August 28, 2018 symplectic manifolds and pseudo holomorphic curves. Lalondejholomorphic curves and symplectic invariants. We give conditions on a compatible almost complex structure j on m. Lectures on pseudoholomorphic curves and the symplectic. Symplectic geometry on moduli spaces of j holomorphic curves. Yonggeun oh, hiro lee tanaka submitted on 10 mar 2020, last revised 16 mar 2020 this version, v2 abstract. This book will be particularly useful to graduate students and researchers who have basic literacy in symplectic geometry and algebraic topology, and would like to learn how to apply standard techniques from holomorphic curve theory without dwelling more than necessary on the analytical details. Basic elements of j holomorphic curve theory let m. Lectures on holomorphic curves in symplectic and contact. Symplectic geometry and holomorphic curves ias school of. A special accent will be made on applications to symplectic geometry in lowdimensional topology. Rational curves on holomorphic symplectic fourfolds. Applications of pseudoholomorphic curves to symplectic topology.
Holomorphic curves in symplectic geometry, m audin and f lafontaine, editors progress in math. Symplectic geometry on moduli spaces of j holomorphic curves j. Other readers will always be interested in your opinion of the books youve read. Gromovwitten invariants and symplectic vortices pdf. Holomorphic curves and global questions in contact geometry. P pansu, pseudo holomorphic curves in symplectic manifolds, from. The restricted threebody problem and holomorphic curves. 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. Applications of pseudo holomorphic curves to symplectic topology. The authors study the restricted threebody problem using recent techniques coming from the theory of pseudo holomorphic curves.
The theory of \j\ holomorphic curves has been of great importance since its introduction by gromov in 1985. The purpose of these notes is a more selfcontained presentation of the results of the authors in siti3. Gromovs compactness of pseudoholomorphic curves and symplectic geometry. This book explains the foundations of holomorphic curve theory in contact geometry. Pseudo holomorphic curves in symplectic manifolds, inventiones math. H2m, z is a symplectic form, and show applications and examples. The book serves as an introduction to holomorphic curves in symplectic manifolds, focusing on the case of fourdimensional symplectizations and symplectic cobordisms, and their applications to celestial mechanics. In mathematics, its applications include many key results in symplectic topology. An algebraic approach to virtual fundamental cycles on moduli spaces of pseudo holomorphic curves pardon, john, geometry. This content was uploaded by our users and we assume good faith they have the permission to share this book. Holomorphic curves in symplectic geometry springerlink. Lectures on holomorphic curves in symplectic and contact geometry.
Holomorphic curves in symplectic geometry book, 1994. An introduction to symplectic topology through sheaf theory princeton, fall 2010new york,spring2011 c. Genericity of positive topological entropy for geodesic flows on s2 knieper, gerhard and weiss, howard, journal of differential geometry, 2002. Nongeneric jholomorphic curves in symplectic 4manifolds. 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. Compactness for punctured holomorphic curves cieliebak, k. Contact and homogeneous symplectic geometry 32 chapter 5. In these lectures, we present the basic results on the theory of pseudoholomorphic. Holomorphic curves in symplectic geometry it seems that youre in usa. Gromovs compactness of pseudoholomorphic curves and. Lectures on holomorphic curves in symplectic and contact geometry work in progressversion 3.
The technology of punctured holomorphic curves and neckstretching allows us to translate between these two worlds and prove a theorem. Pseudoholomorphic curves in symplectic manifolds, inventiones math. This book is devoted to pseudo holomorphic curve methods in symplectic geometry. Removal of singularities and gromov compactness for symplectic vortices ott, andreas, journal of symplectic geometry, 2014. Introduction applications of pseudo holomorphic curves to symplectic topology. Holomorphic curves in low dimensions from symplectic.
Mathematics symplectic geometry, mathematics differential geometry. This is a set of expository lecture notes created originally for a graduate course on holomorphic curves taught at eth zurich and the humboldt university berlin in 20092010. An introduction to symplectic topology through sheaf theory. Citeseerx document details isaac councill, lee giles, pradeep teregowda. T parker, j wolfson, pseudo holomorphic maps and bubble trees, j. Symplectic geometry on moduli spaces of jholomorphic curves. Jholomorphic curves and symplectic topology dusa mcduff, dietmar salamon 2nd ed.
Holomorphic curves and continuation maps in liouville bundles. Citeseerx jholomorphic curves in symplectic geometry. Log symplectic structures are poisson structures that are determined by a symplectic form with logarithmic singularities. Riemannian geometry and pseudoholomorphic curves in symplectic geometry. Pseudo holomorphic curves in symplectic manifolds ihes. Holomorphic curves in symplectic geometry michele audin. The goal of the program is to explore different aspects of the theory of holomorphic curves and their interaction. Click download or read online button to get lectures on symplectic manifolds book now. We construct an unwrapped floer theory for bundles of liouville sectors. The gromovwitten invariants of a symplectic manifold m.
Since their introduction by gromov 4 in the mid1980s j holomorphic 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. Furthermore, away from the forms zero set, the metric and the 2form give a compatible almost complex structure and thus pseudo holomorphic subvarieties. As such it ideally serves as preparation and as entry point for a deeper study of the analysis underlying symplectic. An introduction to symplectic geometry springerlink. This site is like a library, use search box in the widget to get ebook that you want. Pseudoholomorphic curves in symplectic manifolds misha. This is a set of expository lecture notes created originally for a graduate course on holomorphic curves taught at eth. Symplectic geometry is a branch of differential geometry and differential topology that studies. Jholomorphic curves and symplectic topology dusa mcduff. Since their introduction by gromov 4 in the mid1980s j holomorphic 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.
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. Holomorphic curves in logsymplectic manifolds core. Chris wendl submitted on 7 nov 2010, last revised 27 may 2014 this version, v2 abstract. The aim of this part is to give a brief introduction to the symplectic geometry, by. Symplectic geometry on moduli spaces of jholomorphic. This text is an expanded version of the lectures delivered by the authors at the cime summer school symplectic 4manifolds and algebraic surfaces, cetraro italy, september 210, 2003. Darbouxs theorem and the moser deformation trick 6 1. Lectures on symplectic manifolds download ebook pdf. Jholomorphic curves and symplectic invariants springerlink. The goal of this paper is to formulate explicit hodgetheoretic criteria for.
Gauge theory and symplectic geometry, montreal, pq, 1995, nato adv. This note discusses the structure of j holomorphic curves in symplectic 4manifolds m,\om when j\in \jj\ss, the set of \omtame j for which a fixed chain \ss of transversally intersecting embedded spheres of selfintersection \le 2 is j holomorphic. By using a particular geometric problem as a starting point the authors guide the reader into the subject. Since the first edition of this book appeared in 2004, symplectic geometry has developed apace. Given an almost complex structure j on a manifold m, a map f from a riemann surface to m is called a pseudoholomorphic or j holomorphic curve if at each point p of the surface, the ordinary differential is a complex linear map with respect to the complex structures jp and jfp on tangent spaces. It was also one of the main inspirations for the creation of floer homology. 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. A selfdual harmonic 2form on a 4dimensional riemannian manifold is symplectic where it does not vanish. We will recall the relevant setup, the construction of the moduli spaces of curves, and their compactness and smoothness properties. August 28, 2018 symplectic manifolds and pseudoholomorphic curves download pdf. Darbouxs theorem and the moser deformation trick 7 1.
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