Showing papers in "Journal of The Institute of Mathematics of Jussieu in 2016"
TL;DR: In this article, a proper moduli space of twisted canonical divisors is defined, which includes the space of canonical divaris as an open subset, leading to geometric/combinatorial constraints on the closures of the moduli spaces of the devisors on nonsingular curves, and a complete proposal relating the sum of the fundamental classes of all components to a formula of Pixton is proposed.
Abstract: The moduli space of canonical divisors (with prescribed zeros and poles) on nonsingular curves is not compact since the curve may degenerate. We define a proper moduli space of twisted canonical divisors in which includes the space of canonical divisors as an open subset. The theory leads to geometric/combinatorial constraints on the closures of the moduli spaces of canonical divisors.In case the differentials have at least one pole (the strictly meromorphic case), the moduli spaces of twisted canonical divisors on genus curves are of pure codimension in . In addition to the closure of the canonical divisors on nonsingular curves, the moduli spaces have virtual components. In the Appendix A, a complete proposal relating the sum of the fundamental classes of all components (with intrinsic multiplicities) to a formula of Pixton is proposed. The result is a precise and explicit conjecture in the tautological ring for the weighted fundamental class of the moduli spaces of twisted canonical divisors.As a consequence of the conjecture, the classes of the closures of the moduli spaces of canonical divisors on nonsingular curves are determined (both in the holomorphic and meromorphic cases).
89 citations
TL;DR: In this article, a lifting of Joseph ideals for the minimal nilpotent orbit closure to the setting of affine Kac-Moody algebras was considered.
Abstract: We consider a lifting of Joseph ideals for the minimal nilpotent orbit closure to the setting of affine Kac–Moody algebras and find new examples of affine vertex algebras whose associated varieties are minimal nilpotent orbit closures. As an application we obtain a new family of lisse ( -cofinite) -algebras that are not coming from admissible representations of affine Kac–Moody algebras.
42 citations
TL;DR: In this article, the authors give a complete answer to the question by proving a simple necessary and sufficient criterion on the return probability for disorder relevance, which confirms earlier predictions from the literature.
Abstract: The effect of disorder for pinning models is a subject which has attracted much attention in theoretical physics and rigorous mathematical physics. A peculiar point of interest is the question of coincidence of the quenched and annealed critical point for a small amount of disorder. The question has been mathematically settled in most cases in the last few years, giving in particular a rigorous validation of the Harris Criterion on disorder relevance. However, the marginal case, where the return probability exponent is equal to $1/2$, i.e. where the inter-arrival law of the renewal process is given by $K(n)=n^{-3/2}\phi(n)$ where $\phi$ is a slowly varying function, has been left partially open. In this paper, we give a complete answer to the question by proving a simple necessary and sufficient criterion on the return probability for disorder relevance, which confirms earlier predictions from the literature. Moreover, we also provide sharp asymptotics on the critical point shift: in the case of the pinning (or wetting) of a one dimensional simple random walk, the shift of the critical point satisfies the following high temperature asymptotics $$ \lim_{\beta\rightarrow 0}\beta^2\log h_c(\beta)= - \frac{\pi}{2}. $$ This gives a rigorous proof to a claim of B. Derrida, V. Hakim and J. Vannimenus (Journal of Statistical Physics, 1992).
42 citations
TL;DR: In this paper, asymptotics for the range Rn of a random walk on the d-dimensional lattice indexed by a random tree with n vertices were provided.
Abstract: We provide asymptotics for the range Rn of a random walk on the d-dimensional lattice indexed by a random tree with n vertices. Using Kingman’s subadditive ergodic theorem, we prove under general assumptions thatn 1 Rn converges to a constant, and we give conditions ensuring that the limiting constant is strictly positive. On the other hand, in dimension 4 and in the case of a symmetric random walk with exponential moments, we prove that Rn grows like n/ logn. We apply our results to asymptotics for the range of branching random walk when the initial size of the population tends to infinity.
32 citations
TL;DR: In this paper, the authors consider compact sets which are invariant and partially hyperbolic under the dynamics of a diffeomorphism of a manifold and prove that such a set is contained in a locally invariant center submanifold if and only if each strong stable and strong unstable leaf intersects $K$ at exactly one point.
Abstract: We consider compact sets which are invariant and partially hyperbolic under the dynamics of a diffeomorphism of a manifold. We prove that such a set $K$ is contained in a locally invariant center submanifold if and only if each strong stable and strong unstable leaf intersects $K$ at exactly one point.
31 citations
TL;DR: In this article, a cohomology theory using quasi-smooth derived schemes as generators and an analog of the bordism relation using derived fiber products as relations was constructed, and the resulting theory agreed with algebraic cobordism as defined by Levine and Morel.
