Distinguishing symplectic structures 334 chapter 10. Jholomorphic curves and symplectic topology dusa mcduff. Jholomorphic curves and symplectic topology pdf free. What is known about the moduli space of j holomorphic maps in particular, how do we construct a compact moduli space of j holomorphic maps. Holomorphic curves in symplectic geometry springerlink. In other words, jmakes tm into a complex vector bundle and we have the canonical decomposition tm. See 57 for a detailed exposition on pseudo holomorphic curves. Aug 28, 2018 symplectic manifolds and pseudo holomorphic curves 6 transformation groups 4 metric, measure, concentration and isoperimetric inequalities 12 stein manifolds and infinite coverings of kaehler manifolds 11 infinite cartesian products and symbolic geometry 8 formalization of genetic and biomolecular structures 6 ergosystems 7. The second half of the book then extends this program in two complementary directions. Darbouxs theorem and the moser deformation trick 7 1. But be forewarned, serious commitment is required from the reader and rightly so, of course. Witten theory is a basic tool in string theory, algebraic geometry, and symplectic geometry, and understanding its relationship to string topology is an exciting area of current and probably future research. Tabachnikov, editor, differential and symplectic topology of knots and curves.
Applications of pseudoholomorphic curves to symplectic topology. A parallel that one can draw between the two subjects is the analogy between geodesics in riemannian geometry and pseudoholomorphic curves in symplectic geometry. Jholomorphic curves and symplectic topology request pdf. First steps of symplectic topology, viiith international congress on mathemati cal physics marseille, 1986, 116, world sci. Introduction one of the main problems in the theory of irreducible holomorphic symplectic manifolds is the description of the ample cone in the picard group. Compactness for punctured holomorphic curves cieliebak, k. Mosers lemma and local triviality of symplectic differential geometry 17 2. Rational curves on holomorphic symplectic fourfolds brendan. An almost complex structure on a manifold m is a bundle endomorphism j. An almost complex structure on a manifold mis a bundle endomorphism j.
Rational curves on holomorphic symplectic fourfolds by brendan hassett and yuri tschinkel 1. Some questions in the theory of pseudoholomorphic curves. The purpose of these notes is a more selfcontained presentation of the results of the authors in siti3. A selfdual harmonic 2form on a 4dimensional riemannian manifold is symplectic where it does not vanish. P2 integral counts of holomorphic curves in smooth complex projective. Holomorphic curves in symplectic geometry progress in. Geodesics are curves of shortest length locally, while pseudoholomorphic curves are surfaces of minimal area. This theory is regarded as one of the most important tools in symplectic geometry. Holomorphic curves in symplectic geometry michele audin. Another important feature is that it is a 2dimensional geometry that measures the area of complex curves instead of the length of real curves. Lectures on pseudo holomorphic curves and the symplectic isotopy problem bernd siebert and gang tian abstract. Holomorphic curves in low dimensions from symplectic ruled.
It contains an introduction to symplectic geometry and relevant techniques of riemannian geometry, proofs of gromovs compactness theorem, an investigation of local properties of holomorphic curves, including positivity of intersections, and applications to lagrangian embeddings problems. Pseudoholomorphic curves in symplectic manifolds misha. Holomorphic curves, planar open books and symplectic fillings. Symplectic geometry on moduli spaces of jholomorphic curves j. All in all jholomorphic curves and symplectic topology is a scholarly work of great pedagogical value, and an evident sine quo non for entry into the subject of symplectic topology and the hot areas of research it generates and touches. We use cookies to offer you a better experience, personalize content, tailor advertising, provide social media features, and better understand the use of our services. Lectures on holomorphic curves in symplectic and contact geometry work in progressversion 3. Lectures on pseudoholomorphic curves and the symplectic. Other readers will always be interested in your opinion of the books youve read. One important difference is that, although all its concepts are initially expressed in. Holomorphic curves, planar open books and symplectic fillings a minicourse by chris wendl the overarching theme of this minicourse will be the properties of pseudoholomorphic curves and their use in proving global results about symplectic or contact manifolds based on more localized information. It turns out to be very different from the riemannian geometry with which we are familiar. Both concepts play a fundamental role in their respective disciplines. We will recall the relevant setup, the construction of the moduli spaces of curves, and their compactness and smoothness properties.
