The Experts below are selected from a list of 309 Experts worldwide ranked by ideXlab platform
Daisuke Matsushita - One of the best experts on this subject based on the ideXlab platform.
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on subgroups of an automorphism group of an irreducible Symplectic Manifold
arXiv: Algebraic Geometry, 2018Co-Authors: Daisuke MatsushitaAbstract:Let X be an irreducible Symplectic Manifold and L a nef line bundle on X which is isotropic with respect to the Beauville-Bogomolov quadratic form. It is known that a subgroup Aut(X,L) of an automorphism group of X which fix L is almost abelian. We give a formula of the rank of Aut(X,L) in terms of MBM divisors. We also prove that the nef cone of X cut out MBM classes, which is a generalization of Kovac's structure theorem of nef cones of K3 surfaces
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On nef reductions of projective irreducible Symplectic Manifolds
arXiv: Algebraic Geometry, 2006Co-Authors: Daisuke MatsushitaAbstract:Let X be a projective irreducible Symplectic Manifold and L a non trivial nef divisor on X. Assume that the nef dimension of L is strictly less than the dimension of X. We prove that L is semiample
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addendum to on fibre space structures of a projective irreducible Symplectic Manifold
arXiv: Algebraic Geometry, 1999Co-Authors: Daisuke MatsushitaAbstract:In this note, we prove that every fibre space structures of a projective irreducible Symplectic Manifold is a lagrangian fibration.
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on fibre space structures of a projective irreducible Symplectic Manifold
Topology, 1999Co-Authors: Daisuke MatsushitaAbstract:Abstract In this note, we investigate fibre space structures of a projective irreducible Symplectic Manifold. We prove that a 2n-cdimensional projective irreducible Symplectic Manifold admits only an n-dimensional fibration over a Fano variety which has only Q -factorial log-terminal singularities and whose Picard number is one. Moreover we prove that a general fibre is an abelian variety up to finite unramified cover, especially, for 4-fold, a general fibre is an abelian surface and all fibres are equidimensional.
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on fibre space structures of a projective irreducible Symplectic Manifold
arXiv: Algebraic Geometry, 1997Co-Authors: Daisuke MatsushitaAbstract:In this note, we investigate fibre space structures of a projective irreducible Symplectic Manifold. We prove that an 2n-dimensional projective irreducible Symplectic Manifold admits only an n-dimensional fibration over a Fano variety which has only Q-factorial log-terminal singularities and whose Picard number is one. Moreover we prove that a general fibre is an abelian variety up to finite unramified cover, especially, a general fibre is an abelian surface for 4-fold.
Fabian Ziltener - One of the best experts on this subject based on the ideXlab platform.
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Hamiltonian group actions on exact Symplectic Manifolds with proper momentum maps are standard
Transactions of the American Mathematical Society, 2017Co-Authors: Yael Karshon, Fabian ZiltenerAbstract:We give a complete characterization of Hamiltonian actions of compact Lie groups on exact Symplectic Manifolds with proper momentum maps. We deduce that every Hamiltonian action of a compact Lie group on a contractible Symplectic Manifold with a proper momentum map is globally linearizable.
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hofer geometry of a subset of a Symplectic Manifold
Geometriae Dedicata, 2013Co-Authors: Jan Swoboda, Fabian ZiltenerAbstract:To every closed subset X of a Symplectic Manifold (M, ω) we associate a natural group of Hamiltonian diffeomorphisms Ham (X, ω). We equip this group with a semi-norm \({\Vert\cdot\Vert^{X, \omega}}\), generalizing the Hofer norm. We discuss Ham (X, ω) and \({\Vert\cdot\Vert^{X, \omega}}\) if X is a Symplectic or isotropic subManifold. The main result involves the relative Hofer diameter of X in M. Its first part states that for the unit sphere in \({\mathbb{R}^{2n}}\) this diameter is bounded below by \({\frac{\pi}{2}}\) , if n ≥ 2. Its second part states that for n ≥ 2 and d ≥ n there exists a compact subset X of the closed unit ball in \({\mathbb{R}^{2n}}\), such that X has Hausdorff dimension at most d + 1 and relative Hofer diameter bounded below by π / k(n, d), where k(n, d) is an explicitly defined integer.
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coisotropic displacement and small subsets of a Symplectic Manifold
Mathematische Zeitschrift, 2012Co-Authors: Jan Swoboda, Fabian ZiltenerAbstract:We prove a coisotropic intersection result and deduce the following: (a) Lower bounds on the displacement energy of a subset of a Symplectic Manifold, in particular a sharp stable energy-Gromov-width inequality. (b) A stable non-squeezing result for neighborhoods of products of unit spheres. (c) Existence of a “badly squeezable” set in \({\mathbb R^{2n}}\) of Hausdorff dimension at most d, for every n ≥ 2 and d ≥ n. (d) Existence of a stably exotic Symplectic form on \({\mathbb R^{2n}}\) , for every n ≥ 2. (e) Non-triviality of a new capacity, which is based on the minimal action of a regular coisotropic subManifold of dimension d.
