A vanishing theorem for twisted Alexander polynomials with applications to symplectic 4-manifolds

Stefan Friedl; Stefano Vidussi

Displaying similar documents to “A vanishing theorem for twisted Alexander polynomials with applications to symplectic 4-manifolds”

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 $Φ * ω_{Y} = c ω_{X}$ .

Moduli spaces of polarized irreducible symplectic manifolds are not necessarily connected

Apostol Apostolov (2014)

Annales de l’institut Fourier

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We show that the moduli space of polarized irreducible symplectic manifolds of $K 3^{[n]}$ -type, of fixed polarization type, is not always connected. This can be derived as a consequence of Eyal Markman’s characterization of polarized parallel-transport operators of $K 3^{[n]}$ -type.

Fano manifolds of degree ten and EPW sextics

Atanas Iliev, Laurent Manivel (2011)

Annales scientifiques de l'École Normale Supérieure

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O’Grady showed that certain special sextics in $ℙ^{5}$ called EPW sextics admit smooth double covers with a holomorphic symplectic structure. We propose another perspective on these symplectic manifolds, by showing that they can be constructed from the Hilbert schemes of conics on Fano fourfolds of degree ten. As applications, we construct families of Lagrangian surfaces in these symplectic fourfolds, and related integrable systems whose fibers are intermediate Jacobians.

$𝒞^{0}$ -rigidity of characteristics in symplectic geometry

Emmanuel Opshtein (2009)

Annales scientifiques de l'École Normale Supérieure

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The paper concerns a $𝒞^{0}$ -rigidity result for the characteristic foliations in symplectic geometry. A symplectic homeomorphism (in the sense of Eliashberg-Gromov) which preserves a smooth hypersurface also preserves its characteristic foliation.

Symplectic periods of the continuous spectrum of $GL (2 n)$

Shunsuke Yamana (2014)

Annales de l’institut Fourier

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We provide a formula for the symplectic period of an Eisenstein series on $GL (2 n)$ and determine when it is not identically zero.

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.

Projective structure, $\tilde{SL} (3, ℝ)$ and the symplectic Dirac operator

Marie Holíková, Libor Křižka, Petr Somberg (2016)

Archivum Mathematicum

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Inspired by the results on symmetries of the symplectic Dirac operator, we realize symplectic spinor fields and the symplectic Dirac operator in the framework of (the double cover of) homogeneous projective structure in two real dimensions. The symmetry group of the homogeneous model of the double cover of projective geometry in two real dimensions is $\tilde{} (3,)$ .

Fundamental group of $Symp (M, ω)$ with no circle action

Jarek Kędra (2009)

Archivum Mathematicum

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We show that $π_{1} (Symp (M, ω))$ can be nontrivial for $M$ that does not admit any symplectic circle action.

Generalized Conley-Zehnder index

Jean Gutt (2014)

Annales de la faculté des sciences de Toulouse Mathématiques

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The Conley-Zehnder index associates an integer to any continuous path of symplectic matrices starting from the identity and ending at a matrix which does not admit $1$ as an eigenvalue. Robbin and Salamon define a generalization of the Conley-Zehnder index for any continuous path of symplectic matrices; this generalization is half integer valued. It is based on a Maslov-type index that they define for a continuous path of Lagrangians in a symplectic vector space $(W, \bar{Ω})$ , having chosen a given...

On compact homogeneous symplectic manifolds

P. B. Zwart, William M. Boothby (1980)

Annales de l'institut Fourier

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In this paper the authors study compact homogeneous spaces $G / K$ (of a Lie group $G$ ) on which there if defined a $G$ -invariant symplectic form $Ω$ . It is an important feature of the paper that very little is assumed concerning $G$ and $K$ . The essential assumptions are: (1) $G$ is connected and (2) $K$ is uniform (i.e., $G / K$ is compact). Further, for convenience only and with no loss of generality, it is supposed that $G$ is simply connected and $K$ contains no connected normal subgroup of $G$ , i.e., that $G$ acts...

The ℤ₂-cohomology cup-length of real flag manifolds

Július Korbaš, Juraj Lörinc (2003)

Fundamenta Mathematicae

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Using fiberings, we determine the cup-length and the Lyusternik-Shnirel’man category for some infinite families of real flag manifolds $O (n ₁ + . . . + n_{q}) / O (n ₁) \times . . . \times O (n_{q})$ , q ≥ 3. We also give, or describe ways to obtain, interesting estimates for the cup-length of any $O (n ₁ + . . . + n_{q}) / O (n ₁) \times . . . \times O (n_{q})$ , q ≥ 3. To present another approach (combining well with the “method of fiberings”), we generalize to the real flag manifolds Stong’s approach used for calculations in the ℤ₂-cohomology algebra of the Grassmann manifolds.

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...

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...

Holonomy groups of complete flat manifolds

Michał Sadowski (2007)

Banach Center Publications

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We present short direct proofs of two known properties of complete flat manifolds. They say that the diffeomorphism classes of m-dimensional complete flat manifolds form a finite set $S_{C F} (m)$ and that each element of $S_{C F} (m)$ is represented by a manifold with finite holonomy group.