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

$\displaystyle \mathrm{Vect}_{\mathbb{C}}(X) \rightarrow H^*(X; \mathbb{Z}),$

from the complex vector bundles on a space ${X}$ to the cohomology ring. In other words, to each vector bundle ${E \rightarrow X}$, we will have an element ${c(E) \in H^*(X; \mathbb{Z})}$. In order for this to be natural, we are going to want that, for any map ${f: Y \rightarrow X}$ of topological spaces,

$\displaystyle c(f^*E) = f^* c(E) \in H^*(Y; \mathbb{Z}).$

In other words, we are going to want the map ${\mathrm{Vect}_{\mathbb{C}}(X) \rightarrow H^*(X; \mathbb{Z})}$ to be functorial in ${X}$, when both are considered as contravariant functors in ${X}$. It turns out that each functor ${\mathrm{Vect}_{n, \mathbb{C}}}$ (of ${n}$-dimensional complex vector bundles) and ${H^k(X; \mathbb{Z})}$ is representable on the appropriate homotopy category. (more…)