Involutivity of truncated microsupports

Masaki Kashiwara; Térésa Monteiro Fernandes; Pierre Schapira

Displaying similar documents to “Involutivity of truncated microsupports”

Prolongation of Poisson $2$ -form on Weil bundles

Norbert Mahoungou Moukala, Basile Guy Richard Bossoto (2016)

Archivum Mathematicum

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In this paper, $M$ denotes a smooth manifold of dimension $n$ , $A$ a Weil algebra and $M^{A}$ the associated Weil bundle. When $(M, ω_{M})$ is a Poisson manifold with $2$ -form $ω_{M}$ , we construct the $2$ -Poisson form $ω_{M^{A}}^{A}$ , prolongation on $M^{A}$ of the $2$ -Poisson form $ω_{M}$ . We give a necessary and sufficient condition for that $M^{A}$ be an $A$ -Poisson manifold.

Relative subanalytic sheaves

Teresa Monteiro Fernandes, Luca Prelli (2014)

Fundamenta Mathematicae

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Given a real analytic manifold Y, denote by $Y_{s a}$ the associated subanalytic site. Now consider a product Y = X × S. We construct the endofunctor $ℱ \mapsto ℱ^{S}$ on the category of sheaves on $Y_{s a}$ and study its properties. Roughly speaking, $ℱ^{S}$ is a sheaf on $X_{s a} \times S$ . As an application, one can now define sheaves of functions on Y which are tempered or Whitney in the relative sense, that is, only with respect to X.

Poisson geometry and deformation quantization near a strictly pseudoconvex boundary

Eric Leichtnam, Xiang Tang, Alan Weinstein (2007)

Journal of the European Mathematical Society

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Let $X$ be a complex manifold with strongly pseudoconvex boundary $M$ . If $ψ$ is a defining function for $M$ , then $- log ψ$ is plurisubharmonic on a neighborhood of $M$ in $X$ , and the (real) 2-form $σ = i \partial \bar{\partial} (- log ψ)$ is a symplectic structure on the complement of $M$ in a neighborhood of $M$ in $X$ ; it blows up along $M$ . The Poisson structure obtained by inverting $σ$ extends smoothly across $M$ and determines a contact structure on $M$ which is the same as the one induced by the complex structure. When $M$ is compact, the Poisson structure...

Cluster ensembles, quantization and the dilogarithm

Vladimir V. Fock, Alexander B. Goncharov (2009)

Annales scientifiques de l'École Normale Supérieure

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A cluster ensemble is a pair $(𝒳, 𝒜)$ of positive spaces (i.e. varieties equipped with positive atlases), coming with an action of a symmetry group $Γ$ . The space $𝒜$ is closely related to the spectrum of a cluster algebra [12]. The two spaces are related by a morphism $p : 𝒜 \to 𝒳$ . The space $𝒜$ is equipped with a closed $2$ -form, possibly degenerate, and the space $𝒳$ has a Poisson structure. The map $p$ is compatible with these structures. The dilogarithm together with its motivic and quantum avatars plays a central...

Canonical Poisson-Nijenhuis structures on higher order tangent bundles

P. M. Kouotchop Wamba (2014)

Annales Polonici Mathematici

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Let M be a smooth manifold of dimension m>0, and denote by $S_{c a n}$ the canonical Nijenhuis tensor on TM. Let Π be a Poisson bivector on M and $Π^{T}$ the complete lift of Π on TM. In a previous paper, we have shown that $(T M, Π^{T}, S_{c a n})$ is a Poisson-Nijenhuis manifold. Recently, the higher order tangent lifts of Poisson manifolds from M to $T^{r} M$ have been studied and some properties were given. Furthermore, the canonical Nijenhuis tensors on $T^{A} M$ are described by A. Cabras and I. Kolář [Arch. Math. (Brno) 38 (2002),...

On butterfly-points in $β X$ , Tychonoff products and weak Lindelöf numbers

Sergei Logunov (2022)

Commentationes Mathematicae Universitatis Carolinae

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Let $X$ be the Tychonoff product $\prod_{α < τ} X_{α}$ of $τ$ -many Tychonoff non-single point spaces $X_{α}$ . Let $p \in X^{*}$ be a point in the closure of some $G \subset X$ whose weak Lindelöf number is strictly less than the cofinality of $τ$ . Then we show that $β X ∖ {p}$ is not normal. Under some additional assumptions, $p$ is a butterfly-point in $β X$ . In particular, this is true if either $X = ω^{τ}$ or $X = R^{τ}$ and $τ$ is infinite and not countably cofinal.

The subspace of weak $P$ -points of $ℕ^{*}$

Salvador García-Ferreira, Y. F. Ortiz-Castillo (2015)

Commentationes Mathematicae Universitatis Carolinae

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Let $W$ be the subspace of $ℕ^{*}$ consisting of all weak $P$ -points. It is not hard to see that $W$ is a pseudocompact space. In this paper we shall prove that this space has stronger pseudocompact properties. Indeed, it is shown that $W$ is a $p$ -pseudocompact space for all $p \in ℕ^{*}$ .

