logarithmic quasimodes

Abstract: We consider a semiclassical (pseudo)differential operator on a compact surface (M,g), such that the Hamiltonian flow generated by its principal symbol admits a hyperbolic periodic orbit γ at some energy E0. For any ϵ>0, we then explicitly construct families of quasimodes of this operator, satisfying an energy width of order ϵh|logh| in the semiclassical limit, but which still exhibit a "strong scar" on the orbit γ, i.e. that these states have a positive weight in any microlocal neighbourhood of γ. We pay attention to optimizing the constants involved in the estimates. This result generalizes a recent result of Brooks \cite{Br13} in the case of hyperbolic surfaces. Our construction, inspired by the works of Vergini et al. in the physics literature, relies on controlling the propagation of Gaussian wavepackets up to the Ehrenfest time.

The Grunwald problem and approximation properties for homogeneous spaces

Abstract: Given a group $G$ and a number field $K$, the Grunwald problem asks whether given field extensions of completions of $K$ at finitely many places can be approximated by a single field extension of $K$ with Galois group G. This can be viewed as the case of constant groups $G$ in the more general problem of determining for which $K$-groups $G$ the variety $\mathrm{SL}_n/G$ has weak approximation. We show that away from an explicit set of bad places both problems have an affirmative answer for iterated semidirect products with abelian kernel. Furthermore, we give counterexamples to both assertions at bad places. These turn out to be the first examples of transcendental Brauer-Manin obstructions to weak approximation for homogeneous spaces.

A curvature formula associated to a family of pseudoconvex domains

Abstract: We shall give a definition of the curvature operator for a family of weighted Bergman spaces {H-t} associated to a smooth family of smoothly bounded strongly pseudoconvex domains {D-t}. In order to study the "boundary term" in the curvature operator, we shall introduce the notion of geodesic curvature for the associated family of boundaries {delta D-t} As an application, we get a variation formula for the norms of Bergman projections of currents with compact support. A flatness criterion for {H(t)1 and its applications to triviality of fibrations are also given in this paper.

The Li-Yau inequality and applications under a curvature-dimension condition

Abstract: We prove a global Li-Yau inequality for a general Markov semigroup under a curvature-dimension condition. This inequality is stronger than all classical Li-Yau type inequalities known to us. On a Riemannian manifold, it is equivalent to a new parabolic Harnack inequality, both in negative and positive curvature, giving new subsequents bounds on the heat kernel of the semigroup. Under positive curvature we moreover reach ultracontractive bounds by a direct and robust method.

On the regularity problem of complex Monge–Ampere equations with conical singularities

Abstract: In the category of metrics with conical singularities along a smooth divisor with angle in $(0, 2\pi)$, we show that locally defined weak solutions ($C^{1,1}-$solutions) to the Kahler-Einstein equations actually possess maximum regularity, which means the metrics are actually Holder continuous in the singular polar coordinates. This shows the weak Kahler-Einstein metrics constructed by Guenancia-Paun \cite{GP}, and independently by Yao \cite{GT}, are all actually strong-conical Kahler-Einstein metrics. The key step is to establish a Liouville-type theorem for weak-conical Kahler-Ricci flat metrics defined over $\C^{n}$, which depends on a Calderon-Zygmund theory in the conical setting.

Reconstruction formulas for X-ray transforms in negative curvature

Abstract: We give reconstruction formulas inverting the geodesic X-ray transform over functions (call it I_0) and solenoidal vector fields on surfaces with negative curvature and strictly convex boundary. These formulas generalize the Pestov-Uhlmann formulas in [PeUh] (established for simple surfaces) to cases allowing geodesics with infinite length on surfaces with trapping. Such formulas take the form of Fredholm equations, where the analysis of error operators requires deriving new estimates for the normal operator \Pi_0 = I_0^*I_0. Numerical examples are provided at the end.

The Hopf algebra of finite topologies and mould composition

Abstract: We exhibit an internal coproduct on the Hopf algebra of finite topologies recently defined by the second author, C. Malvenuto and F. Patras, dual to the composition of "quasi-ormoulds", which are the natural version of J. Ecalle's moulds in this setting. All these results are displayed in the linear species formalism.

Bounded negativity, Harbourne constants and transversal arrangements of curves

Abstract: The Bounded Negativity Conjecture predicts that for every complex projective surface $X$ there exists a number $b(X)$ such that $C^2\geq -b(X)$ holds for all reduced curves $C\subset X$. For birational surfaces $f:Y\to X$ there have been introduced certain invariants (Harbourne constants) relating to the effect the numbers $b(X)$, $b(Y)$ and the complexity of the map $f$. These invariants have been studied previously when $f$ is the blowup of all singular points of an arrangement of lines in ${\mathbb P}^2$, of conics and of cubics. In the present note we extend these considerations to blowups of ${\mathbb P}^2$ at singular points of arrangements of curves of arbitrary degree $d$. We also considerably generalize and modify the approach witnessed so far and study transversal arrangements of sufficiently positive curves on arbitrary surfaces with the non-negative Kodaira dimension.