Lectures on holomorphic curves in symplectic and contact geometry. Introduced in 1985 by mikhail gromov, pseudoholomorphic curves have since revolutionized the study of symplectic manifolds. The theory of \j\ holomorphic curves has been of great importance since its introduction by gromov in 1985. Jholomorphic curves and symplectic topology 2nd dusa. Jholomorphic curves and symplectic topology mathematical. Lectures on holomorphic curves in symplectic and contact. Symplectic topology is a subfield of symplectic geometry, in which global properties of symplectic. Symplectic structures a new approach to geometry dusa mcduff introduction symplectic geometry is the geometry of a closed skewsymmetric form. The technology of punctured holomorphic curves and neckstretching allows us to translate between these two worlds and prove a theorem. Symplectic geometry on moduli spaces of jholomorphic curves. Furthermore, away from the forms zero set, the metric and the 2form give a compatible almost complex structure and thus pseudoholomorphic subvarieties. The point is that the expression on the right hand side of 1.
Observewhile most of our theorems have been applications of ideas from algebraic geometry to symplectic problems, this theorem is a result in real algebraic geometry which seems to need a symplectic proof. Whether youve loved the book or not, if you give your honest and detailed thoughts then people will find new books that are right for them. Rational curves on holomorphic symplectic fourfolds. Symplectic geometry and holomorphic curves ias school of. Pseudo holomorphic curves in symplectic manifolds ihes. What are symplectic manifolds, what are j holomorphic curves, and why do we care about them. Rational curves on holomorphic symplectic fourfolds brendan hassett and yuri tschinkel march 2001 1 introduction one of the main problems in the theory of irreducible holomorphic symplectic manifolds is the description of the ample cone in the picard group. We give a complete proof for gromovs compactness theorem for pseudo holomorphic curves both in the case of closed curves and curves with boundary. From symplectic geometry to symplectic topology 10 1. An introduction to symplectic topology through sheaf theory. Symplectic geometry originated from classical mechanics, where the canonical symplectic form on phase space appears in hamiltons equation.
Jholomorphic curves in symplectic geometry janko latschev pleinfeld, september 25 28, 2006 since their introduction by gromov 4 in the mid1980s jholomorphic curves have been one of the most widely used tools in symplectic geometry, leading to the formulation of various theories gromovwitten invariants, quantum co. The structure of pseudoholomorphic subvarieties for. It was also one of the main inspirations for the creation of floer homology. The question of how many holomorphic curves there are in a given homology class with constraints possibly is given by gromovwitten invariants. In mathematics, specifically in topology and geometry, a pseudoholomorphic curve or jholomorphic curve is a smooth map from a riemann surface into an almost complex manifold that satisfies the cauchyriemann equation. In mathematics, its applications include many key results in symplectic topology. Since their introduction by gromov 4 in the mid1980s jholomorphic curves have been one of the most widely used tools in symplectic geometry, leading to the formulation of various theories gromovwitten invariants, quantum cohomology, various floer homologies, symplectic field theory, fukaya category, answers to old questions in symplectic geometry various arnold. Jholomorphic curves and symplectic topology american. This book is devoted to pseudo holomorphic curve methods in symplectic geometry. Part 2 i will introduce certain moduli spaces of j holomorphic curves with extra tangency conditions. Applications of pseudo holomorphic curves to symplectic topology.
Abstract let m be a symplectic manifold, and a compact riemann surface. A special accent will be made on applications to symplectic geometry in lowdimensional topology. Thus symplectic geometry is essentially topological in nature. Holomorphic curves in symplectic geometry book, 1994. The goal of this paper is to formulate explicit hodgetheoretic criteria for. It is related to the theory of dynamical systems and via holomorphic curves to algebraic geometry. Sep 01, 2001 the goal of the program is to explore different aspects of the theory of holomorphic curves and their interaction. S i on the space s i of immersed symplectic surfaces in m, and show that the form is closed and nondegenerate, up to reparametrizations. Thus holomorphic symplectic manifolds also called hyperk ahler arebuilding blocksfor manifolds with k trivial, which are themselves building blocks in the classi cation of projective or compact k ahler manifolds.
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