The theory of \j\ holomorphic curves has been of great importance since its introduction by gromov in 1985. In mathematics, its applications include many key results in symplectic topology. The purpose of these notes is a more selfcontained presentation of the results of the authors in siti3. From symplectic geometry to symplectic topology 10 1. Tabachnikov, editor, differential and symplectic topology of knots and curves. Witten theory is a basic tool in string theory, algebraic geometry, and symplectic geometry, and understanding its relationship to string topology is an exciting area of current and probably future research.
One important difference is that, although all its concepts are initially expressed in. 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. An almost complex structure on a manifold mis a bundle endomorphism j. An almost complex structure on a manifold m is a bundle endomorphism j. Another important feature is that it is a 2dimensional geometry that measures the area of complex curves instead of the length of real curves.
Symplectic geometry on moduli spaces of jholomorphic curves. A selfdual harmonic 2form on a 4dimensional riemannian manifold is symplectic where it does not vanish. Rational curves on holomorphic symplectic fourfolds. Holomorphic curves in symplectic geometry michele audin. Both concepts play a fundamental role in their respective disciplines.
We will recall the relevant setup, the construction of the moduli spaces of curves, and their compactness and smoothness properties. Jholomorphic curves and symplectic topology 2nd dusa. Darbouxs theorem and the moser deformation trick 7 1. This book is devoted to pseudo holomorphic curve methods in symplectic geometry. An introduction to symplectic topology through sheaf theory. It is related to the theory of dynamical systems and via holomorphic curves to algebraic geometry. Distinguishing symplectic structures 334 chapter 10. Other readers will always be interested in your opinion of the books youve read. Holomorphic curves, planar open books and symplectic fillings. Abstract let m be a symplectic manifold, and a compact riemann surface. Furthermore, away from the forms zero set, the metric and the 2form give a compatible almost complex structure and thus pseudoholomorphic subvarieties. Lectures on pseudoholomorphic curves and the symplectic. Thus symplectic geometry is essentially topological in nature. Pseudo holomorphic curves in symplectic manifolds ihes.
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. The point is that the expression on the right hand side of 1. Part 2 i will introduce certain moduli spaces of j holomorphic curves with extra tangency conditions. We use cookies to offer you a better experience, personalize content, tailor advertising, provide social media features, and better understand the use of our services. 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 originated from classical mechanics, where the canonical symplectic form on phase space appears in hamiltons equation. 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.
Holomorphic curves in symplectic geometry springerlink. But be forewarned, serious commitment is required from the reader and rightly so, of course. It was also one of the main inspirations for the creation of floer homology. What are symplectic manifolds, what are j holomorphic curves, and why do we care about them. Applications of pseudoholomorphic curves to symplectic topology. Geodesics are curves of shortest length locally, while pseudoholomorphic curves are surfaces of minimal area. 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. Holomorphic curves in symplectic geometry progress in. Jholomorphic curves and symplectic topology pdf free. Compactness for punctured holomorphic curves cieliebak, k. Lectures on holomorphic curves in symplectic and contact geometry work in progressversion 3. A parallel that one can draw between the two subjects is the analogy between geodesics in riemannian geometry and pseudoholomorphic curves in symplectic geometry. P2 integral counts of holomorphic curves in smooth complex projective.
Lectures on holomorphic curves in symplectic and contact geometry. 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. 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 question of how many holomorphic curves there are in a given homology class with constraints possibly is given by gromovwitten invariants. 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. Aug 28, 2018 symplectic manifolds and pseudo holomorphic curves 6 transformation groups 4 metric, measure, concentration and isoperimetric inequalities 12 stein manifolds and infinite coverings of kaehler manifolds 11 infinite cartesian products and symbolic geometry 8 formalization of genetic and biomolecular structures 6 ergosystems 7. The goal of this paper is to formulate explicit hodgetheoretic criteria for. Symplectic geometry on moduli spaces of jholomorphic curves j. Jholomorphic curves and symplectic topology request pdf. First steps of symplectic topology, viiith international congress on mathemati cal physics marseille, 1986, 116, world sci. This theory is regarded as one of the most important tools in symplectic geometry. Pseudoholomorphic curves in symplectic manifolds misha.
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. Mosers lemma and local triviality of symplectic differential geometry 17 2. Jholomorphic curves and symplectic topology mathematical. Lectures on pseudo holomorphic curves and the symplectic isotopy problem bernd siebert and gang tian abstract. The second half of the book then extends this program in two complementary directions. Symplectic structures a new approach to geometry dusa mcduff introduction symplectic geometry is the geometry of a closed skewsymmetric form. Pdf plenary session,invited speaker conference pure. Holomorphic curves in symplectic geometry book, 1994. Rational curves on holomorphic symplectic fourfolds by brendan hassett and yuri tschinkel 1. 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. Jholomorphic curves and symplectic topology dusa mcduff. Jholomorphic curves and symplectic topology american. In other words, jmakes tm into a complex vector bundle and we have the canonical decomposition tm.
Lectures on holomorphic curves in symplectic and contact. Symplectic topology is a subfield of symplectic geometry, in which global properties of symplectic. The technology of punctured holomorphic curves and neckstretching allows us to translate between these two worlds and prove a theorem. 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. Symplectic geometry and holomorphic curves ias school of. We give a complete proof for gromovs compactness theorem for pseudo holomorphic curves both in the case of closed curves and curves with boundary. It turns out to be very different from the riemannian geometry with which we are familiar. Rational curves on holomorphic symplectic fourfolds brendan. Holomorphic curves in low dimensions from symplectic ruled.