Holomorphic Curves in Low Dimensions

1. Front Matter

   Pages i-xiii

   PDF

2. Introduction

   • Chris Wendl

   Pages 1-20

3. Background on Closed Pseudoholomorphic Curves

   • Chris Wendl

   Pages 21-66

4. Blowups and Lefschetz Fibrations

   • Chris Wendl

   Pages 67-98

5. Compactness

   • Chris Wendl

   Pages 99-115

6. Exceptional Spheres

   • Chris Wendl

   Pages 117-122

7. Rational and Ruled Surfaces

   • Chris Wendl

   Pages 123-131

8. Uniruled Symplectic 4-Manifolds

   • Chris Wendl

   Pages 133-189

9. Holomorphic Curves in Symplectic Cobordisms

   • Chris Wendl

   Pages 191-240

10. Contact 3-Manifolds and Symplectic Fillings

    • Chris Wendl

    Pages 241-273

11. Back Matter

    Pages 275-294

    PDF

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

• MSC (2010): 57R17, 32Q65
• symplectic ruled surfaces
• rational and ruled symplectic 4-manifolds
• planar contact manifolds
• pseudoholomorphic curves
• Gromov-Witten invariants
• uniruled symplectic manifolds
• inimal symplectic 4-manifolds
• Lefschetz fibrations
• Lefschetz pencils
• immersed symplectic spheres
• symplectic fillings

“The book is well-written, well-referenced … . Anyone interested in McDuff’s characterization of rational and ruled symplectic 4-manifolds or her theorem that says ‘uniruled => rational/ruled’ should find this book quite useful.” (David E. Hurtubise, zbMATH 1432.57055, 2020)

• Institut für Mathematik, Humboldt-Universität zu Berlin, Berlin, Germany

  Chris Wendl

Chris Wendl is known among symplectic topologists for contributions to the study of symplectic fillability of contact manifolds, and for transversality results in the theory of pseudoholomorphic curves. He is currently Professor of Differential Geometry and Global Analysis at the Humboldt University in Berlin, and is also the author of two other forthcoming books on holomorphic curves and symplectic field theory.

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