Holomorphic Curves in Low Dimensions: From Symplectic Ruled Surfaces to Planar Contact Manifolds (Lecture Notes in Mathematics, 2216)

This monograph provides an accessible introduction to the applications of pseudoholomorphic curves in symplectic and contact geometry, with emphasis on dimensions four and three. The first half of the book focuses on McDuff's characterization of symplectic rational and ruled surfaces, one of the classic early applications of holomorphic curve theory. The proof presented here uses the language of Lefschetz fibrations and pencils, thus it includes some background on these topics, in addition to a survey of the required analytical results on holomorphic curves. Emphasizing applications rather than technical results, the analytical survey mostly refers to other sources for proofs, while aiming to provide precise statements that are widely applicable, plus some informal discussion of the analytical ideas behind them. The second half of the book then extends this program in two complementary directions: (1) a gentle introduction to Gromov-Witten theory and complete proof of the classification of uniruled symplectic 4-manifolds; and (2) a survey of punctured holomorphic curves and their applications to questions from 3-dimensional contact topology, such as classifying the symplectic fillings of planar contact manifolds.This book will be particularly useful to graduate students and researchers who have basic literacy in symplectic geometry and algebraic topology, and would like to learn how to apply standard techniques from holomorphic curve theory without dwelling more than necessary on the analytical details. This book is also part of the __Virtual Series on Symplectic Geometry__ http://www.springer.com/series/16019 Preface......Page 7 Acknowledgments......Page 9 Contents......Page 11 1.1 Some Examples of Symplectic 4-Manifolds and Submanifolds......Page 14 1.2 Results About Symplectically Embedded Spheres......Page 23 1.3 Summary of the Proofs......Page 28 1.3.2 The Case [S] ·[S] = 0......Page 29 1.3.3 The Case [S] ·[S] > 0......Page 30 1.4 Outline of the Remaining Chapters......Page 32 2.1.1 Symplectic and Almost Complex Structures......Page 34 2.1.2 Simple Holomorphic Curves and Multiple Covers......Page 36 2.1.3 Smoothness and Dimension of the Moduli Space......Page 39 2.1.4 Moduli Spaces with Marked Point Constraints......Page 47 2.1.5 Constraints on Derivatives......Page 53 2.1.6 Gromov Compactness and Singularities......Page 58 2.1.7 Gluing......Page 62 2.1.8 Orientations......Page 66 2.2.1 Automatic Transversality......Page 70 2.2.2 Positivity of Intersections and Adjunction......Page 75 2.2.3 An Implicit Function Theorem for Embedded Spheres with Constraints......Page 77 3.1 The Complex Blowup......Page 80 3.2 The Symplectic Blowup......Page 83 3.3 Smooth Topology of Lefschetz Pencils and Fibrations......Page 90 3.4 Symplectic Lefschetz Pencils and Fibrations......Page 98 4.1 Two Compactness Theorems for Spaces of Embedded Spheres......Page 112 4.2 Index Counting in Dimension Four......Page 117 4.3 Proof of the Compactness Theorems When m=0......Page 122 4.4 Proof of the Compactness Theorems with Constraints......Page 123 5.1 Deforming Pseudoholomorphic (-1)-Curves......Page 129 5.2 Proofs of Theorems B and C......Page 132 6.1 Proofs of Theorems F and G......Page 135 6.2 Proofs of Theorems A, D and E......Page 139 7.1 Further Characterizations of Rational or Ruled Surfaces......Page 144 7.2.1 The Invariants in General......Page 150 7.2.2 The Uniruled Condition......Page 154 7.2.3 Pseudocycles and the Four-Dimensional Case......Page 155 7.2.4 Rational/Ruled Implies Uniruled......Page 165 7.3.1 A Brief Word from Seiberg-Witten Theory......Page 166 7.3.2 Outline of the Proof......Page 169 7.3.3 The Universal J-Holomorphic Curve......Page 172 7.3.4 The Moduli Space as a Blown-Up Ruled Surface......Page 174 7.3.5 The Evaluation Map Has Positive Degree......Page 182 7.3.6 Topology of Ruled Surfaces and the Signature Theorem......Page 185 7.3.7 Conclusion of the Proof......Page 195 8.1 The Conjectures of Arnol'd and Weinstein......Page 201 8.2 Symplectic Cobordisms and Fillings......Page 212 8.3.1 Punctures and the Finite Energy Condition......Page 219 8.3.2 Simple and Multiply Covered Curves......Page 226 8.3.3 Smoothness and Dimension of the Moduli Space......Page 228 8.3.4 SFT Compactness......Page 231 8.3.5 Gluing Along Punctures......Page 235 8.3.6 Coherent Orientations......Page 238 8.3.7 Automatic Transversality......Page 240 8.3.8 Intersection Theory......Page 244 9.1 Fillings of S3 and the Weinstein Conjecture......Page 251 9.2 Fillings of the 3-Torus and Giroux Torsion......Page 262 9.3 Planar Contact Manifolds......Page 269 A Generic Nodes Lefschetz Critical Points......Page 284 Bibliography......Page 288 Index......Page 296 This monograph provides an accessible introduction to the applications of pseudoholomorphic curves in symplectic and contact geometry, with emphasis on dimensions four and three. The first half of the book focuses on McDuff's characterization of symplectic rational and ruled surfaces, one of the classic early applications of holomorphic curve theory. The proof presented here uses the language of Lefschetz fibrations and pencils, thus it includes some background on these topics, in addition to a survey of the required analytical results on holomorphic curves. Emphasizing applications rather than technical results, the analytical survey mostly refers to other sources for proofs, while aiming to provide precise statements that are widely applicable, plus some informal discussion of the analytical ideas behind them. The second half of the book then extends this program in two complementary directions: (1) a gentle introduction to Gromov-Witten theory and complete proof of the classification of uniruled symplectic 4-manifolds; and (2) a survey of punctured holomorphic curves and their applications to questions from 3-dimensional contact topology, such as classifying the symplectic fillings of planar contact manifolds. This book will be particularly useful to graduate students and researchers who have basic literacy in symplectic geometry and algebraic topology, and would like to learn how to apply standard techniques from holomorphic curve theory without dwelling more than necessary on the analytical details. This book is also part of the Virtual Series on Symplectic Geometry http://www.springer.com/series/16019 "This monograph provides an accessible introduction to the applications of pseudoholomorphic curves in symplectic and contact geometry, with emphasis on dimensions four and three. The first half of the book focuses on McDuff's characterization of symplectic rational and ruled surfaces, one of the classic early applications of holomorphic curve theory. The proof presented here uses the language of Lefschetz fibrations and pencils, thus it includes some background on these topics, in addition to a survey of the required analytical results on holomorphic curves. Emphasizing applications rather than technical results, the analytical survey mostly refers to other sources for proofs, while aiming to provide precise statements that are widely applicable, plus some informal discussion of the analytical ideas behind them. The second half of the book then extends this program in two complementary directions: (1) a gentle introduction to Gromov-Witten theory and complete proof of the classification of uniruled symplectic 4-manifolds; and (2) a survey of punctured holomorphic curves and their applications to questions from 3-dimensional contact topology, such as classifying the symplectic fillings of planar contact manifolds. This book will be particularly useful to graduate students and researchers who have basic literacy in symplectic geometry and algebraic topology, and would like to learn how to apply standard techniques from holomorphic curve theory without dwelling more than necessary on the analytical details"--Page 4 of cover