Hermitian Forms Meet Several Complex Variables

Minicourse on CR Geometry Using Hermitian Forms

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 free online textbook (e-book in webspeak) is a half-semester mathematics graduate course in CR geometry using Hermitian forms. These are my class notes for Math 595 for the spring of 2010 at University of Illinois at Urbana-Champaign on Several Complex Variables. Prerequisites are one variable complex analysis, linear algebra, real analysis and basic functional analysis. The notes are not meant as a polished finished product. I hope they are usable as a quick and dirty introduction to the subject. Do note that much is missing. For example, neither the Levi form, nor pseudoconvexity is mentioned. The focus is on geometry using Hermitian forms rather than analysis.

Some of the material has been used, and given further polish, in the self contained introductory book Tasty Bits of Several Complex Variables, which is meant as a full semester course the the basics of Several Complex Variables.

The notes are also not completely self contained. There are many proofs in the beginning sections about complex geometry that are left out due to lack of time in the course. Some material from the course did not make it into these notes and vice-versa.

There are most likely many typos and minor errors throughout (hopefully no major ones), so be wary. Do email me at with any errors that you find.

These notes may be modified and customized for a specific purpose if necessary. If you do modify these notes, make sure to mark them 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.

Table of contents:

1. Complex Analytic Varieties
2. CR Geometry
3. Proper Mappings of Balls


Download the book as PDF
(April 8th, 2014, 62 pages)

Look at the change log to see what changed in the newest version (You can download source files of old versions if you wish).


LaTeX source as a tarball. The main file is scv-mini.tex. I compile the pdf with pdflatex. You also want to run makeindex to generate the index (I generally run pdflatex scv-mini three times, then makeindex scv-mini, and then finally pdflatex scv-mini again).

During the writing of these notes, the author was in part supported by NSF grant DMS-0900885.


Creative Commons License

This work is licensed under a Creative Commons Attribution-Noncommercial-Share Alike 3.0 United States License. You can use, print, copy, and share these notes as much as you want. You can base your own notes on these and reuse parts if you keep the license the same. If you plan to use these notes commercially (sell them for more than just production cost), then you need to contact me and we will work something out. If you are printing a course pack for your students, then it is fine if the copy service or bookstore is charging a fee for printing and selling the printed copy. I consider that production cost.

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