MAT 546: Several Complex Variables (Fall 2026)

About the course

The topic of the course is holomorphic functions is several variables and their properties. Two important topics are the solution of the Levi problem, which describes domains of holomorphy in terms of pseudoconvexity; and $L^2$-estimates for the inhomogeneous Cauchy-Riemann equation. Time permitting, we may also talk about complex manifolds, holomorphic vector bundles, and hermitian metrics; and about some of Berndtsson's work on Bergman kernels and positivity of vector bundles.

Please see the syllabus for additional information about the course, including university-wide policies. Your grade will be determined based on homework and class participation.

Time and location

We meet on Tuesday and Thursday, 11:00–12:20 pm, in Frey Hall 328.

My office hours are Friday, 9:00am–12:00pm, in Mathematics 4–110.

Lecture notes

We are mostly going to follow the book Function Theory of Several Complex Variables by Steven G. Krantz. Other useful sources are An Introduction to Things $\bar{\partial}$ by Bo Berndtsson; Several Complex Variables by Zbigniew Błocki; and Lecture Notes on Several Complex Variables by Harold P. Boas. For some topics, we are going to use my lecture notes on complex manifolds.

Schedule

I will provide references for each lecture here. (Kra = Krantz, Sch = Schnell)

Week	Date	Topics	Reference
1	Aug 25	Holomorphic functions	Sch, Class 1

Homework assignments

During most weeks, I will be collecting written homework. For each assignment, please write up your solutions nicely and hand them in by the due date. Please staple your papers together and put your name on the first page.

Number	Due Date	Assignment