# Complex manifolds

Time and place:

Dienstag 10-12 Uhr WSC-S-U-3.01

Donnerstag 8-10 Uhr WSC-S-U-3.0

Exercise class: Thursday  10-12 WSC-O-3.46

In cae of important collisians with other classes, please let me know as soon as possible.

There is a Moodle page for this class  where you can find brief summaries of the lectures, homework assignments, a forum to discuss and ask questions. (Subcribe with: Funktionentheorie2 )

Aimed at:  Students in Bachelor/Master Mathematics starting from 5. Semster. Basic knowledge on complex functions will be assumed.

Content:

In complex analysis you will have seen some surprising propererties of holomorphic functions in one variable. For example you learned that integrals over closed paths often only take values in a discrete set e.g. multiples of $$2\pi i$$ and this can be used to determine whether a region has holes.

In this course we will move on to complex functions in several variables and see that again these can be used to study geometry. We will learn about complex manifolds (spaces that locally look like open subsets of  $$\mathbb{C}^n$$ and will introduce cohomology, which turned out to be an indispensable tool in geometry.

Literature:

There are many beautiful book son the subject. Some examples are:

D. Huybrechts: Complex Geometry, Springer Verlag

H. Grauert, K. Fritzsche: From holomorphic functions to complex manifolds, Springer Verlag.

I will try to follow the first of these books, but add a bit more background than contained in the appendices.