Showing papers on "Symplectic vector space published in 2001"

Uniqueness of Symplectic Canonical Class, Surface Cone and Symplectic Cone of 4-Manifolds with B+ = 1

Abstract: Let M be a closed oriented smooth 4-manifold admitting symplectic structures. If M is minimal and has b+ = 1, we prove that there is a unique symplectic canonical class up to sign, and any real second cohomology class of positive square is represented by symplectic forms. Similar results hold when M is not minimal.

Deformations of Symplectic Lie Algebroids, Deformations of Holomorphic Symplectic Structures, and Index Theorems

Abstract: Recently Kontsevich solved the classification problem for deformation quantizations of all Poisson structures on a manifold In this paper we study those Poisson structures for which the explicit methods of Fedosov can be applied, namely the Poisson structures coming from symplectic Lie algebroids, as well as holomorphic symplectic structures For deformations of these structures we prove the classification theorems and a general a general index theorem

Deformation theory of singular symplectic n-folds

Abstract: By a symplectic manifold (or a symplectic n-fold) we mean a compact Kaehler manifold of even dimension n with a non-degenerate holomorphic 2form ω, i.e. ω is a nowhere-vanishing n-form. This notion is generalized to a variety with singularities. We call X a projective symplectic variety if X is a normal projective variety with rational Gorenstein singularities and if the regular locus U of X admits a non-degenerate holomorphic 2-form ω. A symplectic variety will play an important role together with a singular Calabi-Yau variety in the generalized Bogomolov decomposition conjecture. Now that essentially a few examples of symplectic manifolds are discovered, it seems an important task to seek new symplectic manifolds by deforming symplectic varieties. In this paper we shall study a projective symplectic variety from a view point of deformation theory. If X has a resolution π : X → X such that (X, πω) is a symplectic manifold, we say that X has a symplectic resolution. Our first results are concerned with a birational contraction map of a symplectic manifold.

Fedosov Deformation Quantization as a BRST Theory

Abstract: The relationship is established between the Fedosov deformation quantization of a general symplectic manifold and the BFV-BRST quantization of constrained dynamical systems. The original symplectic manifold ℳ is presented as a second class constrained surface in the fibre bundle ?* ρℳ which is a certain modification of a usual cotangent bundle equipped with a natural symplectic structure. The second class system is converted into the first class one by continuation of the constraints into the extended manifold, being a direct sum of ?* ρℳ and the tangent bundle Tℳ. This extended manifold is equipped with a nontrivial Poisson bracket which naturally involves two basic ingredients of Fedosov geometry: the symplectic structure and the symplectic connection. The constructed first class constrained theory, being equivalent to the original symplectic manifold, is quantized through the BFV-BRST procedure. The existence theorem is proven for the quantum BRST charge and the quantum BRST invariant observables. The adjoint action of the quantum BRST charge is identified with the Abelian Fedosov connection while any observable, being proven to be a unique BRST invariant continuation for the values defined in the original symplectic manifold, is identified with the Fedosov flat section of the Weyl bundle. The Fedosov fibrewise star multiplication is thus recognized as a conventional product of the quantum BRST invariant observables.

A note on symplectic singularities

Abstract: In this paper we shall prove that the singular locus of a symplectic singularity has no codimension 3 irreducible components. As a corollary, a symplectic singularity is terminal if and only if its singular locus has codimension $\geq 4$. It is hoped that a symplectic singularity has much stronger properties.

Symplectic hypersurfaces in the complement of an isotropic submanifold

Abstract: Using Donaldson's approximately holomorphic techniques, we construct symplectic hypersurfaces lying in the complement of any given compact isotropic submanifold of a compact symplectic manifold. We discuss the connection with rational convexity results in the Kahler case and various applications.

Matrix Integrals and the Geometry of Spinors

Abstract: We obtain the collection of symmetric and symplectic matrix integrals and the collection of Pfaffian tau-functions, recently described by Peng and Adler and van Moerbeke, as specific elements in the Spin-group orbit of the vacuum vector of a fermionic Fock space. This fermionic Fock space is the same space as one constructs to obtain the KP and 1-Toda lattice hierarchy.

Blowing Up Symplectic Orbifolds

Abstract: In the first part of this paper we study different blow-upconstructions on symplectic orbifolds. Unlike the manifold case,we can define different blow-ups by using different circleactions. In the second part, we use some of these constructions todescribe the behavior of reduced spaces of a Hamiltonian circleaction on a symplectic orbifold, when passing a critical level ofits Hamiltonian function. Using these descriptions, we generalize,in the manifold case, the wall-crossing theorem of Guillemin and Sternberg to the case of a Hamiltonian torus action not necessarily quasi-free and also the Duistermaat–Heckman theorem to intervalsof values of the Hamiltonian function containing critical values.

