Lectures on Symplectic Geometry - ETH Z

1 Lectures on Symplectic GeometryAna Cannas da Silva1revised January 2006 Published by Springer-Verlag asnumber 1764 of the series Lecture Notes in original publication is available notes approximately transcribe a 15-week course on Symplectic Geometry Itaught at UC Berkeley in the Fall of course at Berkeley was greatly inspired in content and style by VictorGuillemin, whose masterly teaching of beautiful courses on topics related to sym-plectic Geometry at MIT, I was lucky enough to experience as a graduate am very thankful to him!That course also borrowed from the 1997 Park City summer courses on symplec-tic Geometry and topology, and from many talks and discussions of the symplecticgeometry group at MIT. Among the regular participants in the MIT informal sym-plectic seminar 93-96, I would like to acknowledge the contributions of Allen Knut-son, Chris Woodward, David Metzler, Eckhard Meinrenken, Elisa Prato, EugeneLerman, Jonathan Weitsman, Lisa Jeffrey, Reyer Sjamaar, Shaun Martin, StephanieSinger, Sue Tolman and, last but not least, Yael to everyone sitting in Math 242 in the Fall of 1997 for all the commentsthey made, and especially to those who wrote notes on the basis of which I was betterable to reconstruct what went on.

2 Alexandru Scorpan, Ben Davis, David Martinez,Don Barkauskas, Ezra Miller, Henrique Bursztyn, John-Peter Lund, Laura De Marco,Olga Radko, Peter P rib k, Pieter Collins, Sarah Packman, Stephen Bigelow, SusanHarrington, Tolga Etg u and Yi am indebted to Chris Tuffley, Megumi Harada and Saul Schleimer who read thefirst draft of these notes and spotted many mistakes, and to Fernando Louro, GrishaMikhalkin and, particularly, Jo ao Baptista who suggested several improvements andcareful corrections. Of course I am fully responsible for the remaining errors interest of Alan Weinstein, Allen Knutson, Chris Woodward, Eugene Ler-man, Jiang-Hua Lu, Kai Cieliebak, Rahul Pandharipande, Viktor Ginzburg and YaelKarshon was crucial at the last stages of the preparation of this manuscript. I amgrateful to them, and to Mich`ele Audin for her inspiring texts and , many thanks to Faye Yeager and Debbie Craig who typed pages ofmessy notes into neat LATEX, to Jo ao Palhoto Matos for his technical support, andto Catriona Byrne, Ina Lindemann, Ingrid M arz and the rest of the Springer-Verlagmathematics editorial team for their expert Cannas da SilvaBerkeley, November 1998and Lisbon, September 2000vCONTENTSviiContentsForewordvIntrodu ction1I Symplectic Manifolds31 Symplectic Skew-Symmetric Bilinear Maps.

3 Symplectic Vector Spaces .. Symplectic manifolds .. Symplectomorphisms ..7 Homework 1: Symplectic Linear Algebra82 Symplectic Form on the Cotangent Cotangent Bundle .. Tautological and Canonical Forms in Coordinates .. Coordinate-Free Definitions .. Naturality of the Tautological and Canonical Forms ..11 Homework 2: Symplectic Volume13II Symplectomorphisms153 Lagrangian Submanifolds .. Lagrangian Submanifolds ofT X.. Conormal Bundles .. Application to Symplectomorphisms ..18 Homework 3: Tautological Form and Symplectomorphisms204 Generating Constructing Symplectomorphisms .. Method of Generating Functions .. Application to Geodesic Flow ..24 Homework 4: Geodesic Flow27viiiCONTENTS5 Periodic Points .. Billiards .. Poincar e Recurrence.

4 32 III Local Forms356 Preparation for the Local Isotopies and Vector Fields .. Tubular Neighborhood Theorem .. Homotopy Formula ..39 Homework 5: Tubular Neighborhoods inRn417 Moser Notions of Equivalence for Symplectic Structures .. Moser Trick .. Moser Relative Theorem ..458 Darboux-Moser-Weinstein Darboux Theorem .. Lagrangian Subspaces .. Weinstein Lagrangian Neighborhood Theorem ..48 Homework 6: Oriented Surfaces509 Weinstein Tubular Neighborhood Observation from Linear Algebra .. Tubular Neighborhoods .. Application 1:Tangent Space to the Group of Symplectomorphisms .. Application 2:Fixed Points of Symplectomorphisms ..55IV Contact Manifolds5710 Contact Contact Structures .. Examples .. First Properties ..59 Homework 7: manifolds of Contact Elements61 CONTENTSix11 Contact Reeb Vector Fields.