Abstract: We construct a cohomology theory using quasi-smooth derived schemes as generators and an analog of the bordism relation using derived fiber products as relations. This theory has pull-backs along all morphisms between smooth schemes independent of any characteristic assumptions. We prove that, in characteristic zero, the resulting theory agrees with algebraic cobordism as defined by Levine and Morel. We thus obtain a new set of generators and relations for algebraic cobordism.
29 citations
TL;DR: A classical article by Misiurewicz and Ziemian (J. Lond. Math. Soc.40(2) (1989), 490−506) classifies the elements in Homeo has nonempty interior as discussed by the authors.
Abstract: A classical article by Misiurewicz and Ziemian (J. Lond. Math. Soc.40(2) (1989), 490–506) classifies the elements in Homeo has nonempty interior.
29 citations
TL;DR: It is well known that a finitely generated group can be generated with the assistance of computers as mentioned in this paper, and it is also known that the group can then be split into smaller groups.
Abstract: It is well known that a finitely generated group , possibly with the assistance of computers.
28 citations
TL;DR: In this article, the authors studied countable saturation of metric reduced products and introduced continuous fields of metric structures indexed by locally compact, separable, completely metrizable spaces, where the saturation of the reduced product depends both on the underlying index space and the model.
Abstract: We study countable saturation of metric reduced products and introduce continuous fields of metric structures indexed by locally compact, separable, completely metrizable spaces. Saturation of the reduced product depends both on the underlying index space and the model. By using the Gelfand–Naimark duality we conclude that the assertion “Stone–Cech remainder of the half-line has only trivial automorphisms” is independent from ZFC. Consistency of this statement follows from the Proper Forcing Axiom and this is the first known example of a connected space with this property. The present paper has two largely independent parts moving in two opposite directions. The first part (§§1–4) uses model theory of metric structures and it is concerned with the degree of saturation of various reduced products. The second part (§5) uses set-theoretic methods and it is mostly concerned with rigidity of Stone–Cech remainders of locally compact, Polish spaces. (A topological space is Polish if it is separable and completely metrizable.) The two parts are linked by the standard fact that saturated structures have many automorphisms (the continuous case of this fact is given in Theorem 3.1). By βX we denote the Stone–Cech compactification of X and by X∗ we denote its remainder (also called corona), βX \ X. A continuous map Φ: X∗ → Y ∗ is trivial if there are a compact subset K of X and a continuous map f : X\K → Y such that Φ = βf X∗, where βf : βX → βY is the unique continuous extension of f . Continuum Hypothesis (CH) implies that all Stone–Cech remainders of locally compact, zero dimensional, non-compact Polish spaces are homeomorphic. This is a consequence of Parovicenko’s theorem, see e.g., [31]. By using Stone duality, this follows from the fact that all atomless Boolean algebras are Date: June 17, 2014. The first author was partially supported by NSERC. The second author would like to thank the Israel Science Foundation, Grant no. 710/07, and the National Science Foundation, Grant no. DMS 1101597 for partial support of this research. No. 1042 on Shelah’s list of publications. Since in Czech alphabet letter ‘c’ precedes letter ‘s’ some authors write Cech– Stone compactification instead of Stone–Cech compactification.
26 citations
TL;DR: In this paper, it was shown that cyclic homology, topological cyclicity, and algebraic-theory with compact support can be shown to have a compact support.
Abstract: In this paper, we prove that cyclic homology, topological cyclic homology, and algebraic -theory with compact support.
23 citations
TL;DR: In this article, the authors classify the sheets and the rigid nilpotent orbits in reductive Lie algebras over fields of good characteristic and show that the distribution of nilpotents among the sheets remains the same as in the characteristic zero case.
Abstract: We classify the sheets and the rigid nilpotent orbits in reductive Lie algebras over fields of good characteristic and show that the distribution of nilpotent orbits amongst the sheets remains the same as in the characteristic zero case. We use GAP to determine the reachable and strongly reachable nilpotent orbits in all characteristics and provide some information on derived subalgebras of centralisers.
TL;DR: In this article, a dual map on stable tempered characters was derived from the matching of orbital integrals between functions on (Formula presented.) and its endoscopic groups, and the value of this dual map was expressed as a linear combination of twisted characters.
