240156 Übungen zu Spezialisierung Geometrie und Topologie I (Ü) (SoSe 2019)

"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 is the first part in a sequence of 3 Masters courses in algebraic geometry. The next two courses will be taught by Prof. Dr. Vial in the WS19/20 and the SoSe20, and topics will be decided upon according to the students’ tastes.

The course will cover complex and Kähler manifolds as well as their vector bundles (holomorphic for complex and hermitian for Kähler manifolds). A complex manifold is a differential manifold with holomorphic transition functions (or equivalently 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 and 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). We will briefly discuss their associated complex tori, Abel's theorem and Jacobi inversion.

Bibliography

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”."