Function Theory of Several Complex Variables

Aims at: Masters mathematics

Prerequisits: Analysis courses from the Bachelor Mathematics

Goals: Familiarity with the basic concepts in the theory of analytic functions of several complex variables, as well as with a number of fundamental results of the subject.

Contents: Analyticity in several variables, domains of convergence of power series (Reinhardt domains), Cauchy-Riemann equations, Hartogs' extension theorem, domains of holomorphy, pseudoconvexity and the Levi problem. Description of zero sets (Weierstrass preparation theorem), holomorphic mappings. Plurisubharmonic functions, L^2 and Kernel methods for solving inhomogeneous Cauchy Riemann equations. Cousin problems and cohomology.

Semester: I

Form: Course (2 h. per week) If only few students are attending, we switch possibly to a reading course.

Euro credits: 6 EC Possiblity to extend with maximal 3 EC

Extra college! 7 december 13-15 uur, zaal A.1.06

Examination

Take Home execises

date      exercises                       
Sept 6    Chapter 1  no 8,9,12,18,20,24,29 
Sept 13   Chapter 2  no 2,7,12,24,25,26   
Sept 20   Chapter 3  no 6,14 (and develop from earlier exercises what you need), 24
Sept 27   Chapter 4  no 5,7,9,16,21,26,27,28
Oct.4     Chapter 5. no 12,13,20,21,30,33,
Oct 11    Chapter 6. no 3, 4,5,10,11.
Oct 18    Chapter 6  no 16, 21, 23,26
Nov 1     Chapter 7. no 6,10,12,17,24
Nov 15    Chapter 8. no 5,9,15,33,34
Nov 29-Dec 6   Chapter 9. no 2 5,6,7,11,12,13.



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Material Covered

• 6 sep. Chapter 1. (Globally)
• 13 sep. Chapter 2.
• 20 sep. Chapter 3.
• 27 sep. Chapter 4.
• 4 okt. Chapter 5.
• 11 okt. Fatou-Bieberbach gebied, (a la Krantz), Chapter 6. t/m 6.3.k
• 18 okt. Chapter 6 Stelling Cartan Thullen,
• 25 okt. Geen college
• 1 nov. Chapter 7.
• 8 nov. Chapter 6 en 7:Losse einden
• 15 nov Chapter 8
• 22 nov Geen college
• 29 nov. Chapter 9 Pseudoconvexiteit
• 6 dec. Chapter 9, Levi-pseudoconvexiteit, rand van psc gebieden, Chapter 10 Bochner Martinelly representatie.

The Examination

Literature:
• J. Korevaar-J. Wiegerinck: Lecture notes several complex variables, PDF-version. This will be updated and corrected continuously.

Additional reading:
• S. Krantz: Function theory of several complex variables , AMS-chelsea 1992
• L. Hörmander: Complex analysis in several variables, North Holland 1973
• K. Fritzsche - H. Grauert: From Holomorphic functions to complex manifolds Springer, 2002.
• T. Ohsawa Analysis of Several Complex Variables, TAMS 211 AMS, 2002

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