In mathematics, specifically in topology and geometry, a pseudoholomorphic curve or. From symplectic geometry to symplectic topology 10 1. Master classes holomorphic curves and applications to. Lagrangian intersections, symplectic energy, and areas. 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. One might wonder why we care about these objects, and the rest of this post along with part 2 should be a testament to some but certainly not all aspects of the power. It is related to the theory of dynamical systems and via holomorphic curves to algebraic geometry. Furthermore, away from the forms zero set, the metric and the 2form give a compatible almost complex structure and thus pseudo holomorphic subvarieties. Abstract let m be a symplectic manifold, and a compact riemann surface. Citeseerx jholomorphic curves in symplectic geometry. 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. On symplectic fillings of spinal open book decompositions i. Do you know some example of a holomorphic symplectic manifold with a natural lagrangian foliation polarisation other than the cotangent bundle.
One possible option seems to be taking critial points off. The links below marked pdf and ps allow you to download the same preprint. This lecture is an introduction to symplectic topology, that is global problems apearing in symplectic geometry. Citeseerx document details isaac councill, lee giles, pradeep teregowda. Applications of holomorphic curves to symplectic topology via. Pseudo holomorphic curves in symplectic manifolds ihes. Full text of symplectic geometry on moduli spaces of holomorphic bundles over complex surfaces see other formats o o o n g fields inst. Distinguishing symplectic structures 334 chapter 10. Symplectic topology is a subfield of symplectic geometry, in which global properties of symplectic. The goal of the program is to explore different aspects of the theory of holomorphic curves and their interaction. The aim of this part is to give a brief introduction to the symplectic geometry, by.
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. Gauge theory, symplectic geometry and symplectic topology are important areas at the crossroads of several mathematical disciplines. Lectures on holomorphic curves in symplectic and contact. August 28, 2018 symplectic manifolds and pseudoholomorphic curves download pdf. Use features like bookmarks, note taking and highlighting while reading symplectic topology and floer homology. The goal of this paper is to formulate explicit hodgetheoretic criteria for. But be forewarned, serious commitment is required from the reader and rightly so, of course.
The goal of these notes is to provide a fast introduction to symplectic geometry. The purpose of these notes is a more selfcontained presentation of the results of the authors in siti3. 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. Applications of pseudoholomorphic curves to symplectic topology. Lecture notes in mathematics virtual series on symplectic geometry. Kylerec on jholomorphic curves, part 1 the electric.
But then the immersion is not proper and the proof of the monotonicity formula seems to use properness. Symplectic geometry is a branch of differential geometry studying symplectic manifolds and some generalizations. 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. Rational curves on holomorphic symplectic fourfolds. An important aspect of gromovs use of almost complex structure and pseudoholomorphic. Communications, 24 1999 3123 itepth6398 symplectic geometry on moduli spaces of oh holomorphic bundles over complex surfaces boris khesin and alexei rosly october 1998 cp 2 up to homotopy among symplectic branched covers. Chris wendl submitted on 7 nov 2010, last revised 27 may 2014 this version, v2 abstract. Jholomorphic curves and quantum cohomology dusa mcduff. Jholomorphic curves and symplectic topology dusa mcduff. I will illustrate the abovementioned competition and the use of pseudoholomorphic curves on various kind of problems about the topology of symplectic manifolds. Lectures on pseudoholomorphic curves and the symplectic.
Lectures on pseudo holomorphic curves and the symplectic isotopy problem bernd siebert and gang tian abstract. Holomorphic curves in symplectic geometry springerlink. Holomorphic curves in symplectic geometry progress in mathematics, band 117. Introduced in 1985 by mikhail gromov, pseudoholomorphic curves have since revolutionized the study of symplectic manifolds. I will illustrate the abovementioned competition and the use of pseudo holomorphic curves on various kind of problems about the topology of symplectic manifolds. Other readers will always be interested in your opinion of the books youve read. Applications of pseudoholomorphic curves to symplectic topology introduction applications of pseudoholomorphic curves to symplectic topology jacques lafontaine, michele audin. The theory of curves has, of course, been developed extensively both in differential geometry and algebraic geometry. In mathematics, its applications include many key results in symplectic topology. 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 example, the proof in the book holomorphic curves in symplectic geometry uses a compactly supported vector field. In these lectures we will discuss several applications of holomorphic curves to symplectic topology.