Bergman Kernels for a sequence of almost Kähler–Ricci solitons

Abstract: Fundamental Research Funds for the Central Universities; NSFC [11501501, 11271022, 11331001]

Random trees constructed by aggregation

Abstract: We study a general procedure that builds random R-trees by gluing recursively a new branch on a uniform point of the pre-existing tree. The aim of this paper is to see how the asymptotic behavior of the sequence of lengths of branches influences some geometric properties of the limiting tree, such as compactness and Hausdorff dimension. In particular, when the sequence of lengths of branches behaves roughly like n −α for some α ∈ (0, 1], we show that the limiting tree is a compact random tree of Hausdorff dimension α −1 . This encompasses the famous construction of the Brownian tree of Aldous. When α > 1, the limiting tree is thinner and its Hausdorff dimension is always 1. In that case, we show that α −1 corresponds to the dimension of the set of leaves of the tree.

On evil Kronecker sequences and lacunary trigonometric products

Abstract: — An important result of Weyl states that for every sequence (nk)k>1 of distinct positive integers the sequence of fractional parts of (nkα)k>1 is u.d. mod 1 for almost all α. However, in this general case it is usually extremely difficult to measure the speed of convergence of the empirical distribution of ({n1α}, . . . , {nNα}) towards the uniform distribution. In this paper we investigate the case when (nk)k>1 is the sequence of evil numbers, that is the sequence of non-negative integers having an even sum of digits in base 2. We utilize a connection with lacunary trigonometric products ∏L `=0 ∣∣sinπ2`α∣∣, and by giving sharp metric estimates for such products we derive sharp metric estimates for exponential sums of (nkα)k>1 and for the discrepancy of ({nkα})k>1 . Furthermore, we provide some explicit examples of numbers α for which we can give estimates for the discrepancy of ({nkα})k>1. Résumé. — Un résultat important de Weyl nous dit que pour chaque suite (nk)k>1 de nombres entiers positifs différents la suite {nkα}k>1 est équidistribuée modulo 1 pour presque tous les réels α. Dans ce cas, il est d’habitude extrêmement difficile de mesurer la vitesse de convergence de la distribution empirique vers l’équidistribution. Dans cet article, nous étudions le cas ou (nk)k>1 est la suite des nombres entiers « méchants », donc la suite des nombres positifs la une somme de chiffres paire dans la base 2. Nous relions ce probléme aux produits trigonométriques ∏L l=0 ‖ sinπ2 lα‖ en donnant des estimations exactes pour de tels produits et nous obtenons des estimations exactes pour la discrépance de la suite {nkα}k>1. En plus, nous donnons des exemples concrets de réels α pour lesquels nous pouvons obtenir des estimations pour la discrépance de la suite {nkα}k>1.

A combination theorem for cubulation in small cancellation theory over free products

Abstract: We prove that a group obtained as a quotient of the free product of finitely many cubulable groups by a finite set of relators satisfying the classical $C'(1/6)$--small cancellation condition is cubulable. This yields a new large class of relatively hyperbolic groups that can be cubulated, and constitutes the first instance of a cubulability theorem for relatively hyperbolic groups which does not require any geometric assumption on the peripheral subgroups besides their cubulability. We do this by constructing appropriate wallspace structures for such groups, by combining walls of the free factors with walls coming from the universal cover of an associated $2$-complex of groups.

$q$-Analogues of Laplace and Borel transforms by means of $q$-exponentials

Abstract: — The article discusses certain q-analogues of Laplace and Borel transforms, and shows a new inversion formula between q-Laplace and q-Borel transforms. q-Analogues of Watson type lemma and convolution operators are also discussed. These results give a new framework of the summability of formal power series solutions of q-difference equations. Résumé. — Nous considérons certaines q-analogues des transformées de Laplace et Borel et montrons une nouvelle formule d’inversion entre les transformées de q-Laplace et de q-Borel. Des q-analogues des lemmes de type Watson et des opérateurs de convolution sont aussi discutés. Ces résultats donnent un nouveau cadre pour la sommabilité des séries formelles qui sont solutions d’équations aux q-différences.

A reduction of canonical stability index of 4 and 5 dimensional projective varieties with large volume

Abstract: We study the canonical stability index of nonsingular projective varieties of general type with either large canonical volume or large geometric genus. As applications of a general extension theorem established in the first part, we prove some optimal results in dimensions 4 and 5, which are parallel to some well-known results on surfaces and 3-folds.

Sidonicity and variants of Kaczmarz’s problem

Abstract: We prove that a uniformly bounded system of orthonormal functions satisfying the $\psi_2$ condition: (1) must contain a Sidon subsystem of proportional size, (2) must satisfy the Rademacher-Sidon property, and (3) must have its 5-fold tensor satisfy the Sidon property. On the other hand, we construct a uniformly bounded orthonormal system that satisfies the $\psi_2$ condition but which is not Sidon. These problems are variants of Kaczmarz's Scottish book problem (problem 130) which, in its original formulation, was answered negatively by Rudin. A corollary of our argument is a new, elementary proof of Pisier's theorem that a set of characters satisfying the $\psi_2$ condition is Sidon.

