Winter 2016/17: Hodge Theory and Complex Geometry
You will find typed lecture notes, including due exercises, if you follow this link to my Dropbox. Last updated on: February 13, 2017 (final state).
Target audience and prerequisites
This course is intended as a supplement to the lecture series on complex analysis taught by Prof. Greb between the summer term of 2015 and the summer term of 2016 (see Complex Analysis I, Complex Analysis II – Complex Manifolds, and Complex Analysis III – Complex Geometry). So, in case you have followed Prof. Greb’s lectures, you will easily be able to follow this lecture.
In case you have not attended Prof. Greb’s lectures, don’t worry, there is a good chance that you will be able to follow my class nonetheless! In fact, in my lectures I will build upon only a share of what Prof. Greb has discussed in his Complex Analysis I–III. To provide you with some concrete information, here is a (by no means comprehensive) list of things that might come in handy to know:
- the basic theory of holomorphic functions in several variables [1, Chapter I]
- the concept of what a complex manifold is (definition plus basic properties) [1, Section IV.1]
- complex Lie groups and their actions
- holomorphic fiber bundles (in particular, holomorphic vector bundles) [see 1, Section IV.2]
- constructions with complex manifolds: products, quotients
- examples of complex manifolds (in particular, examples of compact ones): projective spaces, complex tori, Hopf manifolds, smooth projective hypersurfaces, etc. [see 1, Section IV.5]
It goes without saying that a thorough understanding of real analysis, including elements of topology, as well as a thorough understanding of linear algebra are indispensable for this course.
This class consists of a two-hour lecture to be held on a weekly basis. The lecture will be complemented by a two-hour tutorial which is to take place every second week. This amounts to a factual 4,5 credit points. It will, however, be possible to attain 6 credit points given that you submit a small term paper.
The language in the class room will be either English or German, depending on your preferences.
Dates and time
The lecture takes place on Thursday at 8:30 am in room WSC-S-U-3.03 starting with October 27, 2016.
The tutorial takes place every second Friday at 10:15 am in room WSC-S-3.14 starting with November 4, 2016.
The entire lecture will be centered around the concept of complex differential forms. Complex differential forms serve as a new means for us to investigate (the geometry of) complex manifolds. In the previous semester we have already seen the space of holomorphic sections in a complex manifold’s canonical bundle appear, and complex differential forms are related to that. Here is a rough outline of what we shall do.
In the first place, we need to define what a complex differential form on a complex manifold is. In doing so we will touch upon almost complex structures and almost complex manifolds. It is important that complex differential forms can be differentiated using the so-called exterior derivative. The latter gives rise to certain cohomology groups, or rather cohomology vector spaces. In particular, we shall introduce (complex) de Rham cohomology, Dolbeault cohomology, and Bott-Chern cohomology.
The remainder of the lecture will be devoted to studying the interplay between the mentioned cohomology vector spaces. It will turn out that on some complex manifolds the ties between the mentioned cohomologies are closer than on others. When, for instance, a compact complex manfold is of Kähler type, these ties are particularly close. This fact manifests itself in the Hodge decomposition theorem. Time permitting, we shall discuss how to prove the latter theorem. Finally, we might take a peek at recent developments and open problems in Kähler geometry.
This lecture will be accompanied by an online course within the university’s Moodle. If you want to enroll, the passphrase is the last name of a French mathematician whose cohomology we intend to study!
Send me an email in case you have problems logging on.😅
- Klaus Fritzsche and Hans Grauert. From Holomorphic Functions to Complex Manifolds. Vol. 213. Graduate Texts in Mathematics. Springer-Verlag, 2002.
Literature dealing with the actual contents of the lecture will be given during or at the very end of the semester.