MATH-473 / 5 crédits

Enseignant:

Langue: Anglais

Remark: Cours donné en alternance tous les deux ans

## Summary

The goal of this course is to help students learn the basic theory of complex manifolds and Hodge theory.

## Required courses

The students are expected to have already taken a course in complex analysis in one variable and a course in differential and Riemannian geometry.
The course will be aim to be as much as possible of a self-contained treatment of the subject – concepts and basic theorems regarding complex manifolds and complex analysis in many variables will be stated and not assumed.
Some knowledge of algebraic topology (homology, cohomology, characteristic classes) would certainly be useful, as part of the course will be devoted to studying the cohomology of algebraic and related concepts.

## Recommended courses

Complex analysis; algebraic topology.

It may be useful if you have already followed a course in algebraic geometry.

## Learning Outcomes

By the end of the course, the student must be able to:

• Knowledge of the basic concepts of complex analysis in several variable
• Ability to work with complex structures and the related differential concepts.
• Knowledge of vector bundles and sheaves and ability to compute their cohomology in basic examples.
• Knowledge of the concept of Hodge structure and the ability to work with and manipulate that.

## Assessment methods

The final grade will be assigned based on the cumulative points of the student obtained from handed in homework solutions and from the written exam. The weights of the two parts are:

25% - homework

75% - written exam

There will be 4 homeworks that students will be required to hand in on dates to be determined at the start of the course.

Dans le cas de l'art. 3 al. 5 du Règlement de section, l'enseignant décide de la forme de l'examen qu'il communique aux étudiants concernés.

## Bibliography

D. Huybrechts, Complex Geometry - an introduction, Springer, 2004.

P. Griffiths and J. Harris, Principles of Algebraic Geometry. Wiley, 1978.

C. Voisin, Complex Geometry and Hodge Theory I. CUP, 2002.

## Dans les plans d'études

• Semestre: Printemps
• Forme de l'examen: Ecrit (session d'été)
• Matière examinée: Complex manifolds
• Cours: 2 Heure(s) hebdo x 14 semaines
• Exercices: 2 Heure(s) hebdo x 14 semaines
• Semestre: Printemps
• Forme de l'examen: Ecrit (session d'été)
• Matière examinée: Complex manifolds
• Cours: 2 Heure(s) hebdo x 14 semaines
• Exercices: 2 Heure(s) hebdo x 14 semaines
• Semestre: Printemps
• Forme de l'examen: Ecrit (session d'été)
• Matière examinée: Complex manifolds
• Cours: 2 Heure(s) hebdo x 14 semaines
• Exercices: 2 Heure(s) hebdo x 14 semaines

## Semaine de référence

 Lu Ma Me Je Ve 8-9 9-10 10-11 11-12 12-13 13-14 14-15 15-16 16-17 17-18 18-19 19-20 20-21 21-22