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Introduction to symplectic topology

TL;DR: In this article, the authors present a survey of the history of classical and modern manifold geometry, from classical to modern, including linear and almost complex structures, and the Arnold conjecture of the group of symplectomorphisms.
Abstract: Introduction I. FOUNDATIONS 1. From classical to modern 2. Linear symplectic geometry 3. Symplectic manifolds 4. Almost complex structures II. SYMPLECTIC MANIFOLDS 5. Symplectic group actions 6. Symplectic fibrations 7. Constructing symplectic manifolds III. SYMPLECTOMORPHISMS 8. Area-preserving diffeomorphisms 9. Generating functions 10. The group of symplectomorphisms IV. SYMPLECTIC INVARIANTS 11. The Arnold conjecture 12. Symplectic capacities 13. New directions
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01 Jan 1999
TL;DR: In this article, the authors introduce surfaces in 4-manifolds complex surfaces and Kirby calculus, a calculus based on handelbodies and Kirby diagrams, which is used for handel bodies and kirby diagrams.
Abstract: 4-manifolds: Introduction Surfaces in 4-manifolds Complex surfaces Kirby calculus: Handelbodies and Kirby diagrams Kirby calculus More examples Applications: Branched covers and resolutions Elliptic and Lefschetz fibrations Cobordisms, $h$-cobordisms and exotic ${\mathbb{R}}^{4,}$s Symplectic 4-manifolds Stein surfaces Appendices: Solutions Notation, important figures Bibliography Index.

1,412 citations


Cites background from "Introduction to symplectic topology..."

  • ...[McS1], so we again focus on the main applications to 4-manifold topology while avoiding unnecessary coverage of other aspects of these theories....

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  • ...While many useful examples come from algebraic geometry [BPV] and symplectic topology [McS1], perhaps the most powerful general technique for existence results (particularly for manifolds with small Betti numbers) is Kirby calculus....

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Posted Content
TL;DR: In this paper, the concept of a generalized Kahler manifold has been introduced, which is equivalent to a bi-Hermitian geometry with torsion first discovered by physicists.
Abstract: Generalized complex geometry, as developed by Hitchin, contains complex and symplectic geometry as its extremal special cases. In this thesis, we explore novel phenomena exhibited by this geometry, such as the natural action of a B-field. We provide new examples, including some on manifolds admitting no known complex or symplectic structure. We prove a generalized Darboux theorem which yields a local normal form for the geometry. We show that there is an elliptic deformation theory and establish the existence of a Kuranishi moduli space. We then define the concept of a generalized Kahler manifold. We prove that generalized Kahler geometry is equivalent to a bi-Hermitian geometry with torsion first discovered by physicists. We then use this result to solve an outstanding problem in 4-dimensional bi-Hermitian geometry: we prove that there exists a Riemannian metric on the complex projective plane which admits exactly two distinct Hermitian complex structures with equal orientation. Finally, we introduce the concept of generalized complex submanifold, and show that such sub-objects correspond to D-branes in the topological A- and B-models of string theory.

1,380 citations

Journal ArticleDOI
TL;DR: A geometrical structure on even-dimensional manifolds is defined in this paper, which generalizes the notion of a Calabi-Yau manifold and also a symplectic manifold.
Abstract: A geometrical structure on even-dimensional manifolds is defined which generalizes the notion of a Calabi–Yau manifold and also a symplectic manifold. Such structures are of either odd or even type and can be transformed by the action both of diffeomorphisms and closed 2-forms. In the special case of six dimensions we characterize them as critical points of a natural variational problem on closed forms, and prove that a local moduli space is provided by an open set in either the odd or even cohomology. We introduce in this paper a geometrical structure on a manifold which generalizes both the concept of a Calabi–Yau manifold—a complex manifold with trivial canonical bundle—and that of a symplectic manifold. This is possibly a useful setting for the background geometry of recent developments in string theory; but this was not the original motivation for the author’s first encounter with this structure. It arose instead as part of a programme (following the papers [ 11, 12]) for characterizing special geometry in low dimensions by means of invariant functionals of differential forms. In this respect, the dimension six is particularly important. This paper has two aims, then: first to introduce the general concept, and then to look at the variational and moduli space problem in the special case of six dimensions. We begin with the definition in all dimensions of what we call generalized complex manifolds and generalized Calabi–Yau manifolds .

1,275 citations


Cites methods from "Introduction to symplectic topology..."

  • ...For example, Moser’s method applied to symplectic forms ω (see for example [ 15 ]) shows that the cohomology class [ω] ∈ H2(M,R) is a local modulus up to diffeomorphism for symplectic manifolds and clearly the cohomology class of the B-field in H2(M,R) defines B up to exact...

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Journal ArticleDOI
TL;DR: Generalized complex geometry encompasses complex and symplectic ge- ometry as its extremal special cases as mentioned in this paper, including generalized complex branes, which interpolate be- tween at bundles on Lagrangian submanifolds and holomorphic bundles on complex sub-mansifolds, and the basic properties of this geometry, including its enhanced symmetry group, elliptic deforma- tion theory, relation to Poisson geometry, and local structure theory.
Abstract: Generalized complex geometry encompasses complex and symplectic ge- ometry as its extremal special cases. We explore the basic properties of this geometry, including its enhanced symmetry group, elliptic deforma- tion theory, relation to Poisson geometry, and local structure theory. We also dene and study generalized complex branes, which interpolate be- tween at bundles on Lagrangian submanifolds and holomorphic bundles on complex submanifolds.

641 citations


Additional excerpts

  • ...ential form ρ= eB+iωΩ, where Ω is decomposable of degree 0 ≤ k≤ nand such that ωn−k ∧ Ω∧ Ω 6= 0 . Weinstein’s proof of the Darboux normal coordinate theorem for a family of symplectic structures (see [33]) can be used to find a leaf-preserving local diffeomorphism ϕtaking ωto a 2-form whose pullback to each leaf is the standard Darboux symplectic form on R 2n− k, i.e. ϕ∗ω R2n−2k×{pt} = ω 0 = dx 1 ∧ dx 2...

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Journal ArticleDOI
TL;DR: In this article, a series of compactness results for moduli spaces of holomorphic curves arising in Symplectic field theory is presented. But these results generalize Gromov's compactness theorem in (8) as well as compactness theorems in Floer homology theory, and in contact geometry, (9, 19).
Abstract: This is one in a series of papers devoted to the foundations of Symplectic Field Theory sketched in (4). We prove compactness results for moduli spaces of holomorphic curves arising in Symplectic Field Theory. The theorems generalize Gromov's compactness theorem in (8) as well as compactness theorems in Floer homology theory, (6, 7), and in contact geometry, (9, 19).

575 citations