Coleff-Herrera currents, duality, and noetherian operators

Mats Andersson (2011)

Bulletin de la Société Mathématique de France

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Let $ℐ$ be a coherent subsheaf of a locally free sheaf $𝒪 (E_{0})$ and suppose that $ℱ = 𝒪 (E_{0}) / ℐ$ has pure codimension. Starting with a residue current $R$ obtained from a locally free resolution of $ℱ$ we construct a vector-valued Coleff-Herrera current $μ$ with support on the variety associated to $ℱ$ such that $φ$ is in $ℐ$ if and only if $μ φ = 0$ . Such a current $μ$ can also be derived algebraically from a fundamental theorem of Roos about the bidualizing functor, and the relation between these two approaches is discussed....

$𝒞^{k}$ -regularity for the $\bar{\partial}$ -equation with a support condition

Shaban Khidr, Osama Abdelkader (2017)

Czechoslovak Mathematical Journal

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Let $D$ be a $𝒞^{d}$ $q$ -convex intersection, $d \geq 2$ , $0 \leq q \leq n - 1$ , in a complex manifold $X$ of complex dimension $n$ , $n \geq 2$ , and let $E$ be a holomorphic vector bundle of rank $N$ over $X$ . In this paper, $𝒞^{k}$ -estimates, $k = 2, 3, \dots, \infty$ , for solutions to the $\bar{\partial}$ -equation with small loss of smoothness are obtained for $E$ -valued $(0, s)$ -forms on $D$ when $n - q \leq s \leq n$ . In addition, we solve the $\bar{\partial}$ -equation with a support condition in $𝒞^{k}$ -spaces. More precisely, we prove that for a $\bar{\partial}$ -closed form $f$ in $𝒞_{0, q}^{k} (X ∖ D, E)$ , $1 \leq q \leq n - 2$ , $n \geq 3$ , with compact support and for $ε$ with $0 < ε < 1$ there...

On the Picard number of divisors in Fano manifolds

Cinzia Casagrande (2012)

Annales scientifiques de l'École Normale Supérieure

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Let $X$ be a complex Fano manifold of arbitrary dimension, and $D$ a prime divisor in $X$ . We consider the image $𝒩_{1} (D, X)$ of $𝒩_{1} (D)$ in $𝒩_{1} (X)$ under the natural push-forward of $1$ -cycles. We show that $ρ_{X} - ρ_{D} \leq codim 𝒩_{1} (D, X) \leq 8$ . Moreover if $codim 𝒩_{1} (D, X) \geq 3$ , then either $X ≅ S \times T$ where $S$ is a Del Pezzo surface, or $codim 𝒩_{1} (D, X) = 3$ and $X$ has a fibration in Del Pezzo surfaces onto a Fano manifold $T$ such that $ρ_{X} - ρ_{T} = 4$ .

Stratified Whitney jets and tempered ultradistributions on the subanalytic site

N. Honda, G. Morando (2011)

Bulletin de la Société Mathématique de France

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In this paper we introduce the sheaf of stratified Whitney jets of Gevrey order on the subanalytic site relative to a real analytic manifold $X$ . Then, we define stratified ultradistributions of Beurling and Roumieu type on $X$ . In the end, by means of stratified ultradistributions, we define tempered-stratified ultradistributions and we prove two results. First, if $X$ is a real surface, the tempered-stratified ultradistributions define a sheaf on the subanalytic site relative to $X$ . Second,...

Weak convergence of mutually independent $X ₙ^{B}$ and $X ₙ^{A}$ under weak convergence of $X ₙ \equiv X ₙ^{B} - X ₙ^{A}$

W. Szczotka (2006)

Applicationes Mathematicae

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For each n ≥ 1, let $v_{n, k}, k \geq 1$ and $u_{n, k}, k \geq 1$ be mutually independent sequences of nonnegative random variables and let each of them consist of mutually independent and identically distributed random variables with means v̅ₙ and u̅̅ₙ, respectively. Let $X ₙ^{B} (t) = (1 / c ₙ) \sum_{j = 1}^{[n t]} (v_{n, j} - v ̅ ₙ)$ , $X ₙ^{A} (t) = (1 / c ₙ) \sum_{j = 1}^{[n t]} (u_{n, j} - u ̅ ̅ ₙ)$ , t ≥ 0, and $X ₙ = X ₙ^{B} - X ₙ^{A}$ . The main result gives conditions under which the weak convergence $X ₙ \overset{}{\to} X$ , where X is a Lévy process, implies $X ₙ^{B} \overset{}{\to} X^{B}$ and $X ₙ^{A} \overset{}{\to} X^{A}$ , where $X^{B}$ and $X^{A}$ are mutually independent Lévy processes and $X = X^{B} - X^{A}$ .

$C^{*}$ -points vs $P$ -points and $P^{♭}$ -points

Jorge Martinez, Warren Wm. McGovern (2022)

Commentationes Mathematicae Universitatis Carolinae

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In a Tychonoff space $X$ , the point $p \in X$ is called a $C^{*}$ -point if every real-valued continuous function on $C ∖ {p}$ can be extended continuously to $p$ . Every point in an extremally disconnected space is a $C^{*}$ -point. A classic example is the space $𝐖^{*} = ω_{1} + 1$ consisting of the countable ordinals together with $ω_{1}$ . The point $ω_{1}$ is known to be a $C^{*}$ -point as well as a $P$ -point. We supply a characterization of $C^{*}$ -points in totally ordered spaces. The remainder of our time is aimed at studying when a point in a product space...