The relevant branch of the classical theory for us here is that of complex or holomorphic curves. Darbouxs theorem and the moser deformation trick 6 1. In mathematics, its applications include many key results in symplectic topology. Quadratic hamiltonians and linear symplectic geometry chapter 3. A symplective vector space v is a pair of nite dimensional real vector space v. Pseudo holomorphic curves in symplectic manifolds ihes. The purpose of these notes is a more selfcontained presentation of the results of the authors in siti3.
Geodesics are curves of shortest length locally, while pseudoholomorphic curves are surfaces of minimal area. A symplectic manifold is a manifold equipped with a symplectic form. The reason is that this one semester course was aiming for students at the beginning of their masters. This lecture is an introduction to symplectic topology, that is global problems apearing in symplectic geometry. It is related to the theory of dynamical systems and via holomorphic curves to algebraic geometry. Holomorphic curves in symplectic geometry springerlink. The workshop will bring together researchers in contact and symplectic geometry, dynamics, lowdimensional topology and physics, who will explore connections between the various approaches people have taken to using holomorphic curves, develop new holomorphic curve techniques and apply them to various questions in contact and symplectic. But then the immersion is not proper and the proof of the monotonicity formula seems to use properness. The theory of \j\holomorphic curves has been of great importance since its introduction by gromov in 1985. The aim of this part is to give a brief introduction to the symplectic geometry, by. An introduction to symplectic topology through sheaf theory. I will illustrate the abovementioned competition and the use of pseudoholomorphic curves on various kind of problems about the topology of symplectic manifolds. Abstract let m be a symplectic manifold, and a compact riemann surface. Lectures on symplectic geometry fraydoun rezakhanlou departmet of mathematics, uc berkeley december 12, 2015 chapter 1.
An introduction to symplectic topology through sheaf theory princeton, fall 2010new york,spring2011 c. The fight to fix symplectic geometry quanta magazine. Mosers lemma and local triviality of symplectic differential geometry 17 2. Symplectic structures have their origin in the study of classical mechanics. Symplectic geometry and holomorphic curves ias school of. Lectures on pseudoholomorphic curves and the symplectic isotopy problem bernd siebert and gang tian abstract. Denis auroux recall from last time the statement of the following lemma. Lectures on holomorphic curves in symplectic and contact. Symplectic manifolds and darbouxs theorem chapter 4. Floer trajectories overview in these lectures we will discuss several applications of holomorphic curves to symplectic topology.
Donaldson 1026 notices of the ams volume 52, number 9 the terminology pseudoholomorphic curve or jholomorphic curve was introduced by gromov in 1986. Do you know some example of a holomorphic symplectic manifold with a natural lagrangian foliation polarisation other than the cotangent bundle. Both concepts play a fundamental role in their respective disciplines. I will illustrate the abovementioned competition and the use of pseudo holomorphic curves on various kind of problems about the topology of symplectic manifolds. 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. 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 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. 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 applications of pseudo holomorphic curves to symplectic topology. Sep 01, 2001 the goal of the program is to explore different aspects of the theory of holomorphic curves and their interaction.
The goal of the program is to explore different aspects of the theory of holomorphic curves and their interaction. Important developments in the early 1980s led to the emergence of new questions about these structures. You cannot define a polarisation for the holomorphic tangent bundle the way you do it in your comment. One important difference is that, although all its concepts are initially expressed in. Holomorphic curves in symplectic geometry it seems that youre in usa. This book is devoted to pseudoholomorphic curve methods in symplectic geometry. These are lecture notes for a course on symplectic geometry in the dutch mastermath program. Symplectic topology is a subfield of symplectic geometry, in which global properties of symplectic. Applications of pseudoholomorphic curves to symplectic topology. The theory of \j\ holomorphic curves has been of great importance since its introduction by gromov in 1985. For example, the fundamental proof by moser of the equivalence under di. Jholomorphic curves in symplectic topology manabu akaho tokyo metropolitan university.
Jholomorphic curves mean value inequality minimal energy removal of singularities convergence i finiteness of singularities. 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. Contact and homogeneous symplectic geometry 32 chapter 5. Applications of pseudo holomorphic curves to symplectic topology. In the final lecture he describes the hprinciple and others ideas of gromov in symplectic geometry, like pseudo holomorphic curves. It turns out that this geometric approach to hamiltonian systems is very fruitful.