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hofer geometry of a subset of a Symplectic Manifold
arXiv: Symplectic Geometry, 2011Co-Authors: Jan Swoboda, Fabian ZiltenerAbstract:To every closed subset $X$ of a Symplectic Manifold $(M,\omega)$ we associate a natural group of Hamiltonian diffeomorphisms $Ham(X,\omega)$. We equip this group with a semi-norm $\Vert\cdot\Vert^{X,\omega}$, generalizing the Hofer norm. We discuss $Ham(X,\omega)$ and $\Vert\cdot\Vert^{X,\omega}$ if $X$ is a Symplectic or isotropic subManifold. The main result involves the relative Hofer diameter of $X$ in $M$. Its first part states that for the unit sphere in $R^{2n}$ this diameter is bounded below by $\frac\pi2$, if $n\geq2$. Its second part states that for $n\geq2$ and $d\geq n+1$ there exists a compact set in $R^{2n}$ of Hausdorff dimension at most $d$, with relative Hofer diameter bounded below by $\pi/k(n,d)$, where $k(n,d)$ is an explicitly defined integer.
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coisotropic displacement and small subsets of a Symplectic Manifold
arXiv: Differential Geometry, 2011Co-Authors: Jan Swoboda, Fabian ZiltenerAbstract:We prove a coisotropic intersection result and deduce the following: 1. Lower bounds on the displacement energy of a subset of a Symplectic Manifold, in particular a sharp stable energy-Gromov-width inequality. 2. A stable non-squeezing result for neighborhoods of products of unit spheres. 3. Existence of a "badly squeezable" set in $\mathbb{R}^{2n}$ of Hausdorff dimension at most $d$, for every $n\geq2$ and $d\geq n$. 4. Existence of a stably exotic Symplectic form on $\mathbb{R}^{2n}$, for every $n\geq2$. 5. Non-triviality of a new capacity, which is based on the minimal Symplectic area of a regular coisotropic subManifold of dimension $d$.
Dan Mangoubi - One of the best experts on this subject based on the ideXlab platform.
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Spectral flexibility of Symplectic Manifolds T ^2 × M
Mathematische Annalen, 2008Co-Authors: Dan MangoubiAbstract:We consider Riemannian metrics compatible with the natural Symplectic structure on T ^2 × M , where T ^2 is a Symplectic 2-torus and M is a closed Symplectic Manifold. To each such metric we attach the corresponding Laplacian and consider its first positive eigenvalue λ_1. We show that λ_1 can be made arbitrarily large by deforming the metric structure, keeping the Symplectic structure fixed. The conjecture is that the same is true for any Symplectic Manifold of dimension ≥ 4. We reduce the general conjecture to a purely Symplectic question.
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Spectral flexibility of Symplectic Manifolds T 2 × M
Mathematische Annalen, 2007Co-Authors: Dan MangoubiAbstract:We consider Riemannian metrics compatible with the natural Symplectic structure on T 2 × M, where T 2 is a Symplectic 2-torus and M is a closed Symplectic Manifold. To each such metric we attach the corresponding Laplacian and consider its first positive eigenvalue λ1. We show that λ1 can be made arbitrarily large by deforming the metric structure, keeping the Symplectic structure fixed. The conjecture is that the same is true for any Symplectic Manifold of dimension ≥ 4. We reduce the general conjecture to a purely Symplectic question.
Jan Swoboda - One of the best experts on this subject based on the ideXlab platform.
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hofer geometry of a subset of a Symplectic Manifold
Geometriae Dedicata, 2013Co-Authors: Jan Swoboda, Fabian ZiltenerAbstract:To every closed subset X of a Symplectic Manifold (M, ω) we associate a natural group of Hamiltonian diffeomorphisms Ham (X, ω). We equip this group with a semi-norm \({\Vert\cdot\Vert^{X, \omega}}\), generalizing the Hofer norm. We discuss Ham (X, ω) and \({\Vert\cdot\Vert^{X, \omega}}\) if X is a Symplectic or isotropic subManifold. The main result involves the relative Hofer diameter of X in M. Its first part states that for the unit sphere in \({\mathbb{R}^{2n}}\) this diameter is bounded below by \({\frac{\pi}{2}}\) , if n ≥ 2. Its second part states that for n ≥ 2 and d ≥ n there exists a compact subset X of the closed unit ball in \({\mathbb{R}^{2n}}\), such that X has Hausdorff dimension at most d + 1 and relative Hofer diameter bounded below by π / k(n, d), where k(n, d) is an explicitly defined integer.
