Complex algebraic geometry & Hodge theory

Course description

This is a course on complex geometry and Hodge theory. We will try to cover the following:

• Lefschetz theorems on the topology of Kaehler manifolds;

• Kodaira vaishing theorem;

• Kodaira embedding theorem;

• Variation of Hodge structure and Noether-Lefschetz theorem.

Some course materials are available during the semester at PKU disk. The password is Hodge?0?4, ?=the smallest prime.

You are more than welcome to send suggestions/comments about this course to the instructor.

Instructor: Zhiyu TIAN

Course Announcement

Projects are available.

Office Hours

By appointment via email. In your email, specify a few times slots convenient for you.

Office: BICMR 77103

Prerequisite

Working knowledge of complex analysis (single variable), topology (fundamental group & cohomology), sheaf, cohomology of sheaves, basics of smooth manifold.

Textbook and reference

Griffiths-Harris.

Typos and errors in GH (by Aleksey Zinger).

Voisin's two volumne books on Hodge theory Vol 1, Vol 2.

Jean-Pierre Demailly (Institut Fourier) has a free textbook where he carefully explains everything (a nice compliment to GH).

Homework problems

You are strongly encouraged to take a look at the homework problems at this link and the exercises in Voisin's books and to try to solve them.

Evaluation and exams

You may choose between:

• writing a report for some readings assigned by the instructor and possibly an oral presentation. If you choose this, you need to contact the instructor by the end of week 10 and discuss with him to find out a suitable topic.

• a take-home exam.

• 15 homework problems (75), a take-home exam (25).