SUBJECT

Complex manifolds

lecture + practical

master

Faculty of Science

Mathematics MSc

4+3

Semesters 1-4

Autumn/Spring semester

Complex and almost complex manifolds, holomorphic fiber bundles and vector bundles, Lie groups and transformation groups, cohomology, Serre duality, quotient and submanifolds, blowup, Hopf-, Grassmann and projective algebraic manifolds, Weierstrass' preparation and division theorem, analytic sets, Remmert-Stein theorem, meromorphic functions, Siegel, Levi and Chow's theorem, rational functions.

Objectives of the course: the intent of the course is to familiarize the students with the most important methods and objects of the theory of complex manifolds and to do this as simply as possible. The course completely  avoids those abstract concepts (sheaves, coherence, sheaf cohomology) that are subjects of Ph.D. courses. Using only elementary methods (power series, vector bundles, one dimensional cocycle) and presenting many examples, the course introduces the students to the theory of complex manifolds and prepares them for possible future Ph.D. studies.

• Klaus Fritzsche, Hans Grauert: From holomorphic functions to complex manifolds, Springer  Verlag, 2002

Further reading:

• K. Kodaira: Complex manifolds and deformations of complex structures, Springer  Verlag, 2004
• Huybrechts: Complex geometry: An introduction, Springer  Verlag, 2004