Abstract let m be a symplectic manifold, and a compact riemann surface. Following seminal ideas of gromov and floer from the 1980s, several of the most powerful tools in symplectic topology revolve around invariants counting pseudoholomorphic curves. It is now understood to arise naturally in algebraic geometry, in lowdimensional topology, in representation theory and in string theory. The interplay between these two facets of symplectic geometry, analysis of pseudoholomorphic curves and hamiltonian dynamics, have been the main driving force in the development of symplectic topology since floers pioneering work on his semiin. First we will describe some problems in symplectic geometry, or more exactly topology, and the way to solve them using pseudo holomorphic curves techniques. A glimpse into symplectic geometry stony brook mathematics.
Holomorphic curves, planar open books and symplectic fillings. From holomorphic curves to knot invariants via the cotangent bundle lenny ng dukeuniversity ams southeastern section meeting. What are symplectic manifolds, what are j holomorphic curves, and why do we care about them. Applications of holomorphic curves to symplectic topology. Holomorphic curves in symplectic geometry it seems that youre in usa.
Jholomorphic curves and symplectic topology request pdf. Jholomorphic curves and symplectic topology second. Siefrings recent extension of the theory to punctured holomorphic curves allowed similarly important results for contact 3manifolds and their symplectic fillings. If youre an algebraic geometer, i hope to give you a avour of some simple techniques in symplectic topology. Salamon, second edition, introduction to symplectic topology, 2nd ed. Rational curves have appeared in important recent works on the positive cone of holomorphic symplectic varieties that culminated in the papers bm and bht. Nj, where nj is the nijenhuis torsion tensor which measures the. Basic elements of j holomorphic curve theory let m. Introduction applications of pseudoholomorphic curves to.
Jholomorphic curves and symplectic topology dusa mcduff, dietmar salamon 2nd ed. Workshop on holomorphic curves and low dimensional topology. Symplectic topology and algebraic geometry interact in many fruitful ways. Symplectic topology and floer homology volume 1 published in two volumes, this is the. Part 2 i will introduce certain moduli spaces of j holomorphic curves with extra tangency conditions. In mathematics, specifically in topology and geometry, a pseudoholomorphic curve or j holomorphic curve is a smooth map from a riemann surface into an almost complex manifold that satisfies the cauchyriemann equation. Symplectic geometry on moduli spaces of jholomorphic curves j. Jholomorphic curves and quantum cohomology people eth. Lectures on holomorphic curves in symplectic and contact geometry. Sep 29, 2014 it is now understood to arise naturally in algebraic geometry, in lowdimensional topology, in representation theory and in string theory. 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. Pseudo holomorphic curves in symplectic manifolds august 28, 2018 symplectic manifolds and pseudo holomorphic curves. 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. Applications of pseudoholomorphic curves to symplectic topology.
Symplectic topology is a subfield of symplectic geometry, in which global properties of symplectic. Almost complex manifold holomorphic curve singularity symplectic geometry. American mathematical society 201 charles street providence, rhode island 0290422 4014554000 or 8003214267 ams, american mathematical society, the tricolored ams logo, and advancing research, creating connections, are trademarks and services marks of the american mathematical society and registered in the u. Salamon erratum to j holomorphic curves and symplectic topology ams 2004, last edited 21 april 2008, available in pdf. Here nj is the nijenhuis tensor of the almost complex structure j. The first volume covered the basic materials of hamiltonian dynamics and symplectic geometry and the analytic foundations of gromovs pseudoholomorphic curve theory.
The purpose of these notes is a more selfcontained presentation of the results of the authors in siti3. J holomorphic curves and symplectic topology second edition erratum dusa mcdu barnard college columbia university dietmar a. Introduction symplectic and contact topology is an active area of mathematics that combines ideas from dynamical systems, analysis, topology, several complex variables, and differential and algebraic geometry. Symplectic geometry on moduli spaces of jholomorphic curves.