The topology of symplectic manifolds

Abstract: A topological structure is introduced that seems likely to provide a complete topological characterization of compact symplectic manifolds. The article begins with a leisurely introduction to symplectic manifolds from a topological viewpoint. It then focuses on Thurston's construction of a symplectic structure on the total space of a fiber bundle. This is generalized to a technique for putting a symplectic structure on the domain of a J-holomorphic map. A topological structure called a hyperpencil on a compact 2n-manifold is then defined; this is motivated by the special case of a linear system of curves on an algebraic manifold, and it generalizes the notion of a Lefschetz pencil on a 4-manifold (although the critical points of a hyperpencil can be much more complicated). A deformation class of hyperpencils determines an isotopy class of symplectic forms, via the above generalization of Thurston's construction. This correspondence seems to be essentially an inverse to the technique of Donaldson and Auroux for constructing linear systems on symplectic manifolds. The likely end result is that any symplectic form whose cohomology class is rational should be realized up to scale by a hyperpencil. This would topologically characterize symplectic manifolds as being those smooth manifolds admitting hyperpencils, and put a dense subset of all symplectic forms on a manifold (up to scale and isotopy) in bijective correspondence with the set of all hyperpencils on it modulo a suitable equivalence relation.

Compactified D=11 supermembranes and symplectic noncommutative gauge theories

Abstract: It is shown that a double compactified D=11 supermembrane with nontrivial wrapping may be formulated as a symplectic noncommutative gauge theory on the world volume The symplectic noncommutative structure is intrinsically obtained from the symplectic two-form on the world volume defined by the minimal configuration of its Hamiltonian The gauge transformations on the symplectic fibration are generated by the area preserving diffeomorphisms on the world volume Geometrically, this gauge theory corresponds to a symplectic fibration over a compact Riemann surface with a symplectic connection

The Guillemin formula and Kähler metrics on toric symplectic manifolds

Abstract: We discuss the construction of toric Kahler metrics on symplectic 2n-manifolds with hamiltonian n-torus action and present a simple derivation of the Guillemin formula for a distinguished Kahler metric on any such manifold. the results also apply to orbifolds.

The symplectic camel and phase space quantization

Abstract: We show that a result of symplectic topology, Gromov's non-squeezing theorem, also known as the `principle of the symplectic camel', can be used to quantize phase space in cells. That quantization scheme leads to the correct energy levels for integrable systems and to Maslov quantization of Lagrangian manifolds by purely topological arguments. We finally show that the argument leading to the proof of the non-squeezing theorem leads to a classical form of Heisenberg's inequalities.

Closed conformal vector fields and Lagrangian submanifolds in complex space forms

Abstract: We study a wide family of Lagrangian submanifolds in non flat complex space forms that we will call pseudoumbilical because of their geometric properties. They are determined by admitting a closed and conformal vector field X such that X is a principal direction of the shape operator AJX , being J the complex structure of the ambient manifold. We emphasize the case X = JH, where H is the mean curvature vector of the immersion, which are known as Lagrangian submanifolds with conformal Maslov form. In this family we offer different global characterizations of the Whitney spheres in the complex projective and hyperbolic spaces. Let M be a Kaehler manifold of complex dimension n. The Kaehler form Ω on M is given by Ω(v, w) = 〈v, Jw〉, being 〈, 〉 the metric and J the complex structure on M . An immersion φ : M −→ M of an n-dimensional manifold M is called Lagrangian if φ∗Ω ≡ 0. This property involves only the symplectic structure of M . In this family of Lagrangian submanifolds, one can study properties of the submanifold involving the Riemannian structure of M . One ∗Research partially supported by a DGICYT grant No. PB97-0785.

G-actions on graphs

Abstract: Let G be an n-dimensional torus and $\tau$ a Hamiltonian action of G on a compact symplectic manifold, M. If M is pre-quantizable one can associate with $\tau$ a representation of G on a virtual vector space, Q(M), by $\spin^{\CC}$-quantization. If M is a symplectic GKM manifold we will show that several well-known theorems about this ``quantum action'' of G: for example, the convexity theorem, the Kostant multiplicity theorem and the ``quantization commutes with reduction'' theorem for circle subgroups of G, are basically just theorems about G-actions on graphs.

Symplectic submanifolds from surface fibrations

Abstract: We give a simple construction yielding homology classes in (non-simply-connected) symplectic four-manifolds which admit infinitely many pairwise non-isotopic symplectic representatives. Examples are constructed in which the symplectic curves can have arbitrarily large genus. The examples are built from surface bundles over surfaces and involve only elementary techniques. As a corollary we see that a blow-up of any simply-connected complex projective surface contains a connected symplectic surface not isotopic to any complex curve.

Symplectic maps to projective spaces and symplectic invariants

Abstract: After reviewing recent results on symplectic Lefschetz pencils and symplectic branched covers of \CP2, we describe a new construction of maps from symplectic manifolds of any dimension to \CP2 and the associated monodromy invariants. We also show that a dimensional induction process makes it possible to describe any compact symplectic manifold by a series of words in braid groups and a word in a symmetric group.

