Prolongation of Poisson $2$-form on Weil bundles

Norbert Mahoungou Moukala; Basile Guy Richard Bossoto

Displaying similar documents to “Prolongation of Poisson $2$ -form on Weil bundles”

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),...

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...

Involutivity of truncated microsupports

Masaki Kashiwara, Térésa Monteiro Fernandes, Pierre Schapira (2003)

Bulletin de la Société Mathématique de France

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Using a result of J.-M. Bony, we prove the weak involutivity of truncated microsupports. More precisely, given a sheaf $F$ on a real manifold and $k \in ℤ$ , if two functions vanish on ${SS}_{k} (F)$ , then so does their Poisson bracket.

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...

Noncommutative del Pezzo surfaces and Calabi-Yau algebras

Pavel Etingof, Victor Ginzburg (2010)

Journal of the European Mathematical Society

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The hypersurface in $ℂ^{3}$ with an isolated quasi-homogeneous elliptic singularity of type ${\tilde{E}}_{r}, r = 6, 7, 8$ , has a natural Poisson structure. We show that the family of del Pezzo surfaces of the corresponding type $E_{r}$ provides a semiuniversal Poisson deformation of that Poisson structure. We also construct a deformation-quantization of the coordinate ring of such a del Pezzo surface. To this end, we first deform the polynomial algebra $ℂ [x_{1}, x_{2}, x_{3}]$ to a noncommutative algebra with generators $x_{1}, x_{2}, x_{3}$ and the following 3 relations...

One-parameter contractions of Lie-Poisson brackets

Oksana Yakimova (2014)

Journal of the European Mathematical Society

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We consider contractions of Lie and Poisson algebras and the behaviour of their centres under contractions. A polynomial Poisson algebra $𝒜 = 𝕂 [𝔸^{n}]$ is said to be of Kostant type, if its centre $Z (𝒜)$ is freely generated by homogeneous polynomials $F_{1}, ..., F_{r}$ such that they give Kostant’s regularity criterion on $𝔸^{n} (d_{x} F_{i}$ are linear independent if and only if the Poisson tensor has the maximal rank at $x$ ). If the initial Poisson algebra is of Kostant type and $F_{i}$ satisfy a certain degree-equality, then the contraction...

The general rigidity result for bundles of $A$ -covelocities and $A$ -jets

Jiří M. Tomáš (2017)

Czechoslovak Mathematical Journal

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Let $M$ be an $m$ -dimensional manifold and $A = 𝔻_{k}^{r} / I = ℝ \oplus N_{A}$ a Weil algebra of height $r$ . We prove that any $A$ -covelocity $T_{x}^{A} f \in T_{x}^{A *} M$ , $x \in M$ is determined by its values over arbitrary $max {width A, m}$ regular and under the first jet projection linearly independent elements of $T_{x}^{A} M$ . Further, we prove the rigidity of the so-called universally reparametrizable Weil algebras. Applying essentially those partial results we give the proof of the general rigidity result $T^{A *} M ≃ T^{r *} M$ without coordinate computations, which improves and generalizes the partial...

On lifts of projectable-projectable classical linear connections to the cotangent bundle

Anna Bednarska (2013)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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We describe all $ℱ^{2} ℳ_{m_{1}, m_{2}, n_{1}, n_{2}}$ -natural operators $D : Q_{p r o j - p r o j}^{τ} ⇝ Q T^{*}$ transforming projectable-projectable classical torsion-free linear connections $\nabla$ on fibred-fibred manifolds $Y$ into classical linear connections $D (\nabla)$ on cotangent bundles $T^{*} Y$ of $Y$ . We show that this problem can be reduced to finding $ℱ^{2} ℳ_{m_{1}, m_{2}, n_{1}, n_{2}}$ -natural operators $D : Q_{p r o j - p r o j}^{τ} ⇝ (T^{*}, \otimes^{p} T^{*} \otimes \otimes^{q} T)$ for $p = 2$ , $q = 1$ and $p = 3$ , $q = 0$ .

Steinness of bundles with fiber a Reinhardt bounded domain

Karl Oeljeklaus, Dan Zaffran (2006)

Bulletin de la Société Mathématique de France

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Let $E$ denote a holomorphic bundle with fiber $D$ and with basis $B$ . Both $D$ and $B$ are assumed to be Stein. For $D$ a Reinhardt bounded domain of dimension $d = 2$ or $3$ , we give a necessary and sufficient condition on $D$ for the existence of a non-Stein such $E$ (Theorem $1$ ); for $d = 2$ , we give necessary and sufficient criteria for $E$ to be Stein (Theorem $2$ ). For $D$ a Reinhardt bounded domain of any dimension not intersecting any coordinate hyperplane, we give a sufficient criterion for $E$ to be Stein (Theorem...

$𝒞^{k}$ -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 a modification of the Poisson integral operator

Dariusz Partyka (2011)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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Given a quasisymmetric automorphism $γ$ of the unit circle $𝕋$ we define and study a modification $P_{γ}$ of the classical Poisson integral operator in the case of the unit disk $𝔻$ . The modification is done by means of the generalized Fourier coefficients of $γ$ . For a Lebesgue’s integrable complexvalued function $f$ on $𝕋$ , $P_{γ} [f]$ is a complex-valued harmonic function in $𝔻$ and it coincides with the classical Poisson integral of $f$ provided $γ$ is the identity mapping on $𝕋$ . Our considerations are motivated by...

The vertical prolongation of the projectable connections

Anna Bednarska (2012)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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We prove that any first order $ℱ_{2} ℳ_{m_{1}, m_{2}, n_{1}, n_{2}}$ -natural operator transforming projectable general connections on an $(m_{1}, m_{2}, n_{1}, n_{2})$ -dimensional fibred-fibred manifold $p = (p, p) : (p_{Y} : Y \to Y) \to (p_{M} : M \to M)$ into general connections on the vertical prolongation $V Y \to M$ of $p : Y \to M$ is the restriction of the (rather well-known) vertical prolongation operator $𝒱$ lifting general connections $\bar{Γ}$ on a fibred manifold $Y \to M$ into $𝒱 \bar{Γ}$ (the vertical prolongation of $\bar{Γ}$ ) on $V Y \to M$ .

Holomorphic line bundles and divisors on a domain of a Stein manifold

Makoto Abe (2007)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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Let $D$ be an open set of a Stein manifold $X$ of dimension $n$ such that $H^{k} (D, 𝒪) = 0$ for $2 \leq k \leq n - 1$ . We prove that $D$ is Stein if and only if every topologically trivial holomorphic line bundle $L$ on $D$ is associated to some Cartier divisor $𝔡$ on $D$ .

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 $ρ_{X} - ρ_{T} = 4$ .

Displaying similar documents to “Prolongation of Poisson 2 -form on Weil bundles”

Displaying similar documents to “Prolongation of Poisson $2$ -form on Weil bundles”