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.

Covariant derivatives on vector bundles

A smooth function with values in ℝk on a manifold M can be viewed as a section of the trivial vector bundle M × ℝk. The theory of connections is an attempt to generalize the notion of directional derivative of (real or vector-valued) functions to sections in vector bundles.

Let π : E → M be a vector bundle. We are interested in operators which assign to each smooth vector field X on M and smooth section σ of E another smooth section of E called the covariant derivative of σ with respect to X. Of course, we would like these operators to be ℝ-linear, tensorial in the first variable and to satisfy the Leibniz rule. Summarizing, we have:

Definition 5.1. A covariant derivative on E is an ℝ-linear operator ∇ : C∞(M) × Γ(E) → Γ(E) denoted by (X, σ) ↦ ∇Xσ such that for all f ∈ C∞(M), X ∈ χ(M), σ ∈ Γ(E) the following conditions are satisfied:

1. (i) (Tensoriality) ∇f Xσ = f∇Xσ.

2. (ii) (Leibniz rule) ∇X(fσ) = f∇Xσ + (∂Xf)σ.

The first condition simply says that given a section σ, the value of ∇Xσ at some p ∈ M depends on only the value of X at p (Proposition 2.3).