Course Syllabus for

# Complex Analysis in Several Variables Komplex analys i flera variabler

## MATP22F, 7.5 credits

Valid from: Spring 2019
Decided by: Professor Thomas Johansson
Date of establishment: 2019-09-12

## General Information

Division: Mathematics
Course type: Course given jointly for second and third cycle
The course is also given at second-cycle level with course code: MATP22
Teaching language: English

## Aim

The aim of the course is to give an overview of the field of complex analysis in several variables, in particular hightlighting the differences from and the similarities with complex analysis in one variable,

## Goals

Knowledge and Understanding

For a passing grade the doctoral student must

• be able to give different conditions for a function of several complex variables to be holomorphic and to explain why they are equivalent.
• be able to account for the theory of domains of holomorphy. In particular be able to give some examples.
• be able to explain how analytic continuation in several variables is different from the situation in one variable, in particular account for some versions of Hartogs’ extension theorem.
• be able to describe the local structure of zero sets of holomorphic functions and how they relate to analytic sets and complex submanifolds.
• be able to explain how pluripotential theory, in particular the study of plurisubharmonic functions, is needed to develop general methods for solving d-bar equations.
• be able to compare and contrast different notions of convexity, in particular geometric convexity and pseudoconvexity.
• be able to give some examples of integral representation formulas for holomorphic functions in several variables.
• be able to give some examples of the relevance of the solvability of d-bar equations.

## Course Contents

Basic properties of holomorphic functions. Analytic continuation and power series in several variables. The inhomogeneous Cauchy-Riemann equation. The Weierstrass preparation theorem, zero sets and singularities. Hartog's theorem. Holomorphic mappings and complex manifolds. Convexity and holomorphic convexity. Domains of holomorphy. The first Cousin problem. The Levi problem. Pluripotential theory. Pseudoconvex domains. Integral representation formulas. Solutions of the d-bar equation for pseudoconvex domains.

## Course Literature

Korevaar, J. & Wiegerinck, J.: Lecture notes in several complex variables. 2017.
Freely available from https://staff.science.uva.nl/j.j.o.o.wiegerinck/edu/scv/scvboek.pdf

## Instruction Details

Type of instruction: Lectures

## Examination Details

Examination format: Oral exam