# Lectures on Counterexamples in Several Complex Variables

### About this Title

**John Erik Fornæss**, *University of Michigan, Ann Arbor, Ann Arbor, MI* and **Berit Stensønes**, *University of Michigan, Ann Arbor, Ann Arbor, MI*

Publication: AMS Chelsea Publishing

Publication Year:
1987; Volume 363

ISBNs: 978-0-8218-4422-9 (print); 978-1-4704-3120-4 (online)

DOI: https://doi.org/10.1090/chel/363

### Table of Contents

**Front/Back Matter**

**Lectures on counterexamples in several complex variables **

- Some notations and definitions
- Holomorphic functions
- Holomorphic convexity and domains of holomorphy
- Stein manifolds
- Subharmonic/Plurisubharmonic functions
- Pseudoconvex domains
- Invariant metrics
- Biholomorphic maps
- Counterexamples to smoothing of plurisubharmonic functions
- Complex Monge Ampère equation
- $H^\infty $-convexity
- CR-manifolds
- Pseudoconvex domains without pseudoconvex exhaustion
- Stein neighborhood basis
- Riemann domains over $\mathbb {C}^n$
- The Kohn-Nirenberg example
- Peak points
- Bloom’s example
- D’Angelo’s example
- Integral manifolds
- Peak sets for A(D)
- Peak sets. Steps 1–4
- Sup-norm estimates for the $\bar {\partial }$-equation
- Sibony’s $\bar {\partial }$-example
- Hypoellipticity for $\bar {\partial }$
- Inner functions
- Large maximum modulus sets
- Zero sets
- Nontangential boundary limits of functions in $H^\infty (\mathbb {B}^n)$
- Wermer’s example
- The union problem
- Riemann domains
- Runge exhaustion
- Peak sets in weakly pseudoconvex boundaries
- The Kobayashi metric