Chern–Weil homomorphism

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 f ∈ C [ g ] G is a homogeneous polynomial function of degree k; i.e., f ( a x ) = a k f ( x ) for any complex number a and x in g , then, viewing f as a symmetric multilinear functional on ∏ 1 k g (see the ring of polynomial functions), let

f ( Ω )

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

f ( Ω ) ( v 1 , … , v 2 k ) = 1 ( 2 k ) ! ∑ σ ∈ S 2 k ϵ σ f ( Ω ( v σ ( 1 ) , v σ ( 2 ) ) , … , Ω ( v σ ( 2 k − 1 ) , v σ ( 2 k ) ) )

where v_i are tangent vectors to P, ϵ σ is the sign of the permutation σ in the symmetric group on 2k numbers S 2 k (see Lie algebra-valued forms#Operations as well as Pfaffian).

If, moreover, f is invariant; i.e., f ( Ad g ⁡ x ) = f ( x ) , then one can show that f ( Ω ) 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 f ( Ω ) is a closed form follows from the next two lemmas:

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

Indeed, Bianchi's second identity says D Ω = 0 and, since D is a graded derivation, D f ( Ω ) = 0. Finally, Lemma 1 says f ( Ω ) satisfies the hypothesis of Lemma 2.

To see Lemma 2, let π : P → M be the projection and h be the projection of T u P onto the horizontal subspace. Then Lemma 2 is a consequence of the fact that d π ( h v ) = d π ( v ) (the kernel of d π is precisely the vertical subspace.) As for Lemma 1, first note

f ( Ω ) ( d R g ( v 1 ) , … , d R g ( v 2 k ) ) = f ( Ω ) ( v 1 , … , v 2 k ) , R g ( u ) = u g ;

which is because R g ∗ Ω = Ad g − 1 ⁡ Ω and f is invariant. Thus, one can define f ¯ ( Ω ) by the formula:

f ¯ ( Ω ) ( v 1 ¯ , … , v 2 k ¯ ) = f ( Ω ) ( v 1 , … , v 2 k )

where v i are any lifts of v i ¯ : d π ( v i ) = v ¯ i .

Next, we show that the de Rham cohomology class of f ¯ ( Ω ) on M is independent of a choice of connection. Let ω 0 , ω 1 be arbitrary connection forms on P and let p : P × R → P be the projection. Put

ω ′ = t p ∗ ω 1 + ( 1 − t ) p ∗ ω 0

where t is a smooth function on P × R given by ( x , s ) ↦ s . Let Ω ′ , Ω 0 , Ω 1 be the curvature forms of ω ′ , ω 0 , ω 1 . Let i s : M → M × R , x ↦ ( x , s ) be the inclusions. Then i 0 is homotopic to i 1 . Thus, i 0 ∗ f ¯ ( Ω ′ ) and i 1 ∗ f ¯ ( Ω ′ ) belong to the same de Rham cohomology class by the homotopy invariance of de Rham cohomology. Finally, by naturality and by uniqueness of descending,

i 0 ∗ f ¯ ( Ω ′ ) = f ¯ ( Ω 0 )

and the same for Ω 1 . Hence, f ¯ ( Ω 0 ) , f ¯ ( Ω 1 ) belong to the same cohomology class.

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

C [ g ] k G → H 2 k ( M ; C ) , f ↦ [ f ¯ ( Ω ) ] .

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

C [ g ] G → H ∗ ( M ; C )

is an algebra homomorphism.

Example: Chern classes and Chern character

Let G = G L n ( C ) and g = g l n ( C ) its Lie algebra. For each x in g , we can consider its characteristic polynomial in t:

det ( I − t x 2 π i ) = ∑ k = 0 n f k ( x ) t k ,

where i is the square root of -1. Then f k are invariant polynomials on g , 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:

c k ( E ) ∈ H 2 k ( M , Z )

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 det ( I − x 2 π i ) = 1 + f 1 ( x ) + ⋯ + f n ( x ) is an invariant polynomial. The total Chern class of E is the image of this polynomial; that is,

c ( E ) = 1 + c 1 ( E ) + ⋯ + c n ( E ) .

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

c t ( E ) = [ det ( I − t Ω / 2 π i ) ]

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 det ( I − t Ω / 2 π i ) = det ( I − t Ω 1 / 2 π i ) ∧ ⋯ ∧ det ( I − t Ω m / 2 π i ) , we have:

c t ( E ) = c t ( E 1 ) ⋯ c t ( E m )

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 Ω E ⊗ E ′ = Ω E ⊗ I E ′ + I E ⊗ Ω E ′ , we also have:

c 1 ( E ⊗ E ′ ) = c 1 ( E ) rk ⁡ ( E ′ ) + rk ⁡ ( E ) c 1 ( E ′ ) .

Finally, the Chern character of E is given by

ch ⁡ ( E ) = [ tr ⁡ ( e − Ω / 2 π i ) ] ∈ H ∗ ( M , Q )

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:

ch ⁡ ( E ⊕ F ) = ch ⁡ ( E ) + ch ⁡ ( F ) , ch ⁡ ( E ⊗ F ) = ch ⁡ ( E ) ch ⁡ ( F ) .

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

c t ( E ) = ∏ j = 0 n ( 1 + λ j t )

where λ_j are in R (they are sometimes called Chern roots.) Then ch ⁡ ( E ) = e λ j .

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:

p k ( E ) = ( − 1 ) k c 2 k ( E ⊗ C ) ∈ H 4 k ( M ; Z )

where we wrote E ⊗ C for the complexification of E. Equivalently, it is the image under the Chern–Weil homomorphism of the invariant polynomial g 2 k on g l n ( R ) given by:

det ⁡ ( I − t x 2 π ) = ∑ k = 0 n g k ( x ) t k .

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 G = G L n ( C ) ,

C [ g ] k → H k , k ( M ; C ) , f ↦ [ f ( Ω ) ] .