# Chern–Weil homomorphism

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. This theory was an important step in the theory of characteristic classes.

Let *G* be a real or complex Lie group with Lie algebra ; and let denote the algebra of -valued polynomials on (exactly the same argument works if we used instead of .) Let be the subalgebra of fixed points in under the adjoint action of *G*; that is, it consists of all polynomials *f* such that for any *g* in *G* and *x* in ,

Given principal G-bundle *P* on *M*, there is an associated
homomorphism of -algebras

called the **Chern–Weil homomorphism**,
where on the right cohomology is de Rham cohomology. 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 *BG* is isomorphic to the algebra of invariant polynomials:

(The cohomology ring of *BG* can still be given in the de Rham sense:

when and are manifolds.)

## Definition of the homomorphism

Choose any connection form ω in *P*, and let Ω be the associated curvature 2-form; i.e., Ω = *D*ω, the exterior covariant derivative of ω. If is a homogeneous polynomial function of degree *k*; i.e., for any complex number *a* and *x* in , then, viewing *f* as a symmetric multilinear functional on (see the ring of polynomial functions), let

be the (scalar-valued) 2*k*-form on *P* given by

where *v*_{i} are tangent vectors to *P*, is the sign of the permutation in the symmetric group on 2*k* numbers (see Lie algebra-valued forms#Operations as well as Pfaffian).

If, moreover, *f* is invariant; i.e., , then one can show that is a closed form, it descends to a unique form on *M* and that the de Rham cohomology class of the form is independent of *ω*. First, that is a closed form follows from the next two lemmas:[1]

- Lemma 1: The form on
*P*descends to a (unique) form on*M*; i.e., there is a form on*M*that pulls-back to . - Lemma 2: If a form φ on
*P*descends to a form on*M*, then dφ = Dφ.

Indeed, Bianchi's second identity says and, since *D* is a graded derivation, Finally, Lemma 1 says satisfies the hypothesis of Lemma 2.

To see Lemma 2, let be the projection and *h* be the projection of onto the horizontal subspace. Then Lemma 2 is a consequence of the fact that (the kernel of is precisely the vertical subspace.) As for Lemma 1, first note

which is because and *f* is invariant. Thus, one can define by the formula:

where are any lifts of : .

Next, we show that the de Rham cohomology class of on *M* is independent of a choice of connection.[2] Let be arbitrary connection forms on *P* and let be the projection. Put

where *t* is a smooth function on given by . Let be the curvature forms of . Let be the inclusions. Then is homotopic to . Thus, and 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 . Hence, belong to the same cohomology class.

The construction thus gives the linear map: (cf. Lemma 1)

In fact, one can check that the map thus obtained:

is an algebra homomorphism.

## Example: Chern classes and Chern character

Let and its Lie algebra. For each *x* in , we can consider its characteristic polynomial in *t*:

where *i* is the square root of -1. Then are invariant polynomials on , since the left-hand side of the equation is. 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 *f*_{k} under the Chern–Weil homomorphism defined by *E* (or more precisely the frame bundle of *E*). If *t* = 1, then is an invariant polynomial. The total Chern class of *E* is the image of this polynomial; that is,

Directly from the definition, one can show *c*_{j}, *c* given above satisfy the axioms of Chern classes. For example, for the Whitney sum formula, we consider

where we wrote Ω for the curvature 2-form on *M* of the vector bundle *E* (so it is the descendent of the curvature form on the frame bundle of *E*). The Chern–Weil homomorphism is the same if one uses this Ω. Now, suppose *E* is a direct sum of vector bundles *E*_{i}'s and Ω_{i} the curvature form of *E*_{i} so that, in the matrix term, Ω is the block diagonal matrix with Ω_{I}'s on the diagonal. Then, since , 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.

Since ,[4] we also have:

Finally, the Chern character of *E* is given by

where Ω is the curvature form of some connection on *E* (since Ω is nilpotent, it is a polynomial in Ω.) Then ch is a ring homomorphism:

Now suppose, in some ring *R* containing the cohomology ring *H*(*M*, **C**), there is the factorization of the polynomial in *t*:

where λ_{j} are in *R* (they are sometimes called Chern roots.) Then .

## Example: Pontrjagin classes

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 on given by:

## The homomorphism for holomorphic vector bundles

Let *E* be a holomorphic (complex-)vector bundle on a complex manifold *M*. The curvature form Ω 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). Hence, the Chern–Weil homomorphism assumes the form: with ,

## Notes

- Kobayashi-Nomizu 1969, Ch. XII.
- The argument for the independent of a choice of connection here is taken from: Akhil Mathew, Notes on Kodaira vanishing "Archived copy" (PDF). Archived from the original (PDF) on 2014-12-17. Retrieved 2014-12-11.CS1 maint: archived copy as title (link). Kobayashi-Nomizu, the main reference, gives a more concrete argument.
- Editorial note: This definition is consistent with the reference except we have
*t*, which is*t*^{−1}there. Our choice seems more standard and is consistent with our "Chern class" article. - Proof: By definition, . Now compute the square of using Leibniz's rule.

## References

- Bott, R. (1973), "On the Chern–Weil homomorphism and the continuous cohomology of Lie groups",
*Advances in Mathematics*,**11**(3): 289–303, doi:10.1016/0001-8708(73)90012-1. - Chern, S.-S. (1951),
*Topics in Differential Geometry*, Institute for Advanced Study, mimeographed lecture notes. - Shiing-Shen Chern,
*Complex Manifolds Without Potential Theory*(Springer-Verlag Press, 1995) ISBN 0-387-90422-0, ISBN 3-540-90422-0.- The appendix of this book: "Geometry of Characteristic Classes" is a very neat and profound introduction to the development of the ideas of characteristic classes.

- Chern, S.-S.; Simons, J (1974), "Characteristic forms and geometric invariants",
*Annals of Mathematics*, Second Series,**99**(1): 48–69, doi:10.2307/1971013, JSTOR 1971013. - Kobayashi, S.; Nomizu, K. (1963),
*Foundations of Differential Geometry, Vol. 2*(new ed.), Wiley-Interscience (published 2004). - Narasimhan, M.; Ramanan, S. (1961), "Existence of universal connections" (PDF),
*Amer. J. Math.*,**83**(3): 563–572, doi:10.2307/2372896, hdl:10338.dmlcz/700905, JSTOR 2372896. - Morita, Shigeyuki (2000), "Geometry of Differential Forms",
*Translations of Mathematical Monographs*,**201**.