Abstract: Suppose that (Formula presented.) is a connected reductive algebraic group defined over (Formula presented.), (Formula presented.) is its group of real points, (Formula presented.) is an automorphism of (Formula presented.), and (Formula presented.) is a quasicharacter of (Formula presented.). Kottwitz and Shelstad defined endoscopic data associated to (Formula presented.), and conjectured a matching of orbital integrals between functions on (Formula presented.) and its endoscopic groups. This matching has been proved by Shelstad, and it yields a dual map on stable distributions. We express the values of this dual map on stable tempered characters as a linear combination of twisted characters, under some additional hypotheses on (Formula presented.) and (Formula presented.).
TL;DR: In this article, the authors consider a family of mixing piecewise expanding unimodal maps with a critical point that is transversal to the topological classes of such maps and show that is not a Lipchitz function on any subset of with positive Lebesgue measure.
Abstract: Consider a family of mixing piecewise expanding unimodal maps , with a critical point , that is transversal to the topological classes of such maps. Given a Lipchitz observable consider the function where is the unique absolutely continuous invariant probability of . Suppose that for every , where We show that converges to where is a dynamically defined function and is the Lebesgue measure on , normalized in such way that . As a consequence, we show that is not a Lipchitz function on any subset of with positive Lebesgue measure.
TL;DR: For a variety with a Whitney stratification by affine spaces, the authors studied categories of motivic sheaves which are constant mixed Tate along the strata and showed that they have a natural weight structure.
Abstract: For a variety with a Whitney stratification by affine spaces, we study categories of motivic sheaves which are constant mixed Tate along the strata. We are particularly interested in those cases where the category of mixed Tate motives over a point is equivalent to the category of finite-dimensional bigraded vector spaces. Examples of such situations include rational motives on varieties over finite fields and modules over the spectrum representing the semisimplification of de Rham cohomology for varieties over the complex numbers. We show that our categories of stratified mixed Tate motives have a natural weight structure. Under an additional assumption of pointwise purity for objects of the heart, tilting gives an equivalence between stratified mixed Tate sheaves and the bounded homotopy category of the heart of the weight structure. Specializing to the case of flag varieties, we find natural geometric interpretations of graded category and Koszul duality.
TL;DR: In this article, the authors consider etale motivic or Lichtenbaum cohomology and its relation to algebraic cycles and show that the usual integral cycle maps extend to maps on integral LCHs.
Abstract: We consider etale motivic or Lichtenbaum cohomology and its relation to algebraic cycles. We give an geometric interpretation of Lichtenbaum cohomology and use it to show that the usual integral cycle maps extend to maps on integral Lichtenbaum cohomology. We also show that Lichtenbaum cohomology, in contrast to the usual motivic cohomology, compares well with integral cohomology theories. For example, we formulate integral etale versions of the Hodge and the Tate conjecture, and show that these are equivalent to the usual rational conjectures.
TL;DR: In this paper, the trace (or zeroth Hochschild homology) of Khovanov's Heisenberg category is identified with a quotient of the algebra on symmetric functions.
Abstract: The trace (or zeroth Hochschild homology) of Khovanov’s Heisenberg category is identified with a quotient of the algebra on symmetric functions.
TL;DR: In this article, it was shown that the critical values of the Rankin-Selberg product are essentially algebraic multiples of the product of the Whittaker periods of and.
Abstract: Let be an imaginary quadratic field. Let and be irreducible generic cohomological automorphic representation of and , respectively. Each of them can be given two natural rational structures over number fields. One is defined by the rational structure on topological cohomology, and the other is given in terms of the Whittaker model. The ratio between these rational structures is called a Whittaker period. An argument presented by Mahnkopf and Raghuram shows that, at least if is cuspidal and the weights of and are in a standard relative position, the critical values of the Rankin–Selberg product are essentially algebraic multiples of the product of the Whittaker periods of and . We show that, under certain regularity and polarization hypotheses, the Whittaker period of a cuspidal can be given a motivic interpretation, and can also be related to a critical value of the adjoint -function of related automorphic representations of unitary groups. The resulting expressions for critical values of the Rankin–Selberg and adjoint -functions are compatible with Deligne’s conjecture.
TL;DR: In this paper, the delta-foncteur de Borel et al. derived the parties ordinaires derivedes d’Emerton relatif a sous-groupe of Borel sur certaines representations induites de $G(F)$ en utilisant une filtration de Bruhat.
Abstract: Soit $G$ un groupe reductif connexe deploye sur une extension finie $F$ de $\mathbb{Q}_{p}$ . Nous determinons les extensions entre series principales continues unitaires $p$ -adiques et lisses modulo $p$ de $G(F)$ dans le cas generique. Pour cela, nous calculons le delta-foncteur $\text{H}^{\bullet }\text{Ord}_{B(F)}$ des parties ordinaires derivees d’Emerton relatif a un sous-groupe de Borel sur certaines representations induites de $G(F)$ en utilisant une filtration de Bruhat. Ces extensions interviennent dans le programme de Langlands $p$ -adique et modulo $p$ .