It contains a few mistakes and many omissions which are detailed in this commentary of october 2007. Symplectic topology and floer homology by yonggeun oh. The notion has transformed the field of sym plectic topology and has a bearing on many other areas such as algebraic geometry, string theory, and 4manifold theory. The book can also serve as an introduction to current work in symplectic topology. Aug 11, 2017 for 30 years, the holomorphic curve invariants introduced by gromov and floer have played a central role in symplectic and contact topology. Symplectic topology, mirror symmetry and integrable.
We construct a combinatorial invariant of legendrian knots in. P pansu, pseudo holomorphic curves in symplectic manifolds, from. Curves lying in the divisor v and 2 genericity results for almost complex structures constrained to. 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. The theory of \j\ holomorphic curves has been of great importance since its introduction by gromov in 1985. Relative methods in symplectic topology a thesis submitted to the faculty of the graduate school. Donaldson 1026 notices of the ams volume 52, number 9 the terminology pseudoholomorphic curve or j holomorphic curve was introduced by gromov in 1986.
Rational curves on holomorphic symplectic fourfolds by brendan hassett and yuri tschinkel 1. J holomorphic curves and symplectic topology dusa mcduff dietmar salamon american mathematical society providence, rhode island. Hamiltonian dynamical systems, in particular their periodic orbits. Floer trajectories overview in these lectures we will discuss several applications of holomorphic curves to symplectic topology. Holomorphic curves in symplectic geometry springerlink. Based on a series of lectures for graduate students in topology, this book begins with an overview of the closed case, and then proceeds to explain the essentials of siefrings. Holomorphic curves in symplectic geometry, m audin and f lafontaine, editors progress in math. I will give an introduction to lagrangian correspondences and holomorphic quilts, their analysis, and some sample applications in both symplectic topology and the construction of 3 and 4manifold invariants. Applications of holomorphic curves to symplectic topology via. Pseudo holomorphic curves and gromovwitten invariants. The goal of this paper is to formulate explicit hodgetheoretic criteria for. Wewill focusonthe latter side and only treat the former side lightly.
The present notes grew out of lectures given at the contact and symplectic topology summer school in budapest in july 2012. Symplectic and contact structures first arose in the study of classical mechanical systems, allowing one to describe the time evolution of both simple and complex. Zalerts allow you to be notified by email about the availability of new books according to your search query. 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. It is related to the theory of dynamical systems and via holomorphic curves to algebraic geometry. In mathematics, in the field of complex geometry, a holomorphic curve in a complex manifold m is a nonconstant holomorphic map f from the complex plane to m nevanlinna theory addresses the question of the distribution of values of a holomorphic curve in the complex projective line. Introduction to symplectic topology y j holomorphic curves and symplectic topology. Holomorphic curves andsymplectic topology assignment 2 in some exercises the following books are referred to. Introduction applications of pseudo holomorphic curves to symplectic topology. The last chapter sketches some recent developments in floer theory. In the last few years, several new exciting directions have emerged, involving amongst others. Conormal bundles, contact homology and knot invariants.
Introduction applications of pseudoholomorphic curves to symplectic topology. Application to symplectic topology in dimension four. A selfdual harmonic 2form on a 4dimensional riemannian manifold is symplectic where it does not vanish. 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. The origins of symplectic geometry are in classical mechanics. Jholomorphic curves and symplectic topology mathematical. 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. Jholomorphic curves and symplectic topology semantic scholar.
Symplectic geometry originated from classical mechanics, where the canonical symplectic form on phase space appears in hamiltons equation. 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. 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. This thesis extends the theory by including some aspects of the symplectic topology of the quantum phase space. Furthermore, away from the forms zero set, the metric and the 2form give a compatible almost complex structure and thus pseudo holomorphic subvarieties. The most natural example of a symplectic manifold is r2m,thephase space in other words the space.
