A special accent will be made on applications to symplectic geometry in lowdimensional topology. 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. In mathematical physics, it provides a natural context in which to define gromovwitten invariants and quantum cohomologytwo important ingredients of the mirror symmetry conjecture. 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. Holomorphic curves in symplectic geometry progress in. One novel aspect of this treatment is the uniform treatment of both closed and open cases and a complete proof of the boundary regularity theorem of weak solutions of pseudo holomorphic curves with totally real boundary conditions. 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. Lectures on morse homology pennsylvania state university. Distinguishing symplectic structures 334 chapter 10. Volume 2 provides a comprehensive introduction to both hamiltonian floer theory and lagrangian floer theory. To specify the trajectory of the object, one requires both the position q and the momentum p, which form a point p, q in the euclidean plane. Holomorphic curves in symplectic geometry progress. Symplectic topology is a subfield of symplectic geometry, in which global properties of symplectic. Symplectic geometry on moduli spaces of jholomorphic curves j.
The goal of this paper is to formulate explicit hodgetheoretic criteria for. Holomorphic curves in low dimensions from symplectic ruled. Symplectic geometry has its origins in the hamiltonian formulation of classical mechanics where the phase space of certain classical systems takes on the structure of a symplectic manifold. Lectures on symplectic manifolds download ebook pdf, epub. It was also one of the main inspirations for the creation of floer homology. From symplectic geometry to symplectic topology 10 1. The first half of the book focuses on mcduffs characterization of symplectic rational and ruled surfaces, one of the classic early applications of holomorphic curve.
Compatible and tame almost complex structures 29 2. Advanced topics in geometry imperial college london. Thus holomorphic symplectic manifolds also called hyperk ahler. Sep 01, 2001 the goal of the program is to explore different aspects of the theory of holomorphic curves and their interaction. J holomorphic curves and symplectic topology 2nd dusa mcduff, dietmar salamon download bok. Applications of pseudoholomorphic curves to symplectic topology. One possible option seems to be taking critial points off. There are two long chapters on applications, one concentrating on classical results in symplectic topology and the other concerned with quantum cohomology. Lectures on holomorphic curves in symplectic and contact. This site is like a library, use search box in the widget to get ebook that you want. The present book started life as a second edition of that book. Introduced in 1985 by mikhail gromov, pseudoholomorphic curves have since revolutionized the study of symplectic manifolds.
Volume 1, symplectic geometry and pseudoholomorphic curves new mathematical monographs pdf, epub, docx and torrent then this site is not for you. Symplectic geometry and holomorphic curves 20012002. Mosers lemma and local triviality of symplectic differential geometry 17 2. Symplectic manifolds are necessarily evendimensional and orientable, since nondegeneracy says that the top exterior power of a symplectic form is a volume form. Intersection theory has played a prominent role in the study of closed symplectic 4manifolds since gromovs famous 1985 paper on pseudoholomorphic curves, leading to myriad beautiful rigidity results that are either inaccessible or not true in higher dimensions. The link between symplectic and almost complex geometry is by the notion of tameness. For example, the proof in the book holomorphic curves in symplectic geometry uses a compactly supported vector field. Lectures on holomorphic curves in symplectic and contact geometry work in progressversion 3. Holomorphic curves in symplectic geometry michele audin. Almost complex manifolds and j holomorphic curves 25 2.
This can be skipped by the readers that are already familiar with the subject. In mathematics, its applications include many key results in symplectic topology. Jholomorphic curves and symplectic topology pdf free. The five appendices of the book provide necessary background related to the classical theory of linear elliptic operators, fredholm theory, sobolev spaces, as well as a discussion of the moduli space of genus zero stable curves and a proof of the positivity of intersections of \j\ holomorphic curves in fourdimensional manifolds. Keywords moduli space symplectic form j holomorphic curve almost complex structure. Jholomorphic curves and symplectic topology dusa mcduff, dietmar salamon. Abstract let m be a symplectic manifold, and a compact riemann surface. So just why must we leave a good thing like a book holomorphic curves in symplectic geometry progress in mathematics. Symplectic geometry originated from classical mechanics, where the canonical symplectic form on phase space appears in hamiltons equation. For a more liegroup focused account, you can try robert bryants lectures on lie groups and symplectic geometry which are available online here. Braam, morsebott theory and equivariant cohomology, the floer memorial volume, 1995, pp. An introduction to symplectic topology through sheaf theory. Siefrings recent extension of the theory to punctured holomorphic curves allowed similarly important results for contact 3manifolds and their symplectic fillings. Holomorphic curves in symplectic geometry springerlink.
The bubbling phenomenon is the primary cause of noncompactness of moduli space of j holomorphic curves, and when. The book holomorphic curves in symplectic geometry progress in mathematics can give more knowledge and information about everything you want. This book is an expanded version of a set of lecture notes for a minicourse i gave at irma strasbourg in october 2012 as part of the master classes on holomorphic curves and applications to enumerative geometry, symplectic and contact topology. An almost complex structure on a manifold mis a bundle endomorphism j. In the process, we prove a local adjunction formula for holomorphic annuli breaking.
There are two long chapters on applications, one concentrating on classical. The book can also serve as an introduction to current work in symplectic topology. Jholomorphic curves and symplectic topology by dusa mcduff. The theory of \j\ holomorphic curves has been of great importance since its introduction by gromov in 1985. This was partly because symplectic geometry began to run out of problems that could be solved independently of the foundational issues, but also because, in 2012, a pair of researchers dusa mcduff, a prominent symplectic geometer at barnard college and author of a pair of canonical textbooks in the field, and katrin wehrheim, a. A few of you have a different opinion about e book. The second lecture started with the basics on almost complex structures on symplectic manifolds and j holomorphic curves, in particular with a proof that the energy of a j holomorphic curve for some tamed j is given by the pullback of the symplectic form. Salamon erratum to jholomorphic curves and symplectic topology ams 2004, last edited 21 april 2008, available in pdf. Symplectic geometry is the geometry of symplectic manifolds. Click download or read online button to get lectures on symplectic manifolds book now. This content was uploaded by our users and we assume good faith they have the permission to share this book. If you lose the symplectic structure, you are entering a very difficult territory the symplectic form is what is traditionally used to give the required a priori bounds. 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.