Workshop and conference on holomorphic curves and low. Symplectic geometry originated from classical mechanics, where the canonical symplectic form on phase space appears in hamiltons equation. The theory of \j\ holomorphic curves has been of great importance since its introduction by gromov in 1985. Geodesics are curves of shortest length locally, while pseudoholomorphic curves are surfaces of minimal area. 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. Full text of symplectic geometry on moduli spaces of. One important difference is that, although all its concepts are initially expressed in. Communications, 24 1999 3123 itepth6398 symplectic geometry on moduli spaces of oh holomorphic bundles over complex surfaces boris khesin and alexei rosly october 1998 symplectic geometry, pseudoholomorphic curves, singularity theory, moduli spaces, braid groups, monodromy, in addition to classical topology and algebraic geometry, combine to make this one of the most vibrant and active areas of research in mathematics. Symplectic geometry and holomorphic curves ias school of. Holomorphic curves in low dimensions from symplectic ruled. Both concepts play a fundamental role in their respective disciplines. A selfdual harmonic 2form on a 4dimensional riemannian manifold is symplectic where it does not vanish.
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. Gromov introduced the study of jholomorphic curves into symplectic geometry in his famous 1985 paper, immediately revolutionizing the field. Holomorphic curves in symplectic geometry michele audin. Jholomorphic curves and quantum cohomology dusa mcduff and. Download it once and read it on your kindle device, pc, phones or tablets. The fight to fix symplectic geometry quanta magazine. 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 cohomology, various floer homologies, symplectic field theory, fukaya category, answers. Jholomorphic curves and symplectic topology, ams colloquium publications, vol. 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. 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. The present book, with expertly written surveys of recent developments in these areas, includes some of the first expository material of seibergwitten theory, which has revolutionised the subjects since its introduction in late 1994. Jholomorphic curves and symplectic topology mathematical.
Eudml pseudo holomorphic curves in symplectic manifolds. It is funded by nsf focused research grant dms1065955. You can cycle through additional notes using the next and previous controls. Pseudoholomorphic curves in symplectic manifolds misha. It was also one of the main inspirations for the creation of floer homology. You cannot define a polarisation for the holomorphic tangent bundle the way you do it in your comment. Volume 1, symplectic geometry and pseudoholomorphic curves new mathematical monographs book 28 kindle edition by yonggeun oh. A symplectic manifold is a manifold equipped with a symplectic form. A special accent will be made on applications to symplectic geometry in lowdimensional topology. Volume 1, symplectic geometry and pseudoholomorphic curves new mathematical monographs book 28.
Applications of holomorphic curves to symplectic topology via hamiltonian dynamical systems ely kerman msri, august 1014, 2009. Aug 11, 2012 the main focus of this workshop will be on holomorphic curve techniques in lowdimensional topology and symplectic geometry. Basic elements of j holomorphic curve theory let m. In symplectic geometry, however, theres less in the way of conventional wisdom, which invites contention, but also creates exciting possibilities. This is a set of expository lecture notes created originally for a graduate course on holomorphic curves taught at eth.
The main focus of this workshop will be on holomorphic curve techniques in lowdimensional topology and symplectic geometry. Seiberg witten and gromov invariants for symplectic 4. Pseudoholomorphic curves in symplectic manifolds, inventiones math. Symplectic structures a new approach to geometry dusa mcduff introduction symplectic geometry is the geometry of a closed skewsymmetric form. This content was uploaded by our users and we assume good faith they have the permission to share this book. Lagrangian intersections, symplectic energy, and areas of holomorphic curves.
In addition, there were ten onehour lectures delivered by some of the foremost researchers in the field of four dimensional smooth and symplectic topology. 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. The theory of \j\holomorphic curves has been of great importance since its introduction by gromov in 1985. An important aspect of gromovs use of almost complex structure and pseudo holomorphic curves was to explain the first two bullets. Pseudo holomorphic curves in symplectic manifolds, inventiones math. Lectures on holomorphic curves in symplectic and contact geometry. In particular, we will develop and apply tools from hamiltonian floer theory which couples the theory of holomorphic curves to techniques from morse theory. Symplectic geometry on moduli spaces of jholomorphic curves j.
Workshop and conference on holomorphic curves and low dimensional topology july 30 to august 11, 2012. The relevant branch of the classical theory for us here is that of complex or holomorphic curves. Lectures on holomorphic curves in symplectic and contact geometry work in progressversion 3. The second simons symposium on geometry over nonclosed fields took place march 2228. Sep 01, 2001 the goal of the program is to explore different aspects of the theory of holomorphic curves and their interaction. Lectures on pseudoholomorphic curves and the symplectic isotopy problem bernd siebert and gang tian abstract. 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. Through quantum cohomology, these curves are now linked to many of the most exciting new ideas in mathematical physics. Symplectic geometry is a branch of differential geometry and differential topology that studies symplectic manifolds. The inaugural speaker for this event was professor clifford henry taubes of harvard university who delivered three lectures on seibergwitten and gromov invariants. It turns out to be very different from the riemannian geometry with which we are familiar.
Jholomorphic curves and symplectic topology pdf free. Introduction applications of pseudoholomorphic curves to symplectic topology. Applications of holomorphic curves to symplectic topology. Darbouxs theorem and the moser deformation trick 6 1. A parallel that one can draw between the two subjects is the analogy between geodesics in riemannian geometry and pseudoholomorphic curves in symplectic geometry.
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