Introduced in 1985 by mikhail gromov, pseudoholomorphic curves have since revolutionized the study of symplectic manifolds. In mathematics, its applications include many key results in symplectic topology. It is shown that the quantum mechanical uncertainty principle is a special case of an inequality from j holomorphic map theory, that is, j holomorphic curves minimize the difference between the quantum covariance matrix determinant and a. There is a tensor nj called the nijenhuis tensor that is made out of the first. Holomorphic curves in low dimensions from symplectic. J holomorphic curves in symplectic topology manabu akaho.
Singularities of jholomorphic curves in almost complex 4manifolds. T holomorphic curves for large t is the symplectic eld theory compactness theorem. An introduction to symplectic topology through sheaf theory princeton, fall 2010new york,spring2011 c. Rational symplectic field theory for legendrian knots lenhard ng abstract. T parker, j wolfson, pseudo holomorphic maps and bubble trees, j. Lectures on holomorphic curves in symplectic and contact geometry work in progressversion 3. Lectures on holomorphic curves in symplectic and contact. Applications of holomorphic curves to symplectic topology via hamiltonian dynamical systems ely kerman msri, august 1014, 2009. Symplectic 4manifolds, singular plane curves, and isotopy problems denisauroux abstract. Symplectic topology and geometric quantum mechanics. The theory of j holomorphic curves has been of great importance since its introduction by gromov in 1985. Jholomorphic curves and symplectic topology american. Introduction since gromov introduced his pseudo holomorphic curve theory in the 80s, pseudo holomorphic curve has soon become an eminent technique in symplectic topology.
Jholomorphic curves and symplectic topology dusa mcduff. The second half of the book then extends this program in two complementary directions. We give an overview of various recent results concerning. I also mentioned that for compatible j, the symplectic form is a calibration, so that j curves are conformal parametrizations of absolutely areaminimizing surfaces. 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. But be forewarned, serious commitment is required from the reader and rightly so, of course. Isbn 0821834851 the theory of j holomorphic curves has been of great importance since its introduction by gromov in 1985. It was also one of the main inspirations for the creation of floer homology. Metric invariants and quantitative topology 28 elliptic operators on open manifolds and infinite covering spaces. The paper begins with symplectic manifolds and their lagrangian submanifolds, covers contact manifolds and their legendrian submanifolds, and indicates the first steps of symplectic and contact topology. Jholomorphic curves and symplectic topology pdf free. There are two long chapters on applications, one concentrating on classical results in symplectic topology and the other concerned with quantum cohomology. An introduction to symplectic topology through sheaf theory. Rational curves on holomorphic symplectic fourfolds.
The first way is as geometric probes to explore symplectic manifolds. Lectures on pseudo holomorphic curves and the symplectic isotopy problem bernd siebert and gang tian abstract. In these lectures we will discuss several applications of holomorphic curves to symplectic topology. The c0compactness theorem for pseudoholomorphic curves. Lectures on contact 3manifolds, holomorphic curves and. We also discuss the origins of this invariant in symplectic topology, via holomorphic curves and a conormal bundle naturally associated. Rational curves on holomorphic symplectic fourfolds brendan. Symplectic topology and floer homology is a comprehensive resource suitable for experts and newcomers alike. American mathematical society colloquium publications volume 52 j holomorphic curves and symplectic topology second edition dusa mcduff dietmar salamon american mathematical society. 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. Aug 11, 2012 pseudoholomorphic quilts and low dimensional topology by katrin wehrheim abstract. Monodromy invariants in symplectic topology denis auroux thistextisasetoflecturenotesforaseriesoffourtalksgivenati.
From dynamics to contact and symplectic topology and back. Pseudo holomorphic curves, rst introduced by gromov in 1985 gr, have since then become the most important tool in modern symplectic topology. Pseudoholomorphic curves in symplectic manifolds misha. From holomorphic curves to knot invariants via the. From symplectic geometry to symplectic topology 10 1. In particular, we will develop and apply tools from hamiltonian floer theory which couples the theory of holomorphic curves to techniques from morse theory. J for which the given curve c is pseudo holomorphic, and considering a. Darbouxs theorem says that locally all symplectic spaces.
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