Complex Geometry
Course Description:
The course can be viewed as an introduction to the complex side of algebraic geometry. The main reference is Daniel Huybrechts' book "Complex geometry an introduction" Chapter 1-4.
The course will cover complex and Kähler manifolds as well as holomorphic and hermitian vector bundles. A complex manifold is a differential manifold with holomorphic transition functions (or equivalently has an integrable almost complex structure). Due to the different behaviors of complex and real analysis, the complex structures impose rigid geometry on complex manifolds. A Kähler manifold is a complex manifold with a Kähler metric. Hodge theory, most importantly the Hodge decomposition, on Kähler manifolds will be discussed.
The foundation of the course will lead up to vast applications in complex algebraic geometry (for example, Chapter 5 of the book). It also leads up to answer the fundamental question: When is a compact complex manifold projective algebraic (i.e, the zero locus of polynomials on a complex projective space, which is the main object of algebraic geometry)? Without discussing the applications in full generality, we will focus on compact Riemann surfaces (compact complex manifolds of dimension 1, which are always projective algebraic) and complex tori.
Supplementary material concerning the course can be found in the following two books: Griffiths and Harris “Principles of algebraic geometry” and Voisin “Complex algebraic geometry I”.
Time and Location for the lecture (eKVV 240155):
Tuesdays 10-12 at U2-205
Fridays 10-12 at U2-147
Time and Location for the tutorial (eKVV 240156)
Thursdays 14-16 at U2-147 (01.04-25.04)
Mondays 16-18 at T2-228 (29.04-12.07)
Studienleistung und Prüfungsleistung:
Studienleistung is obtained by completing at least 50% of homework problems.
Final written exam is on Friday, 12 July 10-12 s.t. at U2-147.
An oral exam can be requested and a date at the end of September will be arranged. An email should be sent to fxie@math.uni-bielefeld.de for the request and arrangement.