Chern-Weil theory

The comprehensive theory of Chern classes can be found in [11], Ch. 12. We will outline here the definition and properties of the first Chern class, which is the only one needed in the sequel. The following proposition can be taken as a definition:

Proposition 16.1. To every complex vector bundle E over a smooth manifold M one can associate a cohomology class c1(E) ∈ H2(M, ℤ) called the first Chern class of E satisfying the following axioms:

• (Naturality) For every smooth map f : M → N and complex vector bundle E over N, one has f*(c1(E)) = (c1(f*E), where the left term denotes the pull-back in cohomology and f*E is the pull-back bundle defined by f*Ex = Ef(x), ∀ x ∈ M.

• (Whitney sum formula) For every bundles E, F over M one has c1(E×F) = c1(E)+,c1(F), where (E×F) is the Whitney sum defined as the pull-back of the bundle (E×F) → (M×M) by the diagonal inclusion of M in (M×M)

• (Normalization) The first Chern class of the tautological bundle of ℂP1is equal to -1 in H2(ℂP1, ℤ) ≃ ℤ, which means that the integral over ℂP1of any representative of this class equals -1.

Let E → M be a complex vector bundle. We will now explain the Chern- Weil theory, which allows one to express the images in real cohomology of the Chern classes of E using the curvature of an arbitrary connection ∇ on E.