Chern–Weil homomorphism

Choose any connection form ω in P, and let Ω be the associated curvature form; i.e., ${\displaystyle \Omega =D\omega }$, the exterior covariant derivative of ω. If ${\displaystyle f\in \mathbb {C} [{\mathfrak {g}}]^{G}}$ is a homogeneous polynomial function of degree k; i.e., ${\displaystyle f(ax)=a^{k}f(x)}$ for any complex number a and x in ${\displaystyle {\mathfrak {g}}}$, then, viewing f as a symmetric multilinear functional on ${\textstyle \prod _{1}^{k}{\mathfrak {g}}}$ (see the ring of polynomial functions), let

${\displaystyle f(\Omega )}$

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

${\displaystyle f(\Omega )(v_{1},\dots ,v_{2k})={\frac {1}{(2k)!}}\sum _{\sigma \in {\mathfrak {S}}_{2k}}\epsilon _{\sigma }f(\Omega (v_{\sigma (1)},v_{\sigma (2)}),\dots ,\Omega (v_{\sigma (2k-1)},v_{\sigma (2k)}))}$

where v_i are tangent vectors to P, ${\displaystyle \epsilon _{\sigma }}$ is the sign of the permutation ${\displaystyle \sigma }$ in the symmetric group on 2k numbers ${\displaystyle {\mathfrak {S}}_{2k}}$ (see Lie algebra-valued forms#Operations as well as Pfaffian).

If, moreover, f is invariant; i.e., ${\displaystyle f(\operatorname {Ad} _{g}x)=f(x)}$, then one can show that ${\displaystyle f(\Omega )}$ 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 ${\displaystyle \omega }$. First, that ${\displaystyle f(\Omega )}$ is a closed form follows from the next two lemmas:[1]

Lemma 1: The form ${\displaystyle f(\Omega )}$ on P descends to a (unique) form ${\displaystyle {\overline {f}}(\Omega )}$ on M; i.e., there is a form on M that pulls-back to ${\displaystyle f(\Omega )}$.

Lemma 2: If a form of ${\displaystyle \varphi }$ on P descends to a form on M, then ${\displaystyle d\varphi =D\varphi }$.

Indeed, Bianchi's second identity says ${\displaystyle D\Omega =0}$ and, since D is a graded derivation, ${\displaystyle Df(\Omega )=0.}$ Finally, Lemma 1 says ${\displaystyle f(\Omega )}$ satisfies the hypothesis of Lemma 2.

To see Lemma 2, let ${\displaystyle \pi \colon P\to M}$ be the projection and h be the projection of ${\displaystyle T_{u}P}$ onto the horizontal subspace. Then Lemma 2 is a consequence of the fact that ${\displaystyle d\pi (hv)=d\pi (v)}$ (the kernel of ${\displaystyle d\pi }$ is precisely the vertical subspace.) As for Lemma 1, first note

${\displaystyle f(\Omega )(dR_{g}(v_{1}),\dots ,dR_{g}(v_{2k}))=f(\Omega )(v_{1},\dots ,v_{2k}),\,R_{g}(u)=ug;}$

which is because ${\displaystyle R_{g}^{*}\Omega =\operatorname {Ad} _{g^{-1}}\Omega }$ and f is invariant. Thus, one can define ${\displaystyle {\overline {f}}(\Omega )}$ by the formula:

${\displaystyle {\overline {f}}(\Omega )({\overline {v_{1}}},\dots ,{\overline {v_{2k}}})=f(\Omega )(v_{1},\dots ,v_{2k}),}$

where ${\displaystyle v_{i}}$ are any lifts of ${\displaystyle {\overline {v_{i}}}}$: ${\displaystyle d\pi (v_{i})={\overline {v}}_{i}}$.

