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/classnotes 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 (SCV). Prerequisites are one variable complex analysis, linear algebra, real analysis and basic functional analysis. These notes are not meant as a complete introduction to SCV. In fact, they focus on one particular aspect of CR geometry. The focus is on geometry using Hermitian forms rather than analysis.

Some of the material has been used, 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 from a more traditional standpoint. But the focus on Hermitian forms is unique to this present set of notes, so the two should be complementary.

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
(July 23rd, 2020, 62 pages)

Look at the change log to see what changed in the newest version.


The source is hosted on GitHub: https://github.com/jirilebl/scv-mini.

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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