CR geometry is nowadays a very broad subject with its scope spanning from the geometric theory of partial differential equations and microlocal analysis to complex and symplectic geometry and foliation theory. This book by Giuseppe Zampieri does not aim to introduce all topics of current interest in CR geometry. Instead, it attempts to be friendly to the novice by moving in a fairly relaxed way from elements of the theory of holomorphic functions in several complex variables to advanced topics of modern CR geometry. The choice of topics provides a good balance between a first exposure to CR geometry and subjects representing current research.
The first chapter of the book covers classical results in several complex variables (Cauchy formulas in polydiscs, Hartogs’ theorems on separability and extendability of holomorphic functions and the logarithmic supermean of the Taylor radius of holomorphic functions). It finishes with the L2 and subelliptic estimates for the ∂-bar operator. The second chapter covers real/complex structures and real/complex symplectic spaces. The Frobenius and Darboux theorems are proved. The third chapter, which constitutes the second half of the book, covers CR structures. A particular emphasis is devoted to analysis of the conormal bundle to a real submanifold of Cn under a canonical transformation. The author then describes the theory of analytic discs attached to real submanifolds and their infinitesimal deformations. The problem of construction of lifts and partial lifts of analytic discs is also addressed. Zampieri also deals with Bloom-Graham normal forms and separate real analyticity. The book is written in a very readable style; every section starts with a short summary and there are a lot of notes and remarks providing a broader context for discussed questions. The reader is supplied with a lot of exercises (hints are provided) and also with suggested research topics, where the author discusses open problems related to the material of the chapter.