Next, we show that the de Rham cohomology class of ${\displaystyle {\overline {f}}(\Omega )}$ on M is independent of a choice of connection.[2] Let ${\displaystyle \omega _{0},\omega _{1}}$ be arbitrary connection forms on P and let ${\displaystyle p\colon P\times \mathbb {R} \to P}$ be the projection. Put

${\displaystyle \omega '=t\,p^{*}\omega _{1}+(1-t)\,p^{*}\omega _{0}}$

where t is a smooth function on ${\displaystyle P\times \mathbb {R} }$ given by ${\displaystyle (x,s)\mapsto s}$. Let ${\displaystyle \Omega ',\Omega _{0},\Omega _{1}}$ be the curvature forms of ${\displaystyle \omega ',\omega _{0},\omega _{1}}$. Let ${\displaystyle i_{s}:M\to M\times \mathbb {R} ,\,x\mapsto (x,s)}$ be the inclusions. Then ${\displaystyle i_{0}}$ is homotopic to ${\displaystyle i_{1}}$. Thus, ${\displaystyle i_{0}^{*}{\overline {f}}(\Omega ')}$ and ${\displaystyle i_{1}^{*}{\overline {f}}(\Omega ')}$ belong to the same de Rham cohomology class by the homotopy invariance of de Rham cohomology. Finally, by naturality and by uniqueness of descending,

${\displaystyle i_{0}^{*}{\overline {f}}(\Omega ')={\overline {f}}(\Omega _{0})}$

and the same for ${\displaystyle \Omega _{1}}$. Hence, ${\displaystyle {\overline {f}}(\Omega _{0}),{\overline {f}}(\Omega _{1})}$ belong to the same cohomology class.

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

${\displaystyle \mathbb {C} [{\mathfrak {g}}]_{k}^{G}\to H^{2k}(M;\mathbb {C} ),\,f\mapsto \left[{\overline {f}}(\Omega )\right].}$

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

${\displaystyle \mathbb {C} [{\mathfrak {g}}]^{G}\to H^{*}(M;\mathbb {C} )}$

is an algebra homomorphism.

Let ${\displaystyle G=\operatorname {GL} _{n}(\mathbb {C} )}$ and ${\displaystyle {\mathfrak {g}}={\mathfrak {gl}}_{n}(\mathbb {C} )}$ its Lie algebra. For each x in ${\displaystyle {\mathfrak {g}}}$, we can consider its characteristic polynomial in t:[3]

${\displaystyle \det \left(I-t{x \over 2\pi i}\right)=\sum _{k=0}^{n}f_{k}(x)t^{k},}$

where i is the square root of -1. Then ${\displaystyle f_{k}}$ are invariant polynomials on ${\displaystyle {\mathfrak {g}}}$, 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:

${\displaystyle c_{k}(E)\in H^{2k}(M,\mathbb {Z} )}$

is given as the image of ${\displaystyle f_{k}}$ under the Chern–Weil homomorphism defined by E (or more precisely the frame bundle of E). If t = 1, then ${\displaystyle \det \left(I-{x \over 2\pi i}\right)=1+f_{1}(x)+\cdots +f_{n}(x)}$ is an invariant polynomial. The total Chern class of E is the image of this polynomial; that is,

${\displaystyle c(E)=1+c_{1}(E)+\cdots +c_{n}(E).}$

Directly from the definition, one can show that ${\displaystyle c_{j}}$ and c given above satisfy the axioms of Chern classes. For example, for the Whitney sum formula, we consider

${\displaystyle c_{t}(E)=[\det \left(I-t{\Omega /2\pi i}\right)],}$

where we wrote ${\displaystyle \Omega }$ 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 ${\displaystyle \Omega }$. Now, suppose E is a direct sum of vector bundles ${\displaystyle E_{i}}$'s and ${\displaystyle \Omega _{i}}$ the curvature form of ${\displaystyle E_{i}}$ so that, in the matrix term, ${\displaystyle \Omega }$ is the block diagonal matrix with Ω_I's on the diagonal. Then, since ${\textstyle \det(I-t{\frac {\Omega }{2\pi i}})=\det(I-t{\frac {\Omega _{1}}{2\pi i}})\wedge \dots \wedge \det(I-t{\frac {\Omega _{m}}{2\pi i}})}$, we have:

