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 particular, only a countable set of points in l can be hit by the limits of the uniruling curves from step 2, a contradiction. The book can also serve as an introduction to current work in symplectic topology. In mathematics, its applications include many key results in symplectic topology. Introduced in 1985 by mikhail gromov, pseudoholomorphic curves have since revolutionized the study of symplectic manifolds. She was the first recipient of the ruth lyttle satter prize in mathematics, was a noether lecturer, and is a fellow of the royal society. Symplectic geometry on moduli spaces of jholomorphic curves. Jholomorphic curves and symplectic topology request pdf. Jholomorphic curves and symplectic topology by dusa mcduff. Request pdf on jan 1, 2004, dusa mcduff and others published j holomorphic curves and symplectic topology find, read and cite all the research you need on researchgate. Relative frames on j holomorphic curves springerlink.
Jholomorphic curves and symplectic topology dusa mcduff. The theory of jholomorphic curves has been of great importance since its introduction by gromov in 1985. There are two long chapters on applications, one concentrating on classical results in symplectic topology and the other concerned with quantum cohomology. Local symmetry on almost kenmotsu threemanifolds cho, jong taek, hokkaido mathematical journal, 2016. Books here is the revised 1995 text of the book jholomorphic curves and quantum homology ams lecture notes, 1994 that i wrote with salamon. J holomorphic curves mean value inequality minimal energy removal of singularities convergence i. Singularities of jholomorphic curves 634 bibliography 643. 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. 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. Lectures on holomorphic curves in symplectic and contact geometry. Dietmar arno salamon wikipedia, a enciclopedia livre. At its publication in 1995, introduction to symplectic topology was the first comprehensive introduction to the subject, and has since become an established text in this fastdeveloping area of mathematics.
After passing to a subsequence, f i converge to a stable jholomorphic curve f. Itis shown inappendix bthatthefollowingspaces obey ourtechnical assumptions. J holomorphic curves in symplectic topology manabu akaho. We will definitely cover part i of mcduff salamon chapters 14. Holomorphic curves, planar open books and symplectic fillings. Pseudo holomorphic curves in symplectic and contact topology c. Introduction to symplectic topology dusa mcduff, dietmar. 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 j p and j fp on tangent spaces. Holomorphic curves in complex spaces drinovec drnovsek, barbara and forstneric, franc, duke mathematical journal, 2007. Symplectic geometry on moduli spaces of jholomorphic curves j. I also mentioned that for compatible j, the symplectic form is a calibration, so that j curves are conformal parametrizations of absolutely areaminimizing surfaces. A simple proof of unique continuation for jholomorphic curves. We shall show in chapter 3 that any bounded energy j. Salamon erratum to jholomorphic curves and symplectic topology ams 2004, last edited 21 april 2008, available in pdf.
Some questions in the theory of pseudoholomorphic curves. An introduction to symplectic topology through sheaf theory princeton, fall 2010new york,spring2011 c. J holomorphic curves and symplectic topology 2nd dusa mcduff, dietmar salamon download bok. Jholomorphic curves and symplectic topology book, 2004. A pseudoholomorphic curve can be defined as a map from a riemann surface to a symplectic manifold that satisfies a nonlinear pde generalizing the standard cauchyriemann equations of complex analysis. Introduction to symplectic topologyintroduction to. Jholomorphic curves and symplectic topology by dusa. This is a collection of results on the topology of toric symplectic manifolds. Symplectic topology is a subfield of symplectic geometry, in which global properties of symplectic. Dusa mcduff frs corrfrse born 18 october 1945 is an english mathematician who works on symplectic geometry. Mcduff and salamon give three proofs of the unique continuation property. As a consequence, we partially extend to nontrivial hamiltonian fibrations a result of lu math. Jholomorphic curves and symplectic topology dusa mcduff, dietmar.
From symplectic geometry to symplectic topology 10 1. Dusa mcduff department of mathematics barnard college columbia university 2990 broadway. The image of such a map in x is called a jholomorphic curve. Kylerec on jholomorphic curves, part 1 the electric. Steele prize for mathematical exposition for the book jholomorphic curves and symplectic topology, which they coauthored. Master classes holomorphic curves and applications to. Jholomorphic curves and symplectic topology american. In the following, we denote by x, w a symplectic manifold of. Books here is the revised 1995 text of the book j holomorphic curves and quantum homology ams lecture notes, 1994 that i wrote with salamon. Over the past number of years powerful new methods in analysis and topology have led to the development of the modern global theory of symplectic topology, including several striking and important results. Almost complex manifolds and j holomorphic curves 25 2. This second editionhas been significantly revised and expanded, with new references and examples added and theorems included or revised. Jun 14, 2017 kylerec on jholomorphic curves, part 1 this tobe2partbecausethisgotlong post is a continuation of the series on kylerec 2017 starting with the previous post, and covers most of the talks from days 23 of kylerec, focusing on the use of jholomorphic curves in the study of fillings. Jholomorphic curves and symplectic invariants springerlink.
The last chapter sketches some recent developments in floer theory. Jholomorphic curves and symplectic topology mathematical. Compatible and tamed almost complex structures 27 2. Proposition holomorphic curves and reeb dynamics let l be a hyperbolic nmanifold, n 3. The symplectic action can be defined on the space of smooth paths in a symplectic manifold p which join two lagrangian submanifolds of p. Lectures on holomorphic curves in symplectic and contact geometry work in progressversion 3. Almost complex manifolds and jholomorphic curves 23. Jholomorphic curves and quantum cohomology people eth. Introduction to symplectic topology oxford scholarship. For 2017 he received, with dusa mcduff, the ams leroy p.
