# ADVANCED GEOMETRY

## Learning outcomes of the course unit

Knowledge and understanding:

the thoery of several complex variables.

Applying knowledge and understanding: basic.

## Prerequisites

Theory of one complex variable.

## Course contents summary

(i) Domains of holomorphy.

- Holomorphic functions of several complex variables.

- Removable singularities.

- Dolbeault complex.

- Notions of convexity.

(ii) Automorphisms of complex manifolds.

- Limits of automorphisms.

- Automorphisms of bounded domains.

- Automorphisms of the disk and polydisk.

(iii) Cauchy-Riemann manifolds (outline).

- Real submanifolds of complex manifolds.

- Levi form and pseudoconvexity.

- Embedding of CR manifolds.

- Traces of holomorphic functions and extension problems.

## Course contents

(i) Domains of olomorphy.

- Holomorphic functions of several complex variables.

- Removable singularities.

- Dolbeault complex.

- Notions of convexity.

(ii) Automorphisms of complex manifolds.

- Limits of automorphisms.

- Automorphisms of bounded domains.

- Automorphisms of the disk and polydisk.

(iii) Cauchy-Riemann manifolds (outline).

- Real submanifolds of complex manifolds.

- Levi form and pseudoconvexity.

- Embedding of CR manifolds.

- Traces of holomorphic functions and extension problems.

## Recommended readings

L. Hörmander:

An introduction to complex analysis in several variables.

North-Holland Publishing Co., Amsterdam, 1990. (ISBN: 0-444-88446-7)

R.C. Gunning:

Introduction to holomorphic functions of several variables. Vol. I. Function theory.

Wadsworth & Brooks/Cole Advanced Books & Software, Pacific Grove, CA, 1990. (ISBN: 0-534-13308-8)

R. Narasimhan:

Several complex variables.

University of Chicago Press, Chicago, IL, 1995. (ISBN: 0-226-56817-2)

R.M. Range:

Holomorphic functions and integral representations in several complex variables.

Springer-Verlag, New York, 1986. (ISBN: 0-387-96259-X)

G. Della Sala, A. Saracco, A. Simioniuc and G. Tomassini:

Lectures on complex analysis and analytic geometry.

Edizioni della Normale, Pisa, 2006. (ISBN: 88-7642-199-8)

A. Boggess:

CR manifolds and the tangential Cauchy-Riemann complex.

CRC Press, Boca Raton, FL, 1991. (ISBN: 0-8493-7152-X)

## Teaching methods

During the lectures will be offered arguments, with by examples and applications.

## Assessment methods and criteria

The examination consists of an interview aimed

to assess the level of knowledge and understanding acquired on the topics of teaching, and the student's ability to present them in such a way

mathematically correct and to communicate them to others. The final evaluation

is expressed by a single vote for the examination as a whole.