${\displaystyle c_{t}(E)=c_{t}(E_{1})\cdots c_{t}(E_{m})}$

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 ${\displaystyle \Omega _{E\otimes E'}=\Omega _{E}\otimes I_{E'}+I_{E}\otimes \Omega _{E'}}$,[4] we also have:

${\displaystyle c_{1}(E\otimes E')=c_{1}(E)\operatorname {rank} (E')+\operatorname {rank} (E)c_{1}(E').}$

Finally, the Chern character of E is given by

${\displaystyle \operatorname {ch} (E)=[\operatorname {tr} (e^{-\Omega /2\pi i})]\in H^{*}(M,\mathbb {Q} )}$

where ${\displaystyle \Omega }$ is the curvature form of some connection on E (since ${\displaystyle \Omega }$ is nilpotent, it is a polynomial in ${\displaystyle \Omega }$.) Then ch is a ring homomorphism:

${\displaystyle \operatorname {ch} (E\oplus F)=\operatorname {ch} (E)+\operatorname {ch} (F),\,\operatorname {ch} (E\otimes F)=\operatorname {ch} (E)\operatorname {ch} (F).}$

Now suppose, in some ring R containing the cohomology ring ${\displaystyle H^{*}(M,\mathbb {C} )}$, there is the factorization of the polynomial in t:

${\displaystyle c_{t}(E)=\prod _{j=0}^{n}(1+\lambda _{j}t)}$

where ${\displaystyle \lambda _{j}}$ are in R (they are sometimes called Chern roots.) Then ${\displaystyle \operatorname {ch} (E)=e^{\lambda _{j}}}$.

If E is a smooth real vector bundle on a manifold M, then the k-th Pontrjagin class of E is given as:

${\displaystyle p_{k}(E)=(-1)^{k}c_{2k}(E\otimes \mathbb {C} )\in H^{4k}(M;\mathbb {Z} )}$

where we wrote ${\displaystyle E\otimes \mathbb {C} }$ for the complexification of E. Equivalently, it is the image under the Chern–Weil homomorphism of the invariant polynomial ${\displaystyle g_{2k}}$ on ${\displaystyle {\mathfrak {gl}}_{n}(\mathbb {R} )}$ given by:

${\displaystyle \operatorname {det} \left(I-t{x \over 2\pi }\right)=\sum _{k=0}^{n}g_{k}(x)t^{k}.}$

Let E be a holomorphic (complex-)vector bundle on a complex manifold M. The curvature form ${\displaystyle \Omega }$ 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 ${\displaystyle G=\operatorname {GL} _{n}(\mathbb {C} )}$,

${\displaystyle \mathbb {C} [{\mathfrak {g}}]_{k}\to H^{k,k}(M;\mathbb {C} ),f\mapsto [f(\Omega )].}$

• Bott, Raoul (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, Shiing-Shen (1951), Topics in Differential Geometry, Institute for Advanced Study, mimeographed lecture notes.
• Chern, Shiing-Shen (1995), Complex Manifolds Without Potential Theory, Springer-Verlag, 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, Shiing-Shen; Simons, James (1974), "Characteristic forms and geometric invariants", Annals of Mathematics, Second Series, 99 (1): 48–69, doi:10.2307/1971013, JSTOR 1971013.
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• Morita, Shigeyuki (2000), "Geometry of Differential Forms", Translations of Mathematical Monographs, 201, MR 1851352.

• Freed, Daniel S.; Hopkins, Michael J. (2013). "Chern-Weil forms and abstract homotopy theory". Bulletin of the American Mathematical Society. (N.S.). 50 (3): 431–468. arXiv:1301.5959. doi:10.1090/S0273-0979-2013-01415-0. MR 3049871. S2CID 51755613.