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coisotropic displacement and small subsets of a Symplectic Manifold
Mathematische Zeitschrift, 2012Co-Authors: Jan Swoboda, Fabian ZiltenerAbstract:We prove a coisotropic intersection result and deduce the following: (a) Lower bounds on the displacement energy of a subset of a Symplectic Manifold, in particular a sharp stable energy-Gromov-width inequality. (b) A stable non-squeezing result for neighborhoods of products of unit spheres. (c) Existence of a “badly squeezable” set in \({\mathbb R^{2n}}\) of Hausdorff dimension at most d, for every n ≥ 2 and d ≥ n. (d) Existence of a stably exotic Symplectic form on \({\mathbb R^{2n}}\) , for every n ≥ 2. (e) Non-triviality of a new capacity, which is based on the minimal action of a regular coisotropic subManifold of dimension d.
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hofer geometry of a subset of a Symplectic Manifold
arXiv: Symplectic Geometry, 2011Co-Authors: Jan Swoboda, Fabian ZiltenerAbstract:To every closed subset $X$ of a Symplectic Manifold $(M,\omega)$ we associate a natural group of Hamiltonian diffeomorphisms $Ham(X,\omega)$. We equip this group with a semi-norm $\Vert\cdot\Vert^{X,\omega}$, generalizing the Hofer norm. We discuss $Ham(X,\omega)$ and $\Vert\cdot\Vert^{X,\omega}$ if $X$ is a Symplectic or isotropic subManifold. The main result involves the relative Hofer diameter of $X$ in $M$. Its first part states that for the unit sphere in $R^{2n}$ this diameter is bounded below by $\frac\pi2$, if $n\geq2$. Its second part states that for $n\geq2$ and $d\geq n+1$ there exists a compact set in $R^{2n}$ of Hausdorff dimension at most $d$, with relative Hofer diameter bounded below by $\pi/k(n,d)$, where $k(n,d)$ is an explicitly defined integer.
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coisotropic displacement and small subsets of a Symplectic Manifold
arXiv: Differential Geometry, 2011Co-Authors: Jan Swoboda, Fabian ZiltenerAbstract:We prove a coisotropic intersection result and deduce the following: 1. Lower bounds on the displacement energy of a subset of a Symplectic Manifold, in particular a sharp stable energy-Gromov-width inequality. 2. A stable non-squeezing result for neighborhoods of products of unit spheres. 3. Existence of a "badly squeezable" set in $\mathbb{R}^{2n}$ of Hausdorff dimension at most $d$, for every $n\geq2$ and $d\geq n$. 4. Existence of a stably exotic Symplectic form on $\mathbb{R}^{2n}$, for every $n\geq2$. 5. Non-triviality of a new capacity, which is based on the minimal Symplectic area of a regular coisotropic subManifold of dimension $d$.
C Di Pietro - One of the best experts on this subject based on the ideXlab platform.
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a spectral sequence associated with a Symplectic Manifold
Doklady Mathematics, 2007Co-Authors: A M Vinogradov, C Di PietroAbstract:With a Symplectic Manifold a spectral sequence converging to its de Rham cohomology is associated. A method of computation of its terms is presented together with some stabilization results. As an application a characterization of Symplectic harmonic Manifolds is given and a relationship with the C–spectral sequence is indicated. Let (M,Ω) be a 2n–dimensional Symplectic Manifold and Λ(M) be the algebra of differential forms onM . Consider the ideal ΛL(M) of Λ(M), composed of all differential forms that vanish when restricted to any Lagrangian subManifold of M . This ideal is differentially closed and its powers constitute the Symplectic filtration in the de Rham complex of M. The corresponding spectral sequence {Ep,q r , d r } is called the Symplectic spectral sequence associated with (M,Ω). A motivation for this construction comes from the theory of C–spectral sequences (see [5]). Moreover, if M = T ∗N , then the Symplectic spectral sequence is nothing but the “classical part” of the C–spectral sequence associated with the differential equation dρ = 0, ρ ∈ Λ(N). 1. Notations and preliminaries In this section the notation is fixed and all necessary facts concerning Symplectic Manifolds (see [1, 2, 6] for further details) are collected. Throughout the paper (M,Ω) stands for a 2n–dimensional Symplectic Manifold , Λ = ∑ k Λ k for the algebra of differential forms on M , H(M) = ∑ k H (M) for the de Rham cohomology of M and D = ∑ k Dk for the algebra of multivectors on M . The isomorphism Γ1 : V ∈ D1 → V Ω ∈ Λ of C∞(M)–modules extends uniquely to a C∞(M)–algebra isomorphism Γ: D → Λ. P = Γ−1(Ω) is called the corresponding to Ω Poisson bivector. C∞(M)–linear operators : Λ → Λ , ω = ω ∧ Ω, ⊥ : Λ → Λk−2 , ⊥ω = P ω, acting on Λ are basic for our purposes. Put Λ = im and Λ = ker⊥. Elements of Λ are called effective forms. Another very useful fact is the Hodge–Lepage expansion (see, for instance, [2]):
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spectral sequence associated with a Symplectic Manifold
arXiv: Symplectic Geometry, 2006Co-Authors: A M Vinogradov, C Di PietroAbstract:A method of computation of its terms is presented together with some stabilization results. As an application a characterization of Symplectic harmonic Manifolds is given and a relationship with the C-spectral sequence is indicated.
