Several Complex Variables and the Geometry of Real Hypersurfaces

John P. D'Angelo

• Publisher: CRC Press
• Year: 2019
• Print ISBN: 9780849382727
• eBook ISBN: 9781351416719
• Topic: Mathematics
• Subtopic: Complex Analysis
• Index: Mathematics

Several Complex Variables and the Geometry of Real Hypersurfaces covers a wide range of information from basic facts about holomorphic functions of several complex variables through deep results such as subelliptic estimates for the ?-Neumann problem on 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. You will learn how to decide whether a real hypersurface contains complex varieties, how closely such varieties can contact the hypersurface, and why it's important. The book concludes with two sets of problems: routine problems and difficult problems (many of which are unsolved). Principal prerequisites for using this book include a thorough understanding of advanced calculus and standard knowledge of complex analysis in one variable. Several Complex Variables and the Geometry of Real Hypersurfaces will be a useful text for advanced graduate students and professionals working in complex analysis.

1

Holomorphic Functions and Mappings

1.1 Preliminaries

1.1.1 Complex n-dimensional space

We denote by ℂⁿ the complex vector space of dimension n with its usual Euclidean topology. A domain in ℂⁿ is an open, connected set. The collection of open polydiscs constitute a basis for the open subsets of ℂⁿ, where a polydisc is a product of discs. Let z =(z¹, …, z²) denote coordinates on ℂⁿ. Our notation for the polydisc of multi-radius r(r¹,…, rⁿ) centered at w will be

Pr(w)={z:|zj-wj|<rjj=1,...,n}.⁢(1)

An alternate basis for the topology is the collection of open balls. The ball of radius r about w is defined by

Br(w)={ z:‖ z−w ‖2<r2 }⁢(2)

where ||z-w||2=∑j=1j|zj-wj|2 denotes the squared Euclidean distance. Notice that we use the same symbol for the radius of a ball and the multi-radius of a polydisc. One of our first tasks will be to show that the polydisc and the ball have very different function theories.

The geometry of ℂⁿ reveals immediately the importance of the interaction between the real and the complex. Consider the underlying real variables defined by

zj=xj+iyj=Re(zj)+iIm(zj).⁢(3)

For a basis of the complex-valued differential one-forms on ℂⁿ, we can take either the underlying real differentials, or the complex differentials { dz1,…,dzn,dz¯1,…,dz¯n }, These are related by the formulas

dxj=12(dzj+dz¯j)⁢dyj=12i(dzj-dz¯j).⁢(4)

The differential of a smooth function f can be written therefore in two ways:

df=∑​nj=1∂f∂xjdxj+∂f∂yjdyj⁢(5)

df=∂f+∂¯f=∑j=1n∂f∂zjdzj+∑j=1n∂f∂z¯jdz¯1.⁢(6)

It is important to observe that there is no choice in the definition of the partial derivative operators

∂∂zj=12(∂∂xj-i∂∂yj)⁢∂∂z¯j=12(∂∂xj+i∂∂yj).⁢(7)

These definitions follow immediately from equating the two formulas for the differential df and plugging in the formulas for the real differentials in terms of the complex differentials.

In this book we will work only with the complex differentials. Differential one-forms that are (functional) combinations of the dz^j are called forms of type (1,0), and those that are combinations of the dz¯j are called forms of type (0,1). The coefficient functions are in general smooth complex-valued functions, although in Chapter 6 the coefficient functions will be more general. We consider also differential forms of type (p, q). It is standard to write dz^j = dz^j1 Λ dz^j2… Λdz^jq for the basic (q, 0) forms. For complex vector fields one uses similar terminology; thus a (functional) combination of the ∂/∂z^j is called a vector fiel...

Table of contents

• Cover
• Half Title
• Title Page
• Copyright Page
• Dedication
• Contents
• Preface
• 1 Holomorphic Functions and Mappings
• 2 Holomorphic Mappings and Local Algebra
• 3 Geometry of Real Hypersurfaces
• 4 Points of Finite Type
• 5 Proper Holomorphic Mappings Between Balls
• 6 Geometry of the ∂¯-Neumann Problem
• 7 Analysis on Finite Type Domains
• Problems
• Index of Notation
• Bibliography
• Index