Click download or read online button to get lectures on symplectic manifolds book now. 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. Workshop on moduli spaces of pseudo holomorphic curves i simons center for geometry and physics, stony brook. Instantons in g2 manifolds from jholomorphic curves in. First notions symplectic geometry is an even dimensional geometry. An almost complex structure on a manifold mis a bundle endomorphism j. 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. Jholomorphic curves and symplectic topology dusa mcduff.
Symplectic geometry on moduli spaces of j holomorphic curves j. J holomorphic curves in symplectic geometry janko latschev pleinfeld, september 25 28, 2006 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 co. It turns out that this geometric approach to hamiltonian systems is very fruitful. The theory of curves has, of course, been developed extensively both in differential geometry and algebraic geometry. Darbouxs theorem and the moser deformation trick 7 1. Holomorphic curves in symplectic geometry springerlink. Lectures on holomorphic curves in symplectic and contact geometry. The goal of the program is to explore different aspects of the theory of holomorphic curves and their interaction. This monograph provides an accessible introduction to the applications of pseudoholomorphic curves in symplectic and contact geometry, with emphasis on dimensions four and three. Originally, holomorphic curves were introduced in symplectic geometry by gromov 14. Holomorphic curves in symplectic geometry ebook, 1994. A selfdual harmonic 2form on a 4dimensional riemannian manifold is symplectic where it does not vanish. The course will focus on the core foundations of symplectic topology, with an emphasis on explicit geometric techniques and examples, and end with an introduction to j holomorphic curves,which are at the heart of modern symplectic topology. Holomorphic curves in low dimensions from symplectic.
Holomorphic curves and global questions in contact geometry. From symplectic geometry to symplectic topology 10 1. Sep 01, 2001 the goal of the program is to explore different aspects of the theory of holomorphic curves and their interaction. An important aspect of gromovs use of almost complex structure and pseudo holomorphic curves.
In 21, instantons in g2 manifolds were constructed from regular j holomorphic curves in coassociative submanifolds. Symplectic topology is a subfield of symplectic geometry, in which global properties of symplectic. Genericity of positive topological entropy for geodesic flows on s2 knieper, gerhard and weiss, howard, journal of differential geometry, 2002. 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. An important aspect of gromovs use of almost complex structure and pseudo holomorphic curves was to explain the first two bullets. Holomorphic curves in symplectic geometry progress in mathematics holomorphic curves in symplectic geometry progress in mathematics this book is devoted to pseudo holomorphic curve methods in symplectic geometry.
Mosers lemma and local triviality of symplectic differential geometry 17 2. This site is like a library, use search box in the widget to get ebook that you want. Symplectic manifolds and darbouxs theorem chapter 4. For example, the proof in the book holomorphic curves in symplectic geometry uses a compactly supported vector field. The goal of this paper is to formulate explicit hodgetheoretic criteria for. Symplectic geometry and isomonodromic deformations philip paul boalch wadham college, oxford d.
Gromovs nonsqueezing theorem using j holomorphic curves. Introduction applications of pseudo holomorphic curves to symplectic topology. It is related to the theory of dynamical systems and via holomorphic curves to algebraic geometry. One possible option seems to be taking critial points off. The question of how many holomorphic curves there are in a given homology class with constraints possibly is given by gromovwitten invariants.
Quadratic hamiltonians and linear symplectic geometry chapter 3. It is standard to denote od the set of holomorphic functions on d. An introduction to symplectic topology through sheaf theory. Quantitative symplectic geometry 3 symplectic geometry is the geometry underlying hamiltonian systems. Applications of holomorphic curves to symplectic topology via. 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. Aug 20 services referee for journal of symplectic geometry, journal of di erential geometry, advances in mathematics. Lectures on pseudo holomorphic curves and the symplectic isotopy problem bernd siebert and gang tian abstract. Thesis submitted trinity term 1999 abstract in this thesis we study the natural symplectic geometry of moduli spaces of mero. This book explains the foundations of holomorphic curve theory in contact geometry. Introduced in 1985 by mikhail gromov, pseudoholomorphic curves have since revolutionized the study of symplectic manifolds. This book is devoted to pseudo holomorphic curve methods in symplectic geometry.
The point is that the expression on the right hand side of 1. 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. Lectures on symplectic manifolds download ebook pdf. Other readers will always be interested in your opinion of the books youve read. The theory of \j\ holomorphic curves has been of great importance since its introduction by gromov in 1985. But then the immersion is not proper and the proof of the monotonicity formula seems to use properness. Hamiltonian dynamical systems, in particular their periodic orbits. Contact and homogeneous symplectic geometry 32 chapter 5. Basic elements of j holomorphic curve theory let m. 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. 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. Explicit examples are discussed in section 2 below.
One important difference is that, although all its concepts are initially expressed in. The relevant branch of the classical theory for us here is that of complex or holomorphic curves. As such it ideally serves as preparation and as entry point for a deeper study of the analysis underlying symplectic. Geodesics are curves of shortest length locally, while pseudoholomorphic curves are surfaces of minimal area. Lectures on symplectic geometry fraydoun rezakhanlou departmet of mathematics, uc berkeley december 12, 2015 chapter 1. 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. It was also one of the main inspirations for the creation of floer homology.
