In mathematics, the Chern–Weil homomorphism is a basic construction in Chern–Weil theory that computes topological invariants of vector bundles and principal bundles on a smooth manifold M in terms of connections and curvature representing classes in the de Rham cohomology rings of M. That is, the theory forms a bridge between the areas of algebraic topology and differential geometry.
It was developed in the late 1940s by Shiing-Shen Chern and André Weil, in the wake of proofs of the generalized Gauss–Bonnet theorem.
under the adjoint action of G; that is, the subalgebra consisting of all polynomials f such that
This homomorphism is obtained by taking invariant polynomials in the curvature of any connection on the given bundle.
If G is either compact or semi-simple, then the cohomology ring of the classifying space for G-bundles,
of invariant polynomials: (The cohomology ring of BG can still be given in the de Rham sense: when
, the exterior covariant derivative of ω.
is a homogeneous polynomial function of degree k; i.e.,
, then, viewing f as a symmetric multilinear functional on
(see the ring of polynomial functions), let be the (scalar-valued) 2k-form on P given by where vi are tangent vectors to P,
(see Lie algebra-valued forms#Operations as well as Pfaffian).
is a closed form follows from the next two lemmas:[1] Indeed, Bianchi's second identity says
Next, we show that the de Rham cohomology class of
belong to the same de Rham cohomology class by the homotopy invariance of de Rham cohomology.
Finally, by naturality and by uniqueness of descending, and the same for
The construction thus gives the linear map: (cf.
Lemma 1) In fact, one can check that the map thus obtained: is an algebra homomorphism.
, we can consider its characteristic polynomial in t:[3] where i is the square root of -1.
The k-th Chern class of a smooth complex-vector bundle E of rank n on a manifold M: is given as the image of
under the Chern–Weil homomorphism defined by E (or more precisely the frame bundle of E).
The total Chern class of E is the image of this polynomial; that is, Directly from the definition, one can show that
and c given above satisfy the axioms of Chern classes.
For example, for the Whitney sum formula, we consider where we wrote
Now, suppose E is a direct sum of vector bundles
, we have: where on the right the multiplication is that of a cohomology ring: cup product.
For the normalization property, one computes the first Chern class of the complex projective line; see Chern class#Example: the complex tangent bundle of the Riemann sphere.
If E is a smooth real vector bundle on a manifold M, then the k-th Pontrjagin class of E is given as: where we wrote
for the complexification of E. Equivalently, it is the image under the Chern–Weil homomorphism of the invariant polynomial
given by: Let E be a holomorphic (complex-)vector bundle on a complex manifold M. The curvature form
of E, with respect to some hermitian metric, is not just a 2-form, but is in fact a (1, 1)-form (see holomorphic vector bundle#Hermitian metrics on a holomorphic vector bundle).