Poisson Geometry in Constrained Systems

Abstract: Constrained Hamiltonian systems fall into the realm of presymplectic geometry. We show, however, that also Poisson geometry is of use in this context. For the case that the constraints form a closed algebra, there are two natural Poisson manifolds associated to the system, forming a symplectic dual pair with respect to the original, unconstrained phase space. We provide sufficient conditions so that the reduced phase space of the constrained system may be identified with a symplectic leaf in one of those. In the second class case the original constrained system may be reformulated equivalently as an abelian first class system in an extended phase space by these methods. Inspired by the relation of the Dirac bracket of a general second class constrained system to the original unconstrained phase space, we address the question of whether a regular Poisson manifold permits a leafwise symplectic embedding into a symplectic manifold. Necessary and sufficient for this is the vanishing of the characteristic form-class of the Poisson tensor, a certain element of the third relative cohomology.

A remark about Donaldson's construction of symplectic submanifolds

Abstract: We describe a simplification of Donaldson's arguments for the construction of symplectic hypersurfaces or Lefschetz pencils that makes it possible to avoid any reference to Yomdin's work on the complexity of real algebraic sets.

Multi-interval linear ordinary boundary value problems and complex symplectic algebra

Abstract: Introduction: Goals, organization Some definitions for multi-interval systems Complex symplectic spaces Single interval quasi-differential systems Multi-interval quasi-differential systems Boundary symplectic spaces for multi-interval systems Finite multi-interval systems Examples of complete Lagrangians Bibliography.

Complex Symplectic Geometry of Quasi-Fuchsian Space

Abstract: We study the complex symplectic geometry of the space QF(S) of quasi-Fuchsian structures of a compact orientable surface S of genus g > 1. We prove that QF(S) is a complex symplectic manifold. The complex symplectic structure is the complexification of the Weil–Petersson symplectic structure of Teichmuller space and is described in terms which look natural from the point of view of hyperbolic geometry.

A construction of symplectic connections through reduction

Abstract: We give an elementary construction of symplectic connections through reduction. This provides an elegant description of a class of symmetric spaces and gives examples of symplectic connections with Ricci type curvature, which are not locally symmetric; the existence of such symplectic connections was unknown.

Hamiltonian Description of Seismic Wave Propagation and Symplectic Method

Abstract: The process of seismic wave propagation virtually is a process of energy dissipation. While in practice, it is often described by elastic or scalar wave equation with the assumption of no dissipation. Under the mathematic frame of Hamiltonian dynamic system, the propagation of seismic wave is the evolution of the infinite dimensional Hamiltonian system. If without dissipation, the propagation essentially is a symplectic transformation with one parameter, and consequently, the numerical calculation methods of the propagation ought to be symplectic too, which is known as symplectic method. For simplicity, only the symplectic method based on scalar wave equation is given in this paper. Let wave field and its derivative construct a phase space, the scalar wave equation as an evolution equation of a linear Hamiltonian system has symplectic property. Consequently, after discretizing the wave field in time and phase space, many explicit, implicit and leap-frog symplectic schemes are deduced for numerical modeling. The scheme of Finite difference (FD) method and symplectic schemes are compared, and FD method is a good approximation of symplectic method. A second order explicit symplectic scheme and FD method are applied in the same conditions to get a wave field in a synthetic model and a single shot record in Marmousi model. The result illustrates that the two method can give the same wave field as long as the time step is enough little, otherwise accuracy of FD method may be questioned. The theory and methods in this paper provide a new approach for the theoretic and applied study of wave propagation.

Numerical Determination of a Canonical Form of a Symplectic Matrix

Abstract: We propose an algorithm that transforms a real symplectic matrix with a stable structure to a block diagonal form composed of three main blocks. The two extreme blocks of the same size are associated respectively with the eigenvalues outside and inside the unit circle. Moreover, these eigenvalues are symmetric with respect to the unit circle. The central block is in turn composed of several diagonal blocks whose eigenvalues are on the unit circle and satisfy a modification of the Krein-Gelfand-Lidskii criterion. The proposed algorithm also gives a qualitative criterion for structural stability.

Homotopy K3's with several symplectic structures

Abstract: In this note we prove that, for any n2 N, there exist a smooth 4{manifold, homotopic to a K3 surface, dened by applying the link surgery method of Fintushel{Stern to a certain 2{component graph link, which admits n inequivalent symplectic structures.

The canonical class of a symplectic four manifold

Abstract: In this article we present examples of simply connected symplectic 4-manifolds X whose canonical classes are represented by complicated disjoint unions of symplectic submanifolds of X: Theorem. Given finite collections {gi}, {mi}, i=1,...,n, of positive integers, there is a minimal symplectic simply connected 4-manifold X whose canonical class is represented by a disjoint union of embedded symplectic surfaces K ~ Sg1,1 « ... « Sg1,m1 « ... « Sgn,1 «... « S{gn,mn} where Sgi,j is a surface of genus gi. Furthermore, c12(X) = ch(X) - (2+ b) where b= S{i=1}n mi is the total number of connected components of the symplectic representative of the canonical class.