# Course Descriptions

## 16:640:523 - Functions of Several Complex Variables I

### Subtitle:

Function Theory of Several Complex Variables

### Course Description:

A function with $$n$$ complex variables $$z\in {\bf C}^n$$ is said to be holomorphic if it can be locally expanded as power series in $$z$$. An even dimensional smooth manifold is called a complex manifold if the transition functions can be chosen as holomorphic functions.
Roughly speaking, a Cauchy-Riemann manifold (or simply, a CR manifold) is a manifold that can be realized as the boundary of a
certain complex manifold. Several Complex Variables is the subject to study the properties and structures of holomorphic functions,
complex manifolds and CR manifolds.

Different from one complex variable, if $$n>1$$ one can never find a holomorphic function over the punctured ball that blows up at its
center. This is the striking phenomenon that Hartogs discovered about 100 years ago, which opened up the firstpage of the subject. Then
Poincar\'e, E. Cartan, Oka, etc, further explored this field and laid down its foundation. Nowadays as the subject is
intensively interacting with other fields, providing important examples, methods and problems, the basic materials in Several
Complex Variables have become mandatory for many investigations in pure mathematics. This class tries to serve such a purpose, by
presenting the following topics from Several Complex Variables.

(a). Holomorphic functions of several complex variables, Hartogs phenomenon, Poincare's in-equivalence of bidisks and balls.

(b). Domains of holomorphy, pseudo-convex domains, hulls of holomorphy.

(c). d-bar equations with compact support, Hormander's L^2-estimate.

(d) Cauchy-Riemann geometry, Webster's pseudo-Hermitian Geometry and subelliptic analysis on CR manifolds

(e) Complex manifolds, holomorphic vector bundles, Sheaf-Cohomology theory, d-bar equations on complex manifolds, Kodaira vanishing theorem and Kodaira embedding theorem, Kodaira-Spencer's deformation theory (if time permits)

### Text:

L. Hormander, {\it An introduction to complex analysis in several variables}, Third edition, North-Holland, 1990.

The course materials will be largely taken from the following:

[1] L. Hormander, {\it An introduction to complex analysis in several variables}, Third edition, North-Holland, 1990.

[2] James Morrow and K. Kodaira, {\it Complex Manifolds}, Rinehart and Winston, 1971.

[3] Xiaojun Huang, {\it Subelliptic analysis in Cauchy-Riemann Geometry and Complex Geometry}, Lecture Notes on the
national summer graduate school of China, 2007.

### Prerequisites:

One complex variable and the basic Hilbert space theory from real analysis

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### Subtitle:

Functions of Several Complex Variables: Selected Topics

### Course Description:

The main themes of this topic-course are $$L^2$$-estimates of the $$\bar{\partial}$$-operator, spectral theory of the complex Laplacians, and their applications to problems in algebraic geometry. Topics include: H\"{o}rmander's $$L^2$$-estimates of the $$\bar{\partial}$$-operator; spectral discreteness of the $$\bar\partial$$-Neumann Laplacian; Holomorphic Morse inequality; vanishing theorems; and relevant recent developments.

### Text:

Complex analytic and algebraic geometry, by J.-P. Demailly, available online at Demailly's website.

### Prerequisites:

Complex Variables I

### Previous Semesters

• Fall 2021 Prof. X.J. Huang
• Spring 2018 S.Q. Fu