scholarly journals Desingularizing $\boldsymbol{b^m}$-Symplectic Structures

2017 ◽  
Vol 2019 (10) ◽  
pp. 2981-2998 ◽  
Author(s):  
Victor Guillemin ◽  
Eva Miranda ◽  
Jonathan Weitsman
Keyword(s):  
Canonical Form ◽  
Symplectic Form ◽  
Poisson Manifold ◽  
Symplectic Structures ◽  
Symplectic Forms ◽  
The Family

Abstract A $2n$-dimensional Poisson manifold $(M ,\Pi)$ is said to be $b^m$-symplectic if it is symplectic on the complement of a hypersurface $Z$ and has a simple Darboux canonical form at points of $Z$ which we will describe below. In this article, we will discuss a desingularization procedure which, for $m$ even, converts $\Pi$ into a family of symplectic forms $\omega_{\epsilon}$ having the property that $\omega_{\epsilon}$ is equal to the $b^m$-symplectic form dual to $\Pi$ outside an $\epsilon$-neighborhood of $Z$ and, in addition, converges to this form as $\epsilon$ tends to zero in a sense that will be made precise in the theorem below. We will then use this construction to show that a number of somewhat mysterious properties of $b^m$-manifolds can be more clearly understood by viewing them as limits of analogous properties of the $\omega_{\epsilon}$’s. We will also prove versions of these results for $m$ odd; however, in the odd case the family $\omega_{\epsilon}$ has to be replaced by a family of “folded” symplectic forms.

scholarly journals Pseudoholomorphic tori in the Kodaira–Thurston manifold

2015 ◽  
Vol 151 (12) ◽  
pp. 2212-2250
Author(s):  
Jonathan David Evans ◽  
Jarek Kędra
Keyword(s):  
Lie Algebra ◽  
Lie Group ◽  
Symplectic Form ◽  
Kähler Manifold ◽  
Witten Invariant ◽  
Nilpotent Lie Group ◽  
Cocompact Lattice ◽  
Symplectic Forms ◽  
The Family ◽  
Genus One

The Kodaira–Thurston manifold is a quotient of a nilpotent Lie group by a cocompact lattice. We compute the family Gromov–Witten invariants which count pseudoholomorphic tori in the Kodaira–Thurston manifold. For a fixed symplectic form the Gromov–Witten invariant is trivial so we consider the twistor family of left-invariant symplectic forms which are orthogonal for some fixed metric on the Lie algebra. This family defines a loop in the space of symplectic forms. This is the first example of a genus one family Gromov–Witten computation for a non-Kähler manifold.

scholarly journals Integrable Systems on Singular Symplectic Manifolds: From Local to Global

2021 ◽  
Author(s):  
Robert Cardona ◽  
Eva Miranda
Keyword(s):  
Global Existence ◽  
Integrable System ◽  
Symplectic Manifold ◽  
Integrable Systems ◽  
Symplectic Form ◽  
Symplectic Manifolds ◽  
Symplectic Structures ◽  
Critical Set ◽  
Symplectic Forms ◽  
Topological Obstructions

Abstract In this article, we consider integrable systems on manifolds endowed with symplectic structures with singularities of order one. These structures are symplectic away from a hypersurface where the symplectic volume goes either to infinity or to zero transversally, yielding either a $b$-symplectic form or a folded symplectic form. The hypersurface where the form degenerates is called critical set. We give a new impulse to the investigation of the existence of action-angle coordinates for these structures initiated in [34] and [35] by proving an action-angle theorem for folded symplectic integrable systems. Contrary to expectations, the action-angle coordinate theorem for folded symplectic manifolds cannot be presented as a cotangent lift as done for symplectic and $b$-symplectic forms in [34]. Global constructions of integrable systems are provided and obstructions for the global existence of action-angle coordinates are investigated in both scenarios. The new topological obstructions found emanate from the topology of the critical set $Z$ of the singular symplectic manifold. The existence of these obstructions in turn implies the existence of singularities for the integrable system on $Z$.

scholarly journals The Homology Class of a Poisson Transversal

2018 ◽  
Vol 2020 (10) ◽  
pp. 2952-2976
Author(s):  
Pedro Frejlich ◽  
Ioan Mărcuț
Keyword(s):  
Homology Class ◽  
Poisson Manifold ◽  
Poisson Manifolds ◽  
Poisson Structures ◽  
Symplectic Structures ◽  
Symplectic Case

Abstract This note is devoted to the study of the homology class of a compact Poisson transversal in a Poisson manifold. For specific classes of Poisson structures, such as unimodular Poisson structures and Poisson manifolds with closed leaves, we prove that all their compact Poisson transversals represent nontrivial homology classes, generalizing the symplectic case. We discuss several examples in which this property does not hold, as well as a weaker version of this property, which holds for log-symplectic structures. Finally, we extend our results to Dirac geometry.

scholarly journals Symplectic Forms on Six-dimensional Real Solvable Lie Algebras I

2009 ◽  
Vol 16 (02) ◽  
pp. 253-266 ◽  
Author(s):  
Rutwig Campoamor-Stursberg
Keyword(s):  
Lie Algebras ◽  
Symplectic Form ◽  
Nilpotent Lie Algebras ◽  
Symplectic Forms ◽  
Direct Sum ◽  
Solvable Lie Algebras

We analyze symplectic forms on six-dimensional real solvable and non-nilpotent Lie algebras. More precisely, we obtain all those algebras endowed with a symplectic form that decompose as the direct sum of two ideals or are indecomposable solvable algebras with a four-dimensional nilradical.

scholarly journals Construction of symplectic structures on 4-manifolds with a free circle action

