So, I’m in a tutorial this summer, planning to write my final paper on the Kodaira embedding theorem, and I’ve been finding my total ignorance of complex algebraic geometry to be something of a problem. One of my goals next year is, coincidentally, to acquire a solid understanding of most of the topics in Griffiths-Harris. To start with, I’d like to spend a few posts on Chern-Weil theory. This gives an analytic method of computing the Chern classes of a complex vector bundle, and more generally a framework for the characteristic classes of a principal bundle over a Lie group. In fact, it tells you what the cohomology of the classifying space of a Lie group is (it’s a certain algebra of invariant polynomials on the Lie algebra), from which — by Yoneda’s lemma — you can associate cohomology classes to a principal bundle on any space.

Today, I’d like to review what Chern classes are like.

**1. Introduction**

To start with, we will need to describe what the Chern classes really are. These are going to be natural maps

from the complex vector bundles on a space to the cohomology ring. In other words, to each vector bundle , we will have an element . In order for this to be *natural*, we are going to want that, for any map of topological spaces,

In other words, we are going to want the map to be *functorial* in , when both are considered as contravariant functors in . It turns out that each functor (of -dimensional complex vector bundles) and is *representable* on the appropriate homotopy category. (more…)