
TEACHING: ACADEMIC YEAR 2022–2023
MA 329: TOPICS IN SEVERAL COMPLEX VARIABLES

Lecture hours
Mondays, Wednesdays, and Fridays 2:00–3:00 p.m.

Classroom
Lecture Hall 4, Department of Mathematics

About this course
This topics course is being run as an experiment in approaching the properties of holomorphic maps in several
complex variables (SCV) in a selfcontained manner: i.e., without requiring any prior exposure to SCV.
The course will begin with a complete and rigorous introduction to holomorphic functions in several variables and their
basic properties. This will pave the way to motivating and studying certain objects that are perhaps entirely indigenous
to SCV, such as invariant metrics and plurisubharmonic functions. This will allow us to establish the inequivalence of
the (Euclidean) ball and the polydisc in higher dimensions, to discuss appropriate analogues of the onevariable Riemann
Mapping Theorem in higher dimensions, and results of a similar nature.
Next, we shall study the properties of the Kobayashi metric (which is one of the invariant metrics mentioned above) and
the Kobayashi distance. This will be used to study the behaviour of automorphisms of bounded domains and refinements of
some of the results hinted at above—to the extent that time permits.
Prerequisites:
 A first course in complex analysis at the level of MA 224: i.e., our first course
in complex analysis.
 Students who are unsure of the contents of MA 224 (e.g., students who completed
their M.Sc. elsewhere) and are interested in this course are encouraged to speak/write to the instructor.
Grading policy:
Click here for the grading policy for this course.

Recommended books
L. Hormander, Complex Analysis in Several Variables, 3rd edition,
NorthHolland Publishing Co., Amsterdam, 1990.
M. Jarnicki and P. Pflug, Invariant Distances and Metrics in Complex Analysis,
de Gruyter Expositions in Mathematics, no. 9, Walter de Gruyter, Berlin, 1993.

Announcements
Nov 12: The endofsemester exam is scheduled for
November 29 at 2:00 p.m. Location: TBA.
Nov 11: The second makeup lecture will be
held at the usual location: i.e., Lecture Hall 4.
Oct 27: The second of the two makeup lectures will be
on November 12 at 2:00 p.m. Location: TBA.
Oct 1: The first of the two makeup lectures will be
held on Saturday, October 22, at 3:00 p.m. Venue: TBA.
Aug 25: As announced earlier in class, I shall be away from August 27 attending a conference. There
will thus be no lectures on August 26 and September 2 and 5 (August 31 is a holiday at the
Institute).
Aug 2: There will be no lecture on August 3.
July 30: The first lecture of the course is on August 1.

Notes
This is the space for occasional notes—usually references to optional reading—that expand upon some point that I
did not go into in detail in my lectures.
 A domain on which the Lempert function does not satisfy the triangle inequality: Example 2.3 of
Chapter III in:
Several Complex Variables III (G.M. Khenkin, ed.), Encyclopedia of Mathematical Sciences
9, SpringerVerlag, 1989.
 The Lempert function equals the Kobayashi pseudodistance on a convex domain: Theorem 4.8.1 in
Hyperbolic Complex Spaces (Shoshichi Kobayashi), Grundlehren der mathematischen Wissenschaften 318,
SpringerVerlag, 1998 [this is a presentation in English of the original result in the article:
Laszlo Lempert, La métrique de Kobayashi et la répresentation des domaines sur la boule,
Bull. Soc. Math. France 109 (1981), 427–474.]

Homework assignments
Homework 3
Homework 2
Homework 1
TEACHING: LAST 5 YEARS

ANALYSIS–II: MEASURE AND INTEGRATION (MA222)
[Spring 2017,
Spring 2020 ]

UNDERGRADUATE ANALYSIS & LINEAR ALGEBRA (UM101)
[Autumn 2017,
"Autumn" 2020 ]

ANALYSIS–I (MA221) [Autumn 2018]

INTRODUCTION TO BASIC ANALYSIS (UM204) [Spring 2019,
Spring 2022 ]

INTRODUCTION TO SEVERAL COMPLEX VARIABLES (MA328) [
Autumn 2019 ]

INTRODUCTION TO COMPLEX DYNAMICS (MA380) [
Autumn 2016,
Spring 2021 ]