2012 ◽  
Vol 142 (2) ◽  
pp. 359-370 ◽  
Author(s):  
Stefan Friedl ◽  
Stefano Vidussi
Keyword(s):  
Circle Action ◽  
Symplectic Structures ◽  
Symplectic Forms

Let M be a closed 4-manifold with a free circle action. If the orbit manifold N3 satisfies an appropriate fibering condition, then we show how to represent a cone in H2(M; ℝ) by symplectic forms. This generalizes earlier constructions by Thurston, Bouyakoub and Fernández et al. In the case that M is the product 4-manifold S1 × N, our construction complements our previous results and allows us to determine completely the symplectic cone of such 4-manifolds.

scholarly journals GENERALIZED CALABI–YAU STRUCTURES, K3 SURFACES, AND B-FIELDS

2005 ◽  
Vol 16 (01) ◽  
pp. 13-36 ◽  
Author(s):  
DANIEL HUYBRECHTS
Keyword(s):  
Moduli Space ◽  
Moduli Spaces ◽  
Symplectic Form ◽  
Weak Form ◽  
K3 Surfaces ◽  
Standard Theory ◽  
General Notion ◽  
Symplectic Structures ◽  
Special Cases ◽  
Torelli Theorem

Generalized Calabi–Yau structures, a notion recently introduced by Hitchin, are studied in the case of K3 surfaces. We show how they are related to the classical theory of K3 surfaces and to moduli spaces of certain SCFT as studied by Aspinwall and Morrison. It turns out that K3 surfaces and symplectic structures are both special cases of this general notion. The moduli space of generalized Calabi–Yau structures admits a canonical symplectic form with respect to which the moduli space of symplectic structures is Lagrangian. The standard theory of K3 surfaces implies surjectivity of the period map and a weak form of the Global Torelli theorem.

scholarly journals On the compatible weakly nonlocal Poisson brackets of hydrodynamic type

2002 ◽  
Vol 32 (10) ◽  
pp. 587-614 ◽  
Author(s):  
Andrei Ya. Maltsev
Keyword(s):  
Integrable Hierarchies ◽  
Riemannian Metrics ◽  
Riemannian Metric ◽  
Hydrodynamic Type ◽  
Poisson Brackets ◽  
Symplectic Structures ◽  
Symplectic Forms ◽  
Hamiltonian Functions ◽  
Hamiltonian Operators ◽  
Nonlocal Part

We consider the pairs of general weakly nonlocal Poisson brackets of hydrodynamic type (Ferapontov brackets) and the corresponding integrable hierarchies. We show that, under the requirement of the nondegeneracy of the corresponding “first” pseudo-Riemannian metricg(0) νμand also some nondegeneracy requirement for the nonlocal part, it is possible to introduce a “canonical” set of “integrable hierarchies” based on the Casimirs, momentum functional and some “canonical Hamiltonian functions.” We prove also that all the “higher” “positive” Hamiltonian operators and the “negative” symplectic forms have the weakly nonlocal form in this case. The same result is also true for “negative” Hamiltonian operators and “positive” symplectic structures in the case when both pseudo-Riemannian metricsg(0) νμandg(1) νμare nondegenerate.

scholarly journals FOLIATIONS ASSOCIATED TO REGULAR POISSON STRUCTURES

2001 ◽  
Vol 03 (03) ◽  
pp. 441-456 ◽  
Author(s):  
MÉLANIE BERTELSON
Keyword(s):  
Poisson Structure ◽  
Symplectic Form ◽  
Poisson Manifold ◽  
Poisson Structures

A regular Poisson manifold can be described as a foliated space carrying a tangentially symplectic form. Examples of foliations are produced here that are not induced by any Poisson structure although all the basic obstructions vanish.

scholarly journals An invitation to singular symplectic geometry

2019 ◽  
Vol 16 (supp01) ◽  
pp. 1940008 ◽  
Author(s):  
Roisin Braddell ◽  
Amadeu Delshams ◽  
Eva Miranda ◽  
Cédric Oms ◽  
Arnau Planas
Keyword(s):  
Celestial Mechanics ◽  
Symplectic Geometry ◽  
Symplectic Structure ◽  
Symplectic Manifolds ◽  
Symplectic Structures ◽  
Contact Manifolds ◽  
Symplectic Forms

In this paper, we analyze in detail a collection of motivating examples to consider [Formula: see text]-symplectic forms and folded-type symplectic structures. In particular, we provide models in Celestial Mechanics for every [Formula: see text]-symplectic structure. At the end of the paper, we introduce the odd-dimensional analogue to [Formula: see text]-symplectic manifolds: [Formula: see text]-contact manifolds.

scholarly journals Deformations of Pre-symplectic Structures and the Koszul L∞-algebra

2018 ◽  
Vol 2020 (14) ◽  
pp. 4191-4237 ◽  
Author(s):  
Florian Schätz ◽  
Marco Zambon
Keyword(s):  
Lie Algebra ◽  
Deformation Theory ◽  
Symplectic Structure ◽  
Poisson Manifold ◽  
Symplectic Structures ◽  
Characteristic Foliation ◽  
Infinitesimal Deformations ◽  
Fixed Rank

Abstract We study the deformation theory of pre-symplectic structures, that is, closed 2-forms of fixed rank. The main result is a parametrization of nearby deformations of a given pre-symplectic structure in terms of an $L_{\infty }$-algebra, which we call the Koszul $L_{\infty }$-algebra. This $L_{\infty }$-algebra is a cousin of the Koszul dg Lie algebra associated to a Poisson manifold. In addition, we show that a quotient of the Koszul $L_{\infty }$-algebra is isomorphic to the $L_{\infty }$-algebra that controls the deformations of the underlying characteristic foliation. Finally, we show that the infinitesimal deformations of pre-symplectic structures and of foliations are both obstructed.

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