Several Complex Variables and the Geometry of Real Hypersurfaces

Intended for mathematicians and algebraic analysts, this book provides a thorough introduction to the geometric and algebraic relationships between real hypersurfaces and complex analytical varieties. It covers a wide range of information, from basic facts about holomorphic functions of several complex variables, to deep results such as subelliptic estimates for the muNeumann problem of pseudoconvex domains with a real analytic boundary. The book focuses on describing the geometry of a real hypersurface in a complex vector space by understanding its relationship with ambient complex analytic varieties. It shows the reader how to decide wheter a real hypersurface contains complex varieties, how closely such varieties can contact the hypersurface and why it is important. The book concludes with two sets of problems: routine problems; and difficult problems (many of which are unsolved). Several concrete examples are included, such as for the local parametization theorem, the jump phenomena for the 1-type of a point on a hypersurface, and Kohn's method for computing ideal type. Principle prerequisites for using this book include a thorough understanding of advanced calculus and standard knowledge of complex analysis in one variable.