The goal of the program is to explore different aspects of the theory of holomorphic curves and their interaction. Basic elements of j holomorphic curve theory let m. 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. Pseudo holomorphic curves in symplectic manifolds, inventiones math. 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. We will recall the relevant setup, the construction of the moduli spaces of curves, and their compactness and smoothness properties. Mathematics symplectic geometry, mathematics differential geometry.
Furthermore, away from the forms zero set, the metric and the 2form give a compatible almost complex structure and thus pseudo holomorphic subvarieties. Introduction applications of pseudo holomorphic curves to symplectic topology. Jholomorphic curves and symplectic topology dusa mcduff. This book explains the foundations of holomorphic curve theory in contact geometry. The gromovwitten invariants of a symplectic manifold m. Holomorphic curves in symplectic geometry, m audin and f lafontaine, editors progress in math. A selfdual harmonic 2form on a 4dimensional riemannian manifold is symplectic where it does not vanish. In these lectures, we present the basic results on the theory of pseudoholomorphic. Pseudoholomorphic curves in symplectic manifolds misha. 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. Lectures on pseudo holomorphic curves and the symplectic isotopy problem bernd siebert and gang tian abstract.
Applications of pseudoholomorphic curves to symplectic topology. Removal of singularities and gromov compactness for symplectic vortices ott, andreas, journal of symplectic geometry, 2014. Chris wendl submitted on 7 nov 2010, last revised 27 may 2014 this version, v2 abstract. The restricted threebody problem and holomorphic curves. The goal of this paper is to formulate explicit hodgetheoretic criteria for. From symplectic geometry to symplectic topology 10 1. Positivity of symplectic area for perturbed jholomorphic. An introduction to symplectic topology through sheaf theory.
Rational curves on holomorphic symplectic fourfolds. Applications of pseudo holomorphic curves to symplectic topology. Symplectic geometry on moduli spaces of jholomorphic. The purpose of these notes is a more selfcontained presentation of the results of the authors in siti3. Gromovs compactness of pseudoholomorphic curves and. August 28, 2018 symplectic manifolds and pseudo holomorphic curves. The technology of punctured holomorphic curves and neckstretching allows us to translate between these two worlds and prove a theorem. Jholomorphic curves and symplectic topology dusa mcduff, dietmar salamon 2nd ed.
Pseudo holomorphic curves in symplectic manifolds ihes. Gromovs compactness of pseudoholomorphic curves and symplectic geometry. Click download or read online button to get lectures on symplectic manifolds book now. 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. August 28, 2018 symplectic manifolds and pseudoholomorphic curves download pdf. 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. Holomorphic curves and continuation maps in liouville bundles. In mathematics, its applications include many key results in symplectic topology. Holomorphic curves in low dimensions from symplectic. We give conditions on a compatible almost complex structure j on m. Holomorphic curves in symplectic geometry springerlink. Compactness for punctured holomorphic curves cieliebak, k. Citeseerx document details isaac councill, lee giles, pradeep teregowda.
An introduction to symplectic geometry springerlink. Contact and homogeneous symplectic geometry 32 chapter 5. Symplectic geometry is a branch of differential geometry and differential topology that studies. Holomorphic curves and global questions in contact geometry. This content was uploaded by our users and we assume good faith they have the permission to share this book. Jholomorphic curves and symplectic topology american. 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. Citeseerx jholomorphic curves in symplectic geometry. H2m, z is a symplectic form, and show applications and examples. An introduction to symplectic topology through sheaf theory princeton, fall 2010new york,spring2011 c. Lectures on holomorphic curves in symplectic and contact geometry work in progressversion 3. Symplectic geometry on moduli spaces of jholomorphic curves. T parker, j wolfson, pseudo holomorphic maps and bubble trees, j. Pseudoholomorphic curves in symplectic manifolds, inventiones math.
Holomorphic curves in logsymplectic manifolds core. 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. Darbouxs theorem and the moser deformation trick 6 1. Gromovwitten invariants and symplectic vortices pdf. Jholomorphic curves and symplectic topology pdf free. The theory of \j\ holomorphic curves has been of great importance since its introduction by gromov in 1985. Holomorphic curves in symplectic geometry progress in. Riemannian geometry and pseudoholomorphic curves in symplectic geometry. It was also one of the main inspirations for the creation of floer homology. Other readers will always be interested in your opinion of the books youve read. A special accent will be made on applications to symplectic geometry in lowdimensional topology.
This site is like a library, use search box in the widget to get ebook that you want. Genericity of positive topological entropy for geodesic flows on s2 knieper, gerhard and weiss, howard, journal of differential geometry, 2002. Since the first edition of this book appeared in 2004, symplectic geometry has developed apace. Symplectic geometry on moduli spaces of j holomorphic curves j. The authors study the restricted threebody problem using recent techniques coming from the theory of pseudo holomorphic curves. By using a particular geometric problem as a starting point the authors guide the reader into the subject.
Log symplectic structures are poisson structures that are determined by a symplectic form with logarithmic singularities. An algebraic approach to virtual fundamental cycles on moduli spaces of pseudo holomorphic curves pardon, john, geometry. The aim of this part is to give a brief introduction to the symplectic geometry, by. Symplectic geometry on moduli spaces of j holomorphic curves. Darbouxs theorem and the moser deformation trick 7 1. Lectures on symplectic manifolds download ebook pdf. Holomorphic curves in symplectic geometry book, 1994. 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. Yonggeun oh, hiro lee tanaka submitted on 10 mar 2020, last revised 16 mar 2020 this version, v2 abstract.
Lectures on holomorphic curves in symplectic and contact geometry. Lectures on holomorphic curves in symplectic and contact. This is a set of expository lecture notes created originally for a graduate course on holomorphic curves taught at eth. Lectures on pseudoholomorphic curves and the symplectic. Symplectic geometry and holomorphic curves ias school of. Jholomorphic curves and symplectic invariants springerlink. 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.
This book is devoted to pseudo holomorphic curve methods in symplectic geometry. 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. We construct an unwrapped floer theory for bundles of liouville sectors. Lalondejholomorphic curves and symplectic invariants. 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. 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. Symplectic geometry enjoys connections to algebraic combinatorics,algebraicgeometry,dynamics,mathemati. 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. P pansu, pseudo holomorphic curves in symplectic manifolds, from. Nongeneric jholomorphic curves in symplectic 4manifolds.