Jholomorphic curves and symplectic topology by dusa. Distinguishing symplectic structures 334 chapter 10. It turns out to be very different from the riemannian geometry with which we are familiar. Applications of holomorphic curves to symplectic topology. The theory of curves has, of course, been developed extensively both in differential geometry and algebraic geometry. Holomorphic curves in symplectic geometry michele audin.
Dk sn describing the configuration of the n deg fk sheets of. Symplectic geometry is a branch of differential geometry studying symplectic manifolds and some generalizations. Darbouxs theorem and the moser deformation trick 7 1. Symplectic geometry 2nd semester 201220 symplectic geometry studies the geometry of manifolds equipped with a nondegenerate, closed 2form. 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. For example, the proof in the book holomorphic curves in symplectic geometry uses a compactly supported vector field. Applications of holomorphic curves to symplectic topology via hamiltonian dynamical systems ely kerman msri, august 1014, 2009. In the final lecture he describes the hprinciple and others ideas of gromov in symplectic geometry, like pseudoholomorphic curves. An important aspect of gromovs use of almost complex structure and pseudo holomorphic curves was to explain the first two bullets. Quantitative symplectic geometry 3 symplectic geometry is the geometry underlying hamiltonian systems. We define a 2form equation on the space equation of immersed symplectic surfaces in.
This book is devoted to pseudo holomorphic curve methods in symplectic geometry. The goal of these notes is to provide a fast introduction to symplectic geometry. It was also one of the main inspirations for the creation of floer homology. If youre an algebraic geometer, i hope to give you a avour of some simple techniques in symplectic topology. Explicit examples are discussed in section 2 below. A special accent will be made on applications to symplectic geometry in lowdimensional topology. Basic elements of j holomorphic curve theory let m. 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.
Higher genus knot contact homology and recursion for colored homflypt polynomials tobias ekholm and lenhard ng abstract. Under a homotopy of the system 2 the curve g, teo, to, with its beginning. In symplectic geometry, however, theres less in the way of conventional wisdom, which invites contention, but also creates exciting possibilities. A parallel that one can draw between the two subjects is the analogy between geodesics in riemannian geometry and pseudoholomorphic curves in symplectic geometry. 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. One possible option seems to be taking critial points off.
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. Symplectic geometry originated from classical mechanics, where the canonical symplectic form on phase space appears in hamiltons equation. Symplectic structures a new approach to geometry dusa mcduff introduction symplectic geometry is the geometry of a closed skewsymmetric form. The gromov school of symplectic geometry the soft theory. The problems that symplectic topology aims to solve are questions such as. From symplectic geometry to symplectic topology 10 1. Symplectic geometry on moduli spaces of jholomorphic curves j. Master classes holomorphic curves and applications to. There are several books on symplectic geometry, but i still took the trouble of writing up lecture notes. X cp 2 up to homotopy among symplectic branched covers. The second simons symposium on geometry over nonclosed fields took place march 2228.
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. Area estimates, pseudoholomorphic curves, and gromovs compactness theorem. Lectures on pseudo holomorphic curves and the symplectic isotopy problem bernd siebert and gang tian abstract. 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. Holomorphic curves in symplectic geometry progress in. An important aspect of gromovs use of almost complex structure and pseudoholomorphic curves was to explain the first two bullets. In this chapter the manifolds and the sheets bounded by them will be. Symplectic topology and algebraic geometry interact in many fruitful ways. Lectures on holomorphic curves in symplectic and contact geometry.
The first symposium in this series focused on rational curves on higherdimensional algebraic varieties and outlined applications of the theory of curves to arithmetic problems. Lectures on holomorphic curves in symplectic and contact geometry work in progressversion 3. Introduced in 1985 by mikhail gromov, pseudoholomorphic curves have since revolutionized the study of symplectic manifolds. Symplectic geometry on moduli spaces of jholomorphic curves. 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. Symplectic geometry is a branch of differential geometry and differential topology that studies symplectic manifolds. For a more liegroup focused account, you can try robert bryants lectures on lie groups and symplectic geometry which are available online here. But be forewarned, serious commitment is required from the reader and rightly so, of course. 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. We sketch a construction of legendrian symplectic field t.
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