Non-solvability of the tangential $ \bar \partial_{M} $-systems

Giuseppe Zampieri

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Non-solvability of the tangential ∂̅-system in manifolds with constant Levi rank

Giuseppe Zampieri (2000)

Annales Polonici Mathematici

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Let M be a real-analytic submanifold of $ℂ^{n}$ whose “microlocal” Levi form has constant rank $s_{M}^{+} + s_{M}^{-}$ in a neighborhood of a prescribed conormal. Then local non-solvability of the tangential ∂̅-system is proved for forms of degrees $s_{M}^{-}$ , $s_{M}^{+}$ (and 0). This phenomenon is known in the literature as “absence of the Poincaré Lemma” and was already proved in case the Levi form is non-degenerate (i.e. $s_{M}^{-} + s_{M}^{+} = n - c o d i m M$ ). We owe its proof to [2] and [1] in the case of a hypersurface and of a higher-codimensional submanifold...

Structure of a neighbourhood of a complex compact submanifold in a complex manifold

Roman Dwilewicz

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CONTENTSIntroduction...................................................................................................................................................................51. Notation and some definitions...................................................................................................................................6 (a) Analytic families.....................................................................................................................................................6 (b)...

Existence and regularity of solutions of the $\bar{δ}$ -system on wedges of $C^{N}$

Giuseppe Zampieri (1999)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

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For a wedge $W$ of $C^{N}$ , we introduce two conditions of weak $q$ -pseudoconvexity, and prove that they entail solvability of the $\bar{δ}$ -system for forms of degree $\geq q + 1$ with coefficients in $C^{\infty} (W)$ and $C^{\infty} (\bar{W})$ respectively. Existence and regularity for $\bar{δ}$ in $W$ is treated by Hörmander [5, 6] (and also by Zampieri [9, 11] in case of piecewise smooth boundaries). Regularity in $W$ is treated by Henkin [4] (strong $q$ -pseudoconvexity by the method of the integral representation), Dufresnoy [3] (full pseudoconvexity), Michel...

Complements of analytic subvarieties and q-complete spaces

Edoardo Ballico (1981)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

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Si dimostra che il complementare $X\setminus Y$ di un sottospazio analitico chiuso localmente intersezione completa di codimensione $q$ di una varietà di Stein è $q$ -completo.

Characterization of diffeomorphisms that are symplectomorphisms

Stanisław Janeczko, Zbigniew Jelonek (2009)

Fundamenta Mathematicae

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Let $(X, ω_{X})$ and $(Y, ω_{Y})$ be compact symplectic manifolds (resp. symplectic manifolds) of dimension 2n > 2. Fix 0 < s < n (resp. 0 < k ≤ n) and assume that a diffeomorphism Φ : X → Y maps all 2s-dimensional symplectic submanifolds of X to symplectic submanifolds of Y (resp. all isotropic k-dimensional tori of X to isotropic tori of Y). We prove that in both cases Φ is a conformal symplectomorphism, i.e., there is a constant c ≠0 such that $Φ * ω_{Y} = c ω_{X}$ .

A Riemann-Roch-Hirzebruch formula for traces of differential operators

Markus Engeli, Giovanni Felder (2008)

Annales scientifiques de l'École Normale Supérieure

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Let $D$ be a holomorphic differential operator acting on sections of a holomorphic vector bundle on an $n$ -dimensional compact complex manifold. We prove a formula, conjectured by Feigin and Shoikhet, giving the Lefschetz number of $D$ as the integral over the manifold of a differential form. The class of this differential form is obtained via formal differential geometry from the canonical generator of the Hochschild cohomology $H H^{2 n} (𝒟_{n}, 𝒟_{n}^{*})$ of the algebra of differential operators on a formal neighbourhood...

Some lagrangian invariants of symplectic manifolds

Michel Nguiffo Boyom (2007)

Banach Center Publications

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The KV-homology theory is a new framework which yields interesting properties of lagrangian foliations. This short note is devoted to relationships between the KV-homology and the KV-cohomology of a lagrangian foliation. Let us denote by $_{F}$ (resp. $V^{F}$ ) the KV-algebra (resp. the space of basic functions) of a lagrangian foliation F. We show that there exists a pairing of cohomology and homology to $V^{F}$ . That is to say, there is a bilinear map $H^{q} (_{F}, V^{F}) \times H_{q} (_{F}, V^{F}) \to V^{F}$ , which is invariant under F-preserving symplectic...

On the spectral sequence fo r the $\bar{\partial}$ -cohomology of a holomorphic bundle with Stein fibres

Guido Lupacciolu (1982)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti

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Si esamina la successione spettrale per la $\bar{\partial}$ -coomologia dello spazio totale di un fibrato olomorfo nel caso in cui le fibre siano varietà di Stein.

A result on extension of C.R. functions

Makhlouf Derridj, John Erik Fornaess (1983)

Annales de l'institut Fourier

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Let $Ω$ an open set in $C^{4}$ near $z_{0} \in \partial Ω$ , $λ$ a suitable holomorphic function near $z_{0}$ . If we know that we can solve the following problem (see [M. Derridj, Annali. Sci. Norm. Pisa, Série IV, vol. IX (1981)]) : $\overline{\partial} u = λ f$ , ( $f$ is a $(0, 1)$ form, $\overline{\partial}$ closed in $U (z_{0})$ in $U (z_{0})$ with supp $(u) \subset \overline{Ω} \cap U (z_{0})$ , then we deduce an extension result for $C . R .$ functions on $\partial Ω \cap U (z_{0})$ , as holomorphic fonctions in $Ω \cap V (z_{0})$ .

Global integral formulas for solving the $\bar{\partial}$ -equation on Stein manifolds

Gennadi M. Henkin, Jürgen Leiterer (1981)

Annales Polonici Mathematici

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Special Lagrangian linear subspaces in product symplectic space

Małgorzata Mikosz (2004)

Banach Center Publications

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The notes consist of a study of special Lagrangian linear subspaces. We will give a condition for the graph of a linear symplectomorphism $f : (ℝ^{2 n}, σ = \sum_{i = 1}^{n} d x_{i} \land d y_{i}) \to (ℝ^{2 n}, σ)$ to be a special Lagrangian linear subspace in $(ℝ^{2 n} \times ℝ^{2 n}, ω = π * ₂ σ - π * ₁ σ)$ . This way a special symplectic subset in the symplectic group is introduced. A stratification of special Lagrangian Grassmannian $S Λ_{2 n} ≃ S U (2 n) / S O (2 n)$ is defined.

Exotic Deformations of Calabi-Yau Manifolds

Paolo de Bartolomeis, Adriano Tomassini (2013)

Annales de l’institut Fourier

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We introduce Quantum Inner State manifolds (QIS manifolds) as (compact) $2 n$ -dimensional symplectic manifolds $(M, κ)$ endowed with a $κ$ -tamed almost complex structure $J$ and with a nowhere vanishing and normalized section $ϵ$ of the bundle $Λ_{J}^{n, 0} (M)$ satisfying the condition ${\bar{\partial}}_{J} ϵ = 0$ . We study the moduli space $𝔐$ of QIS deformations of a given Calabi-Yau manifold, computing its tangent space...

Displaying similar documents to “Non-solvability of the tangential ∂ ¯ M -systems”

Displaying similar documents to “Non-solvability of the tangential ${\bar{\partial}}_{M}$ -systems”