Mathematics Asked by Eduardo Magalhães on January 21, 2021
What are the necessary knowledge in order to learn Several complex variables? In the beginning I thought that complex analysis and multivariable calculus were the only thing, but then I realized that in this branch of mathematics, functions are defined over the $mathbb{C}^n$ space, so I think it would be a good idea to firstunderstand the $mathbb{C}^n$ space and maps between vector spaces over $mathbb{C}$. So what are the list of prerequisites needed to understand Several complex variables in an medium/advanced level?
It turns out that "several complex variables" (popularly "SCV") is significantly different from "one complex variable". The "nothing could go wrong" spirit in one complex variable is not quite warranted in more-than-one complex variable. Yes, there are some things that work better (!!!???) in more than one complex variable (see Hartogs' results, and Bochner's lemma).
The prereqs in terms of other abstract/modern mathematics do become more substantial. It is possible to try to dodge those prereqs, but this makes a mess of its own.
As @TedShifrin commented, L. Hormander's book "Intro to SCV..." is standard. Also S. Krantz's "SCV". R. Gunning wrote 3 volumes on the subject. But/and you will find that at some point these authors see compelling reasons to use non-trivial results from PDE, sheaf theory, abstract algebra, and so on... just to cope with the complications.
In my own experience, Gunning's old "yellow Princeton series" book on Riemann surfaces gives a good intro to rewriting things in terms of sheaves... and seeing the benefits of that machinery. I'd recommend looking at that first.
Also, Griffiths-Harris' book (from about 1978) on "algebraic geometry" is really about algebraic geometry over the complex numbers, and a big part of the advantage of taking that viewpoint is using complex analysis...
Correct answer by paul garrett on January 21, 2021
The best treatment of the subject is, in my opinion, B. V. Shabat's Introduction to Complex Analysis Part II.
Answered by Very Confused on January 21, 2021
It really depends on which direction one takes. But everything from commutative algebra to PDE can be useful. For the very basics, I'd say you need some basic abstract algebra, real-analysis (preferably basic measure theory), and of course complex analysis in one variable (a basic one-semester course or equivalent is good enough for a start). It is good to also have a good introduction to calculus in several variables, preferably analysis on manifolds such as Spivak's book, in particular differential forms.
For a (hopefully) gentle introduction with minimal prerequisites see my online book: Tasty Bits of Several Complex Variables. It has a "further reading" chapter for some more tips on other useful books to look at, and it has several appendices on some useful background.
Answered by Jiri Lebl on January 21, 2021
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