Rational curves on holomorphic symplectic fourfolds. It is related to the theory of dynamical systems and via holomorphic curves to algebraic geometry. If youre looking for a free download links of symplectic topology and floer homology. The book serves as an introduction to holomorphic curves in symplectic manifolds, focusing on the case of fourdimensional symplectizations and symplectic cobordisms.
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. Volume 1 covers the basic materials of hamiltonian dynamics and symplectic geometry and the analytic foundations of gromovs pseudoholomorphic curve theory. For algebraic geometry, one must be familiar not only with the standard theory of algebraic varieties and schemes, but also with derived categories. 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. S2 as what we currently consider, it is the only cause. Symplectic topology and floer homology by yonggeun oh. Dusa mcduff columbia mathematics columbia university. Symplectic geometry on moduli spaces of jholomorphic curves. The c0compactness theorem for pseudoholomorphic curves. 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. In section 3 we introduce log symplectic manifolds, and discuss the relevant notion of holomorphicity in the log setup. The school, the book this book is based on lectures given by the authors of the various chapters in a three week long cimpa summer school, held in sophiaantipolis near nice in july 1992. Jholomorphic curves and symplectic topology dusa mcduff.
Since their introduction by gromov 4 in the mid1980s jholomorphic curves have been one of the most widely used tools in symplectic geometry, leading. This note is an extended version of a 50 min talk given at the indam meeting complex and symplectic geometry, held in cortona from june 12th to june 18th, 2016. One novel aspect of this treatment is the uniform treatment of both closed and open cases and a complete proof of the boundary regularity theorem of weak solutions of pseudo holomorphic. American mathematical society colloquium publications volume 52 j holomorphic curves and symplectic topology second edition dusa mcduff dietmar salamon american mathematical society. Lectures on symplectic manifolds download ebook pdf. But then the immersion is not proper and the proof of the monotonicity formula seems to use properness. 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. Almost complex manifolds and jholomorphic curves 25 2. Holomorphic curves, planar open books and symplectic fillings.
Lectures on holomorphic curves in symplectic and contact geometry. In the final lecture he describes the hprinciple and others ideas of gromov in symplectic geometry, like pseudo holomorphic curves. Symplectic geometry arose from the study of classical mechanics and an example of a symplectic structure is the motion of an object in one dimension. Ana cannas da silva, lectures on symplectic geometry available online here. J holomorphic curves and symplectic topology second edition dusa mcduff dietmar salamon. In this talk we explain the elements of symplectic geometry, and sketch. Holomorphic curves in symplectic geometry book, 1994. 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. This monograph provides an accessible introduction to the applications of pseudoholomorphic curves in symplectic and contact geometry, with emphasis on dimensions four and three. I also mentioned that for compatible j, the symplectic form is a calibration, so that j curves are conformal parametrizations of absolutely areaminimizing surfaces. The fight to fix symplectic geometry quanta magazine. A holomorphic curve is an embedding of a surface complex curve which satisfies the cauchyriemann equations.
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. Applications of pseudo holomorphic curves to symplectic topology. It contains a few mistakes and many omissions which are detailed in this commentary of october 2007. Lectures on symplectic geometry ana cannas da silva1 revised january 2006. Symplectic geometry and holomorphic curves ias school of. But be forewarned, serious commitment is required from the reader and rightly so, of course. Since the first edition of this book appeared in 2004, symplectic geometry. The first week was devoted to the basics of symplectic and riemannian geometry banyaga, audin, lafontaine, gauduchon, the second was the technical one pansu, muller, duval, lalonde and sikorav. Denote by symp2n the category of all symplectic manifolds of dimension 2n, with symplectic embeddings as morphisms. Jholomorphic curves and symplectic topology mathematical.
Symplectic geometry is a branch of differential geometry and differential topology that studies symplectic manifolds. An holomorphic symplectic manifold x is a kahler manifold x with a holomorphic non degenerate closed form. In particular, we deduce sufficient conditions for the existence of jholomorphic. Jholomorphic curves and symplectic topology 2nd dusa. This book is devoted to pseudo holomorphic curve methods in symplectic geometry. Introduction applications of pseudo holomorphic curves to symplectic topology. 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, 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 see also. The goal of the program is to explore different aspects of the theory of holomorphic curves and their interaction.
In other words, jmakes tm into a complex vector bundle and we have the canonical decomposition tm. Darbouxs theorem and the moser deformation trick 6 1. Jholomorphic curves and symplectic topology american. Lectures on contact 3manifolds, holomorphic curves and. Every symplectic manifold has an almostcomplex structure in fact, many, and you can write down the cauchyriemann equations using that structure. An excellent reference for the material of the second lecture is the book 6 in. 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. Its mathematical applications include many key results in symplectic topology. For symplectic geometry, one has to work with pseudo holomorphic curves and fukaya categories. Symplectic geometry on moduli spaces of j holomorphic curves j. There are also other more elementary books such as cannas 2. Holomorphic curves in low dimensions from symplectic.
47 1157 144 1498 630 964 1033 182 875 1240 1052 699 594 1356 1097 244 502 898 178 1212 534 767 430 200 68 1319 1241 737 501 976 828 1088 998 852 721 838 1377 940 952 864 699 371 133 85 1189