A whirlwind tour of the subject
By: Jiří Lebl (website #1 https://www.jirka.org/ (personal), website #2 https://math.okstate.edu/people/lebl/ (work: OSU), email: )
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This book is a polished version of my course notes for Math 6283, Several Complex Variables, given in Spring 2014, Spring 2016, Spring 2019, and Fall 2023 semesters at the Oklahoma State University. There is more material than can fit in a one semester class to allow for several different versions of the course. In fact, I did a different selection each semester I taught it. Quite a few exercises of various difficulty are sprinkled throughout the text, and I hope a reader is at least attempting or thinking about most of them. Many are required later in the text. The reader should attempt exercises in sequence; earlier exercises can help or even be required to solve later ones.
The prerequisites are a decent knowledge of vector calculus, basic real analysis, and a working knowledge of complex analysis in one variable. Measure theory (Lebesgue integral and its convergence theorems) is useful, but it is not essential except in a couple of places later in the book. The first two chapters and most of the third are accessible to beginning graduate students after one semester of a standard single-variable complex analysis graduate course. From time to time, basic knowledge of differential forms is useful, and in the chapter on varieties we use some basic ring theory from algebra. By design, it can replace the second semester of complex analysis, perhaps taught with my one-variable book GCCA.
This book is not intended as an exhaustive reference. It is simply a whirlwind tour of several complex variables. See the end of the book for a list of books for reference and further reading. There are also appendices for a list of one-variable results, an overview of differential forms, some basic algebra, measure theory, and other bits and pieces of analysis.
Do let me know if there are typos or errors. You can email me at .
This book may be modified and customized for a specific purpose if necessary. If you do modify the book, make sure to mark it prominently as such to avoid confusion. This aspect is also important for longevity of the book. The book can be updated and modified even if I happen to drop off the face of the earth. You do not have to depend on any publisher being interested as with traditional textbooks.
1. Holomorphic functions in several variables
2. Convexity and pseudoconvexity
3. CR Geometry
4. The \(\bar{\partial}\)-problem
5. Integral kernels
6. Complex analytic varieties
There are 370 exercises in the book of various difficulty and 48 figures.
Download the book as PDF
(Version 4.1, June 3rd, 2024, 248 pages)
As the exercise numbers have changed and you may want to consult the old version 3.4 from 2020.
See the errata in the current version.
Look at the change log to see what changed in the newest version.
The source is hosted on GitHub: https://github.com/jirilebl/scv
You can get an archive of the source of the released version on github, look under https://github.com/jirilebl/scv/releases, though if you plan to work with it, maybe best to look at just the latest working version as that might have any errata or new additions. Though these might be a work in progress. Perhaps best is to let me know.
The main file is ca.tex. I compile the pdf with pdflatex. You also want to run makeindex to generate the index and makeglossaries to generate the glossary of used notation (and then rerun pdflatex).
The github 'master' version is the current working version, so it will have whatever new changes I make in my tree.
This work is dual licensed under a Creative Commons Attribution-Noncommercial-Share Alike 4.0 License and Creative Commons Attribution-Share Alike 4.0 License. You can use, print, copy, and share this book as much as you want. You can base your own book/notes on these and reuse parts if you keep the license the same (that is, as long as you use at least one of the two licenses).