Topological Methods in Stein Geometry

Abstract

We begin by exploring the homotopy principle for totally real immersions and embeddings. A major part of the chapter is devoted to the study of complex points of real surfaces in complex surfaces. After proving the Lai formulae (on the algebraic number of complex points) and the Eliashberg-Harlamov cancellation theorem, we explore the connections between the generalized adjunction inequality and the existence of a Stein neighborhood basis of an isotopically embedded or immersed surface, showing how Seiberg-Witten theory bears upon these questions by arguments similar to those leading to the generalized Thom conjecture. In the last part we present the Eliashberg-Gompf construction of integrable Stein structures on almost complex manifolds with a suitable handlebody decomposition, and we prove the Soft Oka Principle.