TL;DR: In this paper, the authors studied adjunctions between functors of this sort and introduced a new kind of adjunction relation, called a local adjunction, which is weaker than the standard concept from category theory.
Abstract: Let $E$
be a (right) Hilbert module over a $C^{\ast }$
-algebra $A$
. If $E$
is equipped with a left action of a second $C^{\ast }$
-algebra $B$
, then tensor product with $E$
gives rise to a functor from the category of Hilbert $B$
-modules to the category of Hilbert $A$
-modules. The purpose of this paper is to study adjunctions between functors of this sort. We shall introduce a new kind of adjunction relation, called a local adjunction, that is weaker than the standard concept from category theory. We shall give several examples, the most important of which is the functor of parabolic induction in the tempered representation theory of real reductive groups. Each local adjunction gives rise to an ordinary adjunction of functors between categories of Hilbert space representations. In this way we shall show that the parabolic induction functor has a simultaneous left and right adjoint, namely the parabolic restriction functor constructed in Clare et al. [Parabolic induction and restriction via $C^{\ast }$
-algebras and Hilbert $C^{\ast }$
-modules, Compos. Math. FirstView (2016), 1–33, 2].
TL;DR: In this paper, the p-adic families of the Klingen Eisenstein series and L-functions for cuspforms unramified at an odd prime p on definite unitary groups of signature (r, 0) (for any positive integer r) for a quadratic imaginary field K split at p were constructed.
Abstract: We construct p-adic families of Klingen Eisenstein series and L-functions for cuspforms (not necessarily ordinary) unramified at an odd prime p on definite unitary groups of signature (r, 0) (for any positive integer r) for a quadratic imaginary field K split at p. When r = 2, we show that the constant term of the Klingen Eisenstein family is divisible by a certain p-adic L-function.
TL;DR: In this paper, the authors define a theory of enriched functors from finite pointed simplicial G-sets to symmetric G-spectra, where G is a finite group, and extend a notion of G-linearity suggested by Blumberg to define stably excisive and rho-analytic homotopy functors, as well as a G-differential, in this equivariant context.
Abstract: We define a theory of Goodwillie calculus for enriched functors from finite pointed simplicial G-sets to symmetric G-spectra, where G is a finite group. We extend a notion of G-linearity suggested by Blumberg to define stably excisive and rho-analytic homotopy functors, as well as a G-differential, in this equivariant context. A main result of the paper is that analytic functors with trivial derivatives send highly connected G-maps to G-equivalences. It is analogous to the classical result of Goodwillie that "functors with zero derivative are locally constant". As main example we show that Hesselholt and Madsen's Real algebraic K-theory of a split square zero extension of Wall antistructures defines an analytic functor in the Z/2-equivariant setting. We further show that the equivariant derivative of this Real K-theory functor is Z/2-equivalent to Real MacLane homology.
TL;DR: In this paper, the authors studied the renormalized volume of asymptotically hyperbolic Einstein (AHE) manifolds and showed that the set of ends of AHE manifolds can be viewed as a Lagrangian submanifold in the cotangent space to the space of conformal structures.
Abstract: We study the renormalized volume of asymptotically hyperbolic Einstein (AHE in short) manifolds $(M,g)$ when the conformal boundary $\partial M$ has dimension $n$ even. Its definition depends on the choice of metric $h_0$ on $\partial M$ in the conformal class at infinity determined by $g$, we denote it by ${\rm Vol}_R(M,g;h_0)$. We show that ${\rm Vol}_R(M,g;\cdot)$ is a functional admitting a ''Polyakov type'' formula in the conformal class $[h_0]$ and we describe the critical points as solutions of some non-linear equation $v_n(h_0)={\rm constant}$, satisfied in particular by Einstein metrics. In dimension $n=2$, choosing extremizers in the conformal class amounts to uniformizing the surface, while in dimension $n=4$ this amounts to solving the $\sigma_2$-Yamabe problem. Next, we consider the variation of ${\rm Vol}_R(M,\cdot;\cdot)$ along a curve of AHE metrics $g^t$ with boundary metric $h_0^t$ and we use this to show that, provided conformal classes can be (locally) parametrized by metrics $h$ solving $v_n(h)=\int_{\partial M}v_n(h){\rm dvol}_{h}$, the set of ends of AHE manifolds (up to diffeomorphisms isotopic to the identity) can be viewed as a Lagrangian submanifold in the cotangent space to the space $\mathcal{T}(\partial M)$ of conformal structures on $\partial M$. We obtain as a consequence a higher-dimensional version of McMullen's quasifuchsian reciprocity. We finally show that conformal classes admitting negatively curved Einstein metrics are local minima for the renormalized volume for a warped product type filling.