Fano congruences of index 3 and alternating 3-forms

Abstract: We study congruences of lines $X_\omega$ defined by a sufficiently general choice of an alternating 3-form $\omega$ in $n+1$ dimensions, as Fano manifolds of index $3$ and dimension $n-1$. These congruences include the $\mathrm{G}_2$-variety for $n=6$ and the variety of reductions of projected $\mathbb{P}^2 \times \mathbb{P}^2$ for $n=7$. We compute the degree of $X_\omega$ as the $n$-th Fine number and study the Hilbert scheme of these congruences proving that the choice of $\omega$ bijectively corresponds to $X_\omega$ except when $n=5$. The fundamental locus of the congruence is also studied together with its singular locus: these varieties include the Coble cubic for $n=8$ and the Peskine variety for $n=9$. The residual congruence $Y$ of $X_\omega$ with respect to a general linear congruence containing $X_\omega$ is analysed in terms of the quadrics containing the linear span of $X_\omega$. We prove that $Y$ is Cohen-Macaulay but non-Gorenstein in codimension $4$. We also examine the fundamental locus $G$ of $Y$ of which we determine the singularities and the irreducible components.

The equivariant Minkowski problem in Minkowski space

Abstract: The classical Minkowski problem in Minkowski space asks, for a positive function on H d , for a convex set K in Minkowski space with C 2 space-like boundary S, such that ( ) 1 is the Gauss{Kronecker curvature at the point with normal . Analogously to the Euclidean case, it is possible to formulate a weak version of this problem: given a Radon measure on H d the generalized Minkowski problem in Minkowski space asks for a convex subset K such that the area measure of K is . In the present paper we look at an equivariant version of the problem: given a uniform lattice of isometries of H d , given a invariant Radon measure , given a isometry group of Minkowski space, with as linear part, there exists a unique convex set with area measure , invariant under the action of . The proof uses a functional which is the covolume associated to every invariant convex set. This result translates as a solution of the Minkowski problem in at space times with compact hyperbolic Cauchy surface. The uniqueness part, as well as regularity results, follow from properties of the Monge{Amp

Compact presentability of tree almost automorphism groups

Abstract: We establish compact presentability, i.e. the locally compact version of finite presentability, for an infinite family of tree almost automorphism groups. Examples covered by our results include Neretin’s group of spheromorphisms, as well as the topologically simple group containing the profinite completion of the Grigorchuk group constructed by Barnea, Ershov and Weigel. We additionally obtain an upper bound on the Dehn function of these groups in terms of the Dehn function of an embedded Higman-Thompson group. This, combined with a result of Guba, implies that the Dehn function of the Neretin group of the regular trivalent tree is polynomially bounded.

Del Pezzo surfaces of degree four violating the Hasse principle are Zariski dense in the moduli scheme

Abstract: We show that, over every number field, the degree four del Pezzo surfaces that violate the Hasse principle are Zariski dense in the moduli scheme.

Selmer groups and central values of $L$-functions for modular forms

Abstract: — In this article, we construct an Euler system using CM cycles on Kuga–Sato varieties over Shimura curves and show a relation with the central values of Rankin–Selberg L-functions for elliptic modular forms and ring class characters of an imaginary quadratic field. As an application, we prove that the non-vanishing of the central values of Rankin–Selberg L-functions implies the finiteness of Selmer groups associated to the corresponding Galois representation of modular forms under some assumptions. Résumé. — Dans cet article, nous construisons un système d’Euler en utilisant les cycles CM sur les variétés de Kuga–Sato au-dessus de courbes de Shimura, et montrons une relation avec les valeurs centrales de fonctions L de Rankin–Selberg associées aux formes modulaires de poids 2 et aux caractères de classes d’un corps quadratique imaginaire. Comme application, nous prouvons que la non-annulation des valeurs centrales de fonctions L de Rankin–Selberg implique la finitude des groupes de Selmer associés à la représentation galoisienne de la forme modulaire sous certaines hypothèses.

Geometry and arithmetic of certain log K3 surfaces

Abstract: Let $k$ be a field of characteristic $0$. In this paper we describe a classification of smooth log K3 surfaces $X$ over $k$ whose geometric Picard group is trivial and which can be compactified into del Pezzo surfaces. We show that such an $X$ can always be compactified into a del Pezzo surface of degree $5$, with a compactifying divisor $D$ being a cycle of five $(-1)$-curves, and that $X$ is completely determined by the action of the absolute Galois group of $k$ on the dual graph of $D$. When $k=\mathbb{Q}$ and the Galois action is trivial, we prove that for any integral model $\mathcal{X}/\mathbb{Z}$ of $X$, the set of integral points $\mathcal{X}(\mathbb{Z})$ is not Zariski dense. We also show that the Brauer-Manin obstruction is not the only obstruction for the integral Hasse principle on such log K3 surfaces, even when their compactification is "split".