The course is aimed at Master students in Mathematics and Physics with an interest in Geometry.
We will begin by recalling the definition of manifold and of its tangent space; the latter will be our first example of a fibration. In general, one can imagine a fibration over a manifold as a collection of objects (groups, vector spaces, topological spaces, etc.) varying smoothly (or holomorphically, in the complex case) with the points of the manifold. All the regular operations on the objects in question can be transferred to fibrations; we will speak about dual fibrations, sum of fibrations, homeomorphisms, and so on. We will conclude this first part of the course by discussing holomorphic line bundles, the Picard group, and tautological line bundles on the projective space.
An important concept is the one of section of a fibration, which evolves the definition of function. As we can consider collection of functions, we will also consider collections of sections; this gives rise to the theory of sheaves. We will review the standard De Rham cohomology under this new point of view, introducing sheaf cohomology, Cech cohomology, the abstract De Rham Theorem, Mayer-Vietoris, and applications.
If time permits, we will conclude with metrics, curvatures, and characteristic classes, including vanishing theorems.
- Kursleiter/in: Riccardo Ugolini