TL;DR: Chaudouard et al. as discussed by the authors present a contribution a la fois au calcul du nombre de fibres de Hitchin sur une courbe projective and a luiexplicitation de the partie nilpotente de la formule des traces d'Arthur-Selberg for a fonction test tres simple.
Abstract: Cet article est une contribution a la fois au calcul du nombre de fibres de Hitchin sur une courbe projective et a l’explicitation de la partie nilpotente de la formule des traces d’Arthur-Selberg pour une fonction test tres simple. Le lien entre les deux questions a ete etabli dans [Chaudouard, Sur le comptage des fibres de Hitchin. A paraitre aux actes de la conference en l’honneur de Gerard Laumon]. On decompose cette partie nilpotente en une somme d’integrales adeliques indexees par les orbites nilpotentes. Pour les orbites de type «regulieres par blocs», on explicite completement ces integrales en termes de la fonction zeta de la courbe.
TL;DR: In this paper, it was shown that non-commutative localizations of arbitrary additive categories (generalizing those defined by Cohn in the setting of rings) are closely related to weight structures.
Abstract: In this paper we demonstrate that non-commutative localizations of arbitrary additive categories (generalizing those defined by Cohn in the setting of rings) are closely (and naturally) related to weight structures. Localizing an arbitrary triangulated category .
TL;DR: In this article, the authors improved a previous result about the local energy decay for the damped wave equation on R^d and proved the limiting absorption principle for the powers of the resolvent with inserted weights.
Abstract: We improve a previous result about the local energy decay for the damped wave equation on R^d. The problem is governed by a Laplacian associated with a long range perturbation of the flat metric and a short range absorption index. Our purpose is to recover the decay O(t^{−d+e}) in the weighted energy spaces. The proof is based on uniform resolvent estimates, given by an improved version of the dissipative Mourre theory. In particular we have to prove the limiting absorption principle for the powers of the resolvent with inserted weights.
TL;DR: In this paper, the authors completely characterize solvable real Lie groups definable in ominimal expansions of the real field and show that these groups are solvable in polynomial time.
Abstract: In this paper, we completely characterize solvable real Lie groups definable in o-minimal expansions of the real field.
TL;DR: In this article, an infinitesimal version of the Jacquet-Rallis trace formula for general linear groups is established by integrating a kernel truncated a la Arthur multiplied by the absolute value of the determinant to the power in terms of relative orbital integrals regularised by means of zeta functions.
Abstract: We establish an infinitesimal version of the Jacquet-Rallis trace formula for general linear groups. Our formula is obtained by integrating a kernel truncated a la Arthur multiplied by the absolute value of the determinant to the power in terms of relative orbital integrals regularised by means of zeta functions.
TL;DR: In this article, the ♯-functor, which takes a semiabelian variety G over a separably closed field K of characteristic p > 0 and finite degree of imperfection, is studied.
Abstract: Given a separably closed field K of characteristic p > 0 and finite degree of imperfection we study the ♯-functor which takes a semiabelian variety G over K to the maximal divisible subgroup of G(K). Our main result is an example where G ♯ , as a “type-definable group” in K, does not have “relative Morley rank”, yielding a counterexample to a claim in [Hr]. Our methods involve studying the question of the preservation of exact sequences by the ♯-functor, and relating this to issues of descent as well as model theoretic properties of G ♯ . We mention some characteristic 0 analogues of these “exactness-descent” results, where differential algebraic methods are more prominent. We also develop the notion of an iterative D-structure on a group scheme over an iterative Hasse field, which is interesting in its own right,
TL;DR: In this paper, the Koopman representation of a positive entropy probability measure-preserving action of a countable, discrete, sofic group is deduced and a formula for entropy in the presence of a Polish, but a priori noncompact, model is given.
Abstract: We deduce properties of the Koopman representation of a positive entropy probability measure-preserving action of a countable, discrete, sofic group. Our main result may be regarded as a ‘representation-theoretic’ version of Sinaǐ’s factor theorem. We show that probability measure-preserving actions with completely positive entropy of an infinite sofic group must be mixing and, if the group is nonamenable, have spectral gap. This implies that if is a nonamenable group and is a probability measure-preserving action which is not strongly ergodic, then no action orbit equivalent to has completely positive entropy. Crucial to these results is a formula for entropy in the presence of a Polish, but a priori noncompact, model.