Chern–Weil homomorphism

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In mathematics, the Chern–Weil homomorphism is a basic construction in the 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 ,

The Chern–Weil homomorphism is a homomorphism of -algebras

where on the right cohomology is de Rham cohomology. Such a homomorphism exists and is uniquely defined for every principal G-bundle P on M. If G is compact, then under the homomorphism, 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.) For non-compact groups like SL(n,R), there may be cohomology classes that are not represented by invariant polynomials.

Definition of the homomorphism

Choose any connection form ω in P, and let Ω be the associated curvature 2-form; i.e., Ω = Dω, 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) 2k-form on P given by

where vi are tangent vectors to P, is the sign of the permutation in the symmetric group on 2k 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:

[3]

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 fk 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 cj, 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 Ei's and Ωi the curvature form of Ei so that, in the matrix term, Ω is the block diagonal matrix with ΩI'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 λj 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

  1. Template:Harvnb
  2. The argument for the independent of a choice of connection here is taken from: Akhil Mathew, Notes on Kodaira vanishing [1]. Kobayashi-Nomizu, the main reference, gives a more concrete argument.
  3. 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.
  4. Proof: By definition, . Now compute the square of using Leibniz's rule.

References

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  • 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.
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