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.

Indeed, an {n}-dimensional complex vector bundle is the same as a principal {\mathrm{GL}_n(\mathbb{C})} bundle over {X}, and such are classified by homotopy classes of maps {X \rightarrow B \mathrm{GL}_n(\mathbb{C})}. One can explicitly write down what the space {B \mathrm{GL}_n(\mathbb{C})} is: it is the infinite Grassmannian {\mathrm{Gr}_n(\mathbb{C}^\infty)} of {n}-planes in {\mathbb{C}^\infty}. There is a canonical {n}-dimensional vector bundle on this Grassmannian, consisting of pairs

\displaystyle (x,v) \in \mathrm{Gr}_n(\mathbb{C}^\infty) \times \mathbb{C}^{\infty},

where {v} belongs to the plane corresponding to {x}. This bundle is universal. Although it does not matter for our purposes, the functors {H^k(X; \mathbb{Z})} are also representable, by the Eilenberg-Maclane spaces {K(\mathbb{Z}; k)}.

By Yoneda’s lemma, to give such a “characteristic class” is to give an element in the cohomology ring of each Grassmannian {\mathrm{Gr}_n(\mathbb{C}^\infty)}. We could explicitly do this, but for now, let’s not. Let us just state the axioms that we want Chern classes to satisfy:

  1. {c} of the trivial bundle is {1}.
  2. {c} of an {n}-dimensional bundle has terms in the cohomology ring only in even degrees {\leq 2n}.
  3. {c(E \oplus E') = c(E) c(E')}, for {E, E'} vector bundles.
  4. {c} of the tautological line bundle on {\mathbb{C}\mathbb{P}^1} is a fixed generator of {H^2(\mathbb{C}\mathbb{P}^1; \mathbb{Z})}.

These conditions are actually going to determine the Chern classes. We shall simply assume that they exist, and satisfy these axioms. They are constructed in Milnor and Stasheff’s book, for instance. (We will construct them in the smooth category.)

2. Chern classes of a line bundle

However, let’s step back and try now to construct the Chern class {c(L)} for a line bundle {L} explicitly. Note that {c(L) = 1 + c_2(L)} for {c_2} homogeneous of degree two: this is because of the axioms, and the fact that the total Chern class {c(E)} of a vector bundle {E} is always invertible (because any vector bundle has a complement {E'} such that {E \oplus E'} is trivial). Let {X} be a space, and {\mathcal{A}} the sheaf of complex-valued continuous functions on {X}. Then line bundles on {X} can be described as elements of {H^1(X; \mathcal{A}^*)}, because a line bundle can be constructed by “gluing,” and this is what Cech 1-cocycles measure. There is an exact sequence

\displaystyle 0 \rightarrow \mathbb{Z} \rightarrow \mathcal{A} \stackrel{f \mapsto e^{2\pi i f}}{\rightarrow} \mathcal{A}^* \rightarrow 0

of sheaves, which leads to a map

\displaystyle \mathrm{Pic}(X) = H^1(X; \mathcal{A}^*) \rightarrow H^2(X; \mathbb{Z}).

The claim is that we can describe the first Chern class of a complex line bundle in this way. Let us call the above class {c_2'(L)}; we need to show that {c_2(L) = c_2'(L)}.

Proof: Given a line bundle {L} over {X}, the above construction of an element of {H^2(X; \mathbb{Z})} via the coboundary map is clearly natural in {L}, because pulling back {L} corresponds to pulling back the 1-cocycle that defines it. As a result, we just need to show that {c_2'} of the tautological line bundle on {\mathbb{CP}^\infty} is a generator of the cohomology ring, because of the universality of this line bundle. In fact, because there is a map

\displaystyle \mathbb{CP}^1 \rightarrow \mathbb{CP}^\infty,

which induces an isomorphism on {H^2(\cdot; \mathbb{Z})}, it suffices to do this for the tautological line bundle on {\mathbb{CP}^1}. \Box

3. The splitting principle

We now discuss a technique that often enables questions about Chern classes of vector bundles to be reduced to the case of line bundles. In particular, it implies that if we have two sets of “Chern classes” that agree on line bundles, they agree totally. As a result, we will easily find that Chern classes are unique.

Theorem 1 Let {E \rightarrow X} be a vector bundle. Then there is a space {p: Y \rightarrow X} such that {p^*: H^*(X) \rightarrow H^*(Y)} is injective and {p^* E} splits as a sum of line bundles.

In general, there is no reason to expect a vector bundle to split as a sum of line bundles. Incidentally, on {\mathbb{CP}^1}, this is the case (even in the holomorphic or algebraic category) by a theorem of Grothendieck, which is a fun application of the sheaf cohomology of line bundles on {\mathbb{CP}^1}.

Proof: We shall find a map {p: Y \rightarrow X} such that {p^* E} splits as a sum of a line bundle and another bundle, and {p^*} is injective in cohomology. Repeating this construction, we shall get the claim.

To do this, we take {Y = \mathbf{P}(E)}, the projectivization of {E}. This is a fiber bundle over {E}, whose fiber over {x \in X} is the projective space {\mathbf{P}(E_x)} of lines through the origin in the {\mathbb{C}}-vector space {E_x}. There is a tautological line bundle on {Y}: namely, the subset of {Y \times_X E} consisting of pairs {(y, e)} (lying over, say, {x \in X}), such that {e \in E_x} belongs to the line corresponding to {y}. This is clearly a complex line bundle, and it is a subbundle of {p^*E} by construction.

So all we need to see is that the map in cohomology is injective. This follows from the Leray-Hirsch theorem. That is, the cohomology {H^*(Y)} is a free module over {H^*(X)}, generated by (ironically) the first Chern class (to be defined) of the tautological line bundle on {Y}. \Box

As a result, we may see quickly that the Chern classes, if they exist, are uniquely determined. That is, any two natural transformations c, c' from vector bundles to the cohomology ring satisfying the required axioms coincide.  Namely, because c = c' on the universal line bundle on \mathbb{CP}^\infty, it follows that c=c' on any line bundle (by naturality, since any line bundle is a pull-back of the universal one). It follows that c=c' on any vector bundle which is a sum of line bundles. By the splitting principle, it follows now that the same holds for all vector bundles.

Advertisements