Chern classes I

Dmitri Panov.

We start by defining vector bundles with examples, such as the tautological bundle $O(-1)$. Further we define the notion of pull-back, formulate a first definition of characteristic classes, and illustrate it in the case of the first Stiefel-Whitney class and the Euler class. We specialise further to manifolds, give a first definition of the Chern classes and discuss the Whitney formula. Finally we discuss the Chern classes of the tautological bundle of the Grassmanian $G(2,4)$ which can be explicitly calculated. This will be done with an intention to explain why a smooth cubic surface in $\mathbb CP^3$ contains exactly $27$ lines.

There are some notes from previous years.