From Stein to Weinstein and Back: Symplectic Geometry of Affine Complex Manifolds (Colloquium Publications, 59)

A beautiful and comprehensive introduction to this important field. —Dusa McDuff, Barnard College, Columbia University This excellent book gives a detailed, clear, and wonderfully written treatment of the interplay between the world of Stein manifolds and the more topological and flexible world of Weinstein manifolds. Devoted to this subject with a long history, the book serves as a superb introduction to this area and also contains the authors'new results. —Tomasz Mrowka, MIT This book is devoted to the interplay between complex and symplectic geometry in affine complex manifolds. Affine complex (a.k.a. Stein) manifolds have canonically built into them symplectic geometry which is responsible for many phenomena in complex geometry and analysis. The goal of the book is the exploration of this symplectic geometry (the road from “Stein to Weinstein”) and its applications in the complex geometric world of Stein manifolds (the road “back”). This is the first book which systematically explores this connection, thus providing a new approach to the classical subject of Stein manifolds. It also contains the first detailed investigation of Weinstein manifolds, the symplectic counterparts of Stein manifolds, which play an important role in symplectic and contact topology. Assuming only a general background from differential topology, the book provides introductions to the various techniques from the theory of functions of several complex variables, symplectic geometry, $h$-principles, and Morse theory that enter the proofs of the main results. The main results of the book are original results of the authors, and several of these results appear here for the first time. The book will be beneficial for all students and mathematicians interested in geometric aspects of complex analysis, symplectic and contact topology, and the interconnections between these subjects. Pt. 1. J-convexity. J-convex Functions And Hypersurfaces -- Smoothing -- Shapes For I-convex Hypersurfaces -- Some Complex Analysis -- Pt. 2. Existence Of Stein Structures. Symplectic And Contact Preliminaries -- The H-principles -- The Existence Theorem -- Pt. 3. Morse-smale Theory For J-convex Functions. Recollections From Morse Theory -- Modifications Of J-convex Morse Functions -- Pt. 4. From Stein To Weinstein And Back. Weinstein Structures -- Modifications Of Weinstein Structures -- Existence Revisited -- Deformations Of Flexible Weinstein Structures -- Deformations Of Stein Structures -- Pt. 5. Stein Manifolds And Symplectic Topology. Stein Manifolds Of Complex Dimension Two -- Exotic Stein Structures -- Appendix A: Some Algebraic Topology -- Appendix B: Obstructions To Formal Legendrian Isotopies -- Appendix C: Biographical Notes On The Main Characters. Kai Cieliebak, Yakov Eliashberg. Includes Bibliographical References And Index. Content: pt. 1. J-convexity. J-convex functions and hypersurfaces -- Smoothing -- Shapes for I-convex hypersurfaces -- Some complex analysis -- pt. 2. Existence of Stein structures. Symplectic and contact preliminaries -- The h-principles -- The existence theorem -- pt. 3. Morse-Smale theory for J-convex functions. Recollections from Morse theory -- Modifications of J-convex Morse functions -- pt. 4. From Stein to Weinstein and back. Weinstein structures -- Modifications of Weinstein structures -- Existence revisited -- Deformations of flexible Weinstein structures -- Deformations of Stein structures -- pt. 5. Stein manifolds and symplectic topology. Stein manifolds of complex dimension two -- Exotic Stein structures -- Appendix A: Some algebraic topology -- Appendix B: Obstructions to formal Legendrian isotopies -- Appendix C: Biographical notes on the main characters. Cieliebak (Ludwig Maximilians U., Germany) and Eliashberg (Stanford U., US) add refinements to the existence theorem for Stein structures and present new results concerning surrounding of subsets by J-convex hypersurfaces. The opening chapters explore the basic properties of J-convex functions and hypersurfaces, construct special hypersurfaces needed for extending J-convex functions over handles, and review several h-principles. Later chapters develop a Morse-Smale type theory for J-convex functions and introduce Weinstein manifolds and their deformations. Annotation 2013 Book News, Inc., Portland, OR (booknews.com) A beautiful and comprehensive introduction to this important field.-Dusa McDuff, Barnard College, Columbia University This excellent book gives a detailed, clear, and wonderfully written treatment of the interplay between the world of Stein manifolds and the more topological and flexible world of Weinstein manifolds. Devoted to this subject with a long history, the book serves as a superb introduction to this area and also contains the authors' new results.-Tomasz Mrowka, MIT This book is devoted to the interplay between complex and symplectic geometry in affine complex manifolds. Affine co