Jholomorphic curves and symplectic topology in searchworks. Introduction applications of pseudoholomorphic curves to symplectic topology. This is put together in chapter 4 by showing that the moduli space of holomorphic curves is cobordant to the a moduli space of objects called j. D salamon over the last number of years powerful new methods in analysis and topology have led to the development of the modern global theory of symplectic topology, including several striking and important. Symplectic topology and floer homology is a comprehensive resource suitable for experts and newcomers alike. In the study of jholomorphic curves and symplectic topology as presented by mcduff. The first volume covered the basic materials of hamiltonian dynamics and symplectic geometry and the analytic foundations of gromovs pseudoholomorphic curve theory.
The theory of j holomorphic curves has been of great importance since its introduction by gromov in 1985. This note concerns the structure of singularities of mapsf from a neighborhood of 0 in the complex plane. This is explained for example in mcduff salomon s jholomorphic curves in symplectic topology, in one of the first sections. It is related to the theory of dynamical systems and via holomorphic curves to algebraic geometry. Numerous and frequentlyupdated resource results are available from this search. Jholomorphic curves and symplectic topology 2nd dusa mcduff, dietmar salamon download bok.
We will also hopefully cover portions of part ii chapters 57 and discuss additional topics. Jholomorphic curves and symplectic topology semantic scholar. Compatible and tame almost complex structures 29 2. Jholomorphic curves and symplectic topology 2nd dusa mcduff, dietmar salamon. Jholomorphic curves and symplectic topology 2nd dusa. Intersection theory on the moduli space of holomorphic curves. The theory of jholomorphic curves has been of great importance to symplectic topologists ever since its inception in gromovs paper 26 of 1985. All in all j holomorphic 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. Reliable information about the coronavirus covid19 is available from the world health organization current situation, international travel. Mcduff, hamiltonian s1manifolds are uniruled, duke math. But be forewarned, serious commitment is required from the reader and rightly so, of course. The book was the first comprehensive introduction to the subject and became a key text in the area. The origins of symplectic geometry are in classical mechanics.
J holomorphic curves and symplectic topology 2nd dusa mcduff, dietmar salamon. Almost complex manifolds and jholomorphic curves 23 2. Pseudoholomorphic curves in symplectic geometry author. J holomorphic curves and symplectic topology dusa mcduff dietmar salamon american mathematical society providence, rhode island. This seminar will be a first course in symplectic geometry and topology. Holomorphic curves in symplectic geometry it seems that youre in usa. It was also one of the main inspirations for the creation of floer homology. This note discusses the structure of jholomorphic 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 jholomorphic. Steele por divulgacao da matematica por seu livro com dusa mcduff jholomorphic curves and symplectic topology. In mathematical physics, it provides a natural context in which to define gromovwitten invariants and quantum cohomology, two important ingredients of the mirror symmetry conjecture. He has been a member of academia europaea since 2011.
Nongeneric jholomorphic curves in symplectic 4manifolds. Almost complex manifolds and jholomorphic curves 25 2. The most natural example of a symplectic manifold is r2m,thephase space in other words the space. Jholomorphic curves and symplectic topology second. Solomon, a differential equation for the open gromovwitten potential, preprint 2007. Salamon, jholomorphic curves and symplectic topology. For generic j, the moduli space of simple punctured holomorphic curves in tl is discrete 0dimensional. To pursue a new approach to the variational theory of this. Therefore the integral curves of the hamiltonian system are precisely the vectors. American mathematical society colloquium publications volume 52 jholomorphic curves and symplectic topology dusa mcduff dietmar salamon american mathematical society. Its applications include many key results in symplectic topology. Symplectic geometry originated from classical mechanics, where the canonical symplectic form on phase space appears in hamiltons equation. Oclcs webjunction has pulled together information and resources to assist library staff as they consider how to handle coronavirus.
Lectures on holomorphic curves in symplectic and contact. Jholomorphic curves and symplectic topology dusa mcduff dietmar salamon american mathematical society providence, rhode island. Abstract let m be a symplectic manifold, and a compact riemann surface. American mathematical society colloquium publications volume 52 jholomorphic curves and symplectic topology second edition dusa mcduff dietmar salamon american mathematical society. Request pdf on jan 1, 2004, dusa mcduff and others published jholomorphic curves and symplectic topology find, read and cite all the research you need on researchgate. Singularities of j holomorphic curves 634 bibliography 643. Using an idea of borisov, we show that a closed symplectic manifold supports at most a finite number of toric structures. Singularities of jholomorphic curves in almost complex 4. Jholomorphic curves and symplectic topology second edition erratum dusa mcdu barnard college columbia university dietmar a. Symplectic topology and floer homology by yonggeun oh. Jholomorphic curves in symplectic topology manabu akaho tokyo metropolitan university.
It was also one of the main inspirations for the creation of. The theory of \ j\ holomorphic curves has been of great importance since its introduction by gromov in 1985. It contains a few mistakes and many omissions which are detailed in this commentary of october 2007. The first edition of introduction to symplectic topology was published in 1995. Holomorphic curves in symplectic geometry michele audin. The main result is that whenv has dimension 4, the topology of these singularities is the same as in the case whenj is integrable. In particular if it is a closed curve it is minimizing in its homology class.