Symplectic geometry is a branch of differential geometry studying symplectic manifolds and some generalizations. In particular, we will develop and apply tools from hamiltonian floer theory which couples the theory of holomorphic curves to techniques from morse theory. 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. In mathematics, its applications include many key results in symplectic topology. Applications of pseudo holomorphic curves to symplectic topology. Applications of pseudoholomorphic curves to symplectic topology.
Symplectic geometry on moduli spaces of jholomorphic curves. The technology of punctured holomorphic curves and neckstretching allows us to translate between these two worlds and prove a theorem. An algebraic approach to virtual fundamental cycles on moduli spaces of pseudo holomorphic curves pardon, john, geometry. In these lectures we will discuss several applications of holomorphic curves to symplectic topology. For me personally, what was exciting in symplectic geometry is that whatever problem you look at, its completely unclear from the beginning what would be the answer, he said. The overflow blog coming together as a community to connect. 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. In mathematical physics, it provides a natural context in which to define gromovwitten invariants and quantum cohomologytwo important ingredients of the mirror symmetry conjecture. Lectures on holomorphic curves in symplectic and contact. Holomorphic curves in symplectic geometry progress in. We will recall the relevant setup, the construction of the moduli spaces of curves, and their compactness and smoothness properties. Symplectic geometry has its origins in the hamiltonian formulation of classical mechanics where the.
Rational curves on holomorphic symplectic fourfolds. Holomorphic curves, planar open books and symplectic fillings. Jholomorphic curves and symplectic topology 2nd dusa. Holomorphic legendrian curves compositio mathematica. In symplectic geometry, however, theres less in the way of conventional wisdom, which invites contention, but also creates exciting possibilities. Darbouxs theorem and the moser deformation trick 6 1. Mar 2014 workshop on the topology and invariants of smooth 4manifolds university of minnesota, twin cities.
Distinguishing symplectic structures 334 chapter 10. To discover new applications of symplectic and contact geometry in mathematics and physics. It turns out to be very different from the riemannian geometry with which we are familiar. Furthermore, away from the forms zero set, the metric and the 2form give a compatible almost complex structure and thus pseudo holomorphic subvarieties. On a symplectic manifold x one can always nd an almost complex structure j. Holomorphic curves in symplectic geometry michele audin. We prove that two simple, closed, realanalytic curves in c 2 that are polynomially convex are equivalent under the group of symplectic holomorphic auto morphisms of c 2n if and only if the two curves. Symplectic structures a new approach to geometry dusa mcduff introduction symplectic geometry is the geometry of a closed skewsymmetric form. Removal of singularities and gromov compactness for symplectic vortices ott, andreas, journal of symplectic geometry, 2014. 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. Symplectic geometry is a branch of differential geometry and differential topology that studies symplectic manifolds.
Dec 11, 2015 to discover new applications of symplectic and contact geometry in mathematics and physics. By using a particular geometric problem as a starting point the authors guide the reader into the subject. But be forewarned, serious commitment is required from the reader and rightly so, of course. To give the junior participants a broad perspective of the subject and a working knowledge of its diverse techniques. Symplectic geometry originated from classical mechanics, where the canonical symplectic form on phase space appears in hamiltons equation. In other words, jmakes tm into a complex vector bundle and we have the canonical decomposition tm. Symplectic geometry on moduli spaces of jholomorphic curves j.
For very many moduli spaces of j holomorphic curves m of interest in symplectic geometry, including singular curves, curves with lagrangian boundary conditions, marked points, etc. A parallel that one can draw between the two subjects is the analogy between geodesics in riemannian geometry and pseudoholomorphic curves in symplectic geometry. Both concepts play a fundamental role in their respective disciplines. This book is devoted to pseudoholomorphic curve methods in symplectic geometry. Holomorphic curves in symplectic geometry it seems that youre in usa. I hope to add the chapter on gromov compactness in the next revision, along with. Jholomorphic curves and symplectic topology mathematical. 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 pseudoholomorphic curves and the symplectic. We give a complete proof for gromovs compactness theorem for pseudo holomorphic curves both in the case of closed curves and curves with boundary. Holomorphic curves in cn the main subject of these notes is a certain interplay between symplectic structures and complex or rather almost complex structures on smooth. Symplectic geometry and holomorphic curves ias school of. This is a very general introductory article published in the notices of the ams 1998. Holomorphic curves in symplectic geometry progress.
The purpose of these notes is a more selfcontained presentation of the results of the authors in siti3. The fight to fix symplectic geometry quanta magazine. It contains an introduction to symplectic geometry and relevant techniques of riemannian geometry, proofs of gromovs. 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. A special accent will be made on applications to symplectic geometry in lowdimensional topology. An introduction to symplectic topology through sheaf theory princeton, fall 2010new york,spring2011 c. Abstract let m be a symplectic manifold, and a compact riemann surface. Kuranishi structures on moduli spaces of j holomorphic curves. A theorem in complex symplectic geometry by franc forstneric abstract. Lectures on holomorphic curves in symplectic and contact geometry work in progressversion 3.
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