The subject of Several Complex Variables (SCV) is too large for us to be able to cover in a semester (or even in a year). I will try to cover the "Classical" basis of the theory.
Before semester starts, please read the paper
by Chakrabarti. This should be fun to read and help you get into the mood for this course.
There are several possible texts we could follow. You can take a look at some good ones:
Bers
Shabat
Wiegerinck and Korevaar
There is also a very nice book by Grauert and Fritzsche, "From Holomorphic Functions to Complex Manifolds", but I haven't found a digital version online.
If you find it, please let me know. In any case, you can look at a preview of it on amazon.com.
Please browse these books before the first class, and try to form an opinion about which one you
might want to use for a text.
We will discuss this in the first class.
In the first week we proved the
Hartogs Theorem on separate analyticity. You can imagine that such a basic theorem will have led to a lot
of further mathematical progress. A very scholarly treatment of this is given by
Jarnicki and Pflug. The general prototype
here is the general question: "If a function is Regular in each variable separately, does it follow that the function
is Regular in all variables jointly?"
In the familiar case where 'Regular' = 'continuous', you know that the answer is "No". In the first page of Jarnicki and
Pflug, they make some historical comments. In particular, the great Cauchy thought that the answer was "Yes". You may
be interested to read this.
In connection with Hartogs Theorem, here are some
questions and comments
about possible generalizations.
questions1
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