5 Symplectization .. Conjectures of Seifert and Weinstein ..65V Compatible Almost Complex Structures6712 Almost Complex Three Geometries .. Complex Structures on Vector Spaces .. Compatible Structures ..70 Homework 8: Compatible Linear Structures7213 Compatible Compatibility .. Triple of Structures .. First Consequences ..75 Homework 9: Contractibility7714 Dolbeault Splittings .. Forms of Type(`,m).. Functions .. Dolbeault Cohomology ..81 Homework 10: Integrability82VI K ahler Manifolds8315 Complex Complex Charts .. Forms on Complex manifolds .. Differentials ..86 Homework 11: Complex Projective Space8916 K ahler K ahler Forms .. An Application .. Recipe to Obtain K ahler Forms .. Local Canonical Form for K ahler Forms ..94 Homework 12: The Fubini-Study Structure96xCONTENTS17 Compact K ahler Hodge Theory.

6 Immediate Topological Consequences .. Compact Examples and Counterexamples .. Main K ahler manifolds ..103 VII Hamiltonian Mechanics10518 Hamiltonian Vector Hamiltonian and Symplectic Vector Fields .. Classical Mechanics .. Brackets .. Integrable Systems ..109 Homework 13: Simple Pendulum11219 Variational Equations of Motion .. Principle of Least Action .. Variational Problems .. Solving the Euler-Lagrange Equations .. Minimizing Properties ..117 Homework 14: Minimizing Geodesics11920 Legendre Strict Convexity .. Legendre Transform .. Application to Variational Problems ..122 Homework 15: Legendre Transform125 VIII Moment Maps12721 One-Parameter Groups of Diffeomorphisms .. Lie Groups .. Smooth Actions .. Symplectic and Hamiltonian Actions .. Adjoint and Coadjoint Representations.

7 130 Homework 16: Hermitian Matrices132 CONTENTSxi22 Hamiltonian Moment and Comoment Maps .. Orbit Spaces .. Preview of Reduction .. Classical Examples ..137 Homework 17: Coadjoint Orbits139IX Symplectic Reduction14123 The Marsden-Weinstein-Meyer Statement .. Ingredients .. Proof of the Marsden-Weinstein-Meyer Theorem ..14524 Noether Principle .. Elementary Theory of Reduction .. Reduction for Product Groups .. Reduction at Other Levels .. Orbifolds ..150 Homework 18: Spherical Pendulum152X Moment Maps Revisited15525 Moment Map in Gauge Connections on a Principal Bundle .. Connection and Curvature Forms .. Symplectic Structure on the Space of Connections .. Action of the Gauge Group .. Case of Circle Bundles ..159 Homework 19: Examples of Moment Maps16226 Existence and Uniqueness of Moment Lie Algebras of Vector Fields.

8 Lie Algebra Cohomology .. Existence of Moment Maps .. Uniqueness of Moment Maps ..167 Homework 20: Examples of Reduction168xiiCONTENTS27 Convexity Theorem .. Effective Actions .. Examples ..172 Homework 21: Connectedness174XI Symplectic Toric Manifolds17728 Classification of Symplectic Toric Delzant Polytopes .. Delzant Theorem .. Sketch of Delzant Construction ..18029 Delzant Algebraic Set-Up .. The Zero-Level .. Conclusion of the Delzant Construction .. Idea Behind the Delzant Construction ..186 Homework 22: Delzant Theorem18930 Duistermaat-Heckman Duistermaat-Heckman Polynomial .. Local Form for Reduced Spaces .. Variation of the Symplectic Volume ..195 Homework 23:S1-Equivariant Cohomology197 References199 Index207 IntroductionThe goal of these notes is to provide a fast introduction to Symplectic Symplectic form is a closed nondegenerate 2-form.

9 A Symplectic manifold isa manifold equipped with a Symplectic form. Symplectic Geometry is the geometryof Symplectic manifolds . Symplectic manifolds are necessarily even-dimensional andorientable, since nondegeneracy says that the top exterior power of a symplecticform is a volume form. The closedness condition is a natural differential equation,which forces all Symplectic manifolds to being locally indistinguishable. (Theseassertions will be explained in Lecture 1 and Homework 2.)The list of questions on Symplectic forms begins with those of existence anduniqueness on a given manifold. For specific Symplectic manifolds , one would liketo understand the Geometry and the topology of special submanifolds, the dynamicsof certain vector fields or systems of differential equations, the symmetries and extrastructure, centuries ago, Symplectic Geometry provided a language for classical me-chanics.

10 Through its recent huge development, it conquered an independent andrich territory, as a central branch of differential Geometry and topology. To mentionjust a few key landmarks, one may say that Symplectic Geometry began to take itsmodern shape with the formulation of the Arnold conjectures in the 60 s and withthe foundational work of Weinstein in the 70 s. A paper of Gromov [49] in the 80 sgave the subject a whole new set of tools: pseudo-holomorphic curves. Gromov alsofirst showed that important results from complex K ahler Geometry remain true in themore general Symplectic category, and this direction was continued rather dramati-cally in the 90 s in the work of Donaldson on the topology of Symplectic manifoldsand their Symplectic submanifolds, and in the work of Taubes in the context ofthe Seiberg-Witten invariants.