Chern–Weil homomorphism

<h2 id="definition-of-the-homomorphism">Definition of the homomorphism</h2>
Choose any <a href="/facts/Connection_form/Hnn6FJFN">connection form</a> ω in P, and let Ω be the associated <a href="/facts/Curvature_form/LWp7DMDM">curvature form</a>; i.e., 
 
 
 
 Ω
 =
 D
 ω
 
 
 {\displaystyle \Omega =D\omega }
 
, the <a href="/facts/Exterior_covariant_derivative/eKHqDq6Q">exterior covariant derivative</a> of ω. If 
 
 
 
 f
 ∈
 
 C
 
 [
 
 
 g
 
 
 
 ]
 
 G
 
 
 
 
 {\displaystyle f\in \mathbb {C} [{\mathfrak {g}}]^{G}}
 
 is a homogeneous polynomial function of degree k; i.e., 
 
 
 
 f
 (
 a
 x
 )
 =
 
 a
 
 k
 
 
 f
 (
 x
 )
 
 
 {\displaystyle f(ax)=a^{k}f(x)}
 
 for any complex number a and x in 
 
 
 
 
 
 g
 
 
 
 
 {\displaystyle {\mathfrak {g}}}
 
, then, viewing f as a symmetric multilinear functional on 
 
 
 
 
 ∏
 
 1
 
 
 k
 
 
 
 
 g
 
 
 
 
 {\textstyle \prod _{1}^{k}{\mathfrak {g}}}
 
 (see the <a href="/facts/Ring_of_polynomial_functions/q99CvhPj">ring of polynomial functions</a>), let

f
        (
        Ω
        )
      
    
    {\displaystyle f(\Omega )}

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
            )
          
        
        )
        )
      
    
    {\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 vi are tangent vectors to P, 
 
 
 
 
 ϵ
 
 σ
 
 
 
 
 {\displaystyle \epsilon _{\sigma }}
 
 is the sign of the permutation 
 
 
 
 σ
 
 
 {\displaystyle \sigma }
 
 in the symmetric group on 2k numbers 
 
 
 
 
 
 
 S
 
 
 
 2
 k
 
 
 
 
 {\displaystyle {\mathfrak {S}}_{2k}}
 
 (see <a href="/facts/Lie_algebra-valued_forms/ot9XbLEk">Lie algebra-valued forms#Operations</a> as well as <a href="/facts/Pfaffian/3CmddqLu">Pfaffian</a>).
If, moreover, f is invariant; i.e., 
 
 
 
 f
 (
 
 Ad
 
 g
 
 
 ⁡
 x
 )
 =
 f
 (
 x
 )
 
 
 {\displaystyle f(\operatorname {Ad} _{g}x)=f(x)}
 
, then one can show that 
 
 
 
 f
 (
 Ω
 )
 
 
 {\displaystyle f(\Omega )}
 
 is a <a href="/facts/Closed_and_exact_differential_forms/uvEUbzUK">closed form</a>, it descends to a unique form on M and that the <a href="/facts/De_Rham_cohomology/4qGKeJq0">de Rham cohomology</a> class of the form is independent of 
 
 
 
 ω
 
 
 {\displaystyle \omega }
 
. First, that 
 
 
 
 f
 (
 Ω
 )
 
 
 {\displaystyle f(\Omega )}
 
 is a closed form follows from the next two lemmas:<a class="footnote-ref" id="fnref:1" href="#fn:1">1</a>

Lemma 1: The form 
 
 
 
 f
 (
 Ω
 )
 
 
 {\displaystyle f(\Omega )}
 
 on P descends to a (unique) form 
 
 
 
 
 
 f
 ¯
 
 
 (
 Ω
 )
 
 
 {\displaystyle {\overline {f}}(\Omega )}
 
 on M; i.e., there is a form on M that pulls-back to 
 
 
 
 f
 (
 Ω
 )
 
 
 {\displaystyle f(\Omega )}
 
.
Lemma 2: If a form of 
 
 
 
 φ
 
 
 {\displaystyle \varphi }
 
 on P descends to a form on M, then 
 
 
 
 d
 φ
 =
 D
 φ
 
 
 {\displaystyle d\varphi =D\varphi }
 
.
Indeed, <a href="/facts/Bianchi%27s_second_identity/LWp7DMDM">Bianchi's second identity</a> says 
 
 
 
 D
 Ω
 =
 0
 
 
 {\displaystyle D\Omega =0}
 
 and, since D is a graded derivation, 
 
 
 
 D
 f
 (
 Ω
 )
 =
 0.
 
 
 {\displaystyle Df(\Omega )=0.}
 
 Finally, Lemma 1 says 
 
 
 
 f
 (
 Ω
 )
 
 
 {\displaystyle f(\Omega )}
 
 satisfies the hypothesis of Lemma 2.
To see Lemma 2, let 
 
 
 
 π
 :
 P
 →
 M
 
 
 {\displaystyle \pi \colon P\to M}
 
 be the projection and h be the projection of 
 
 
 
 
 T
 
 u
 
 
 P
 
 
 {\displaystyle T_{u}P}
 
 onto the horizontal subspace. Then Lemma 2 is a consequence of the fact that 
 
 
 
 d
 π
 (
 h
 v
 )
 =
 d
 π
 (
 v
 )
 
 
 {\displaystyle d\pi (hv)=d\pi (v)}
 
 (the kernel of 
 
 
 
 d
 π
 
 
 {\displaystyle d\pi }
 
 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
        ;
      
    
    {\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 
 
 
 
 
 R
 
 g
 
 
 ∗
 
 
 Ω
 =
 
 Ad
 
 
 g
 
 −
 1
 
 
 
 
 ⁡
 Ω
 
 
 {\displaystyle R_{g}^{*}\Omega =\operatorname {Ad} _{g^{-1}}\Omega }
 
 and f is invariant. Thus, one can define 
 
 
 
 
 
 f
 ¯
 
 
 (
 Ω
 )
 
 
 {\displaystyle {\overline {f}}(\Omega )}
 
 by the formula:

f
            ¯
          
        
        (
        Ω
        )
        (
        
          
            
              v
              
                1
              
            
            ¯
          
        
        ,
        …
        ,
        
          
            
              v
              
                2
                k
              
            
            ¯
          
        
        )
        =
        f
        (
        Ω
        )
        (
        
          v
          
            1
          
        
        ,
        …
        ,
        
          v
          
            2
            k
          
        
        )
        ,
      
    
    {\displaystyle {\overline {f}}(\Omega )({\overline {v_{1}}},\dots ,{\overline {v_{2k}}})=f(\Omega )(v_{1},\dots ,v_{2k}),}

where 
 
 
 
 
 v
 
 i
 
 
 
 
 {\displaystyle v_{i}}
 
 are any lifts of 
 
 
 
 
 
 
 v
 
 i
 
 
 ¯
 
 
 
 
 {\displaystyle {\overline {v_{i}}}}
 
: 
 
 
 
 d
 π
 (
 
 v
 
 i
 
 
 )
 =
 
 
 
 v
 ¯
 
 
 
 i
 
 
 
 
 {\displaystyle d\pi (v_{i})={\overline {v}}_{i}}
 
.
Next, we show that the de Rham cohomology class of 
 
 
 
 
 
 f
 ¯
 
 
 (
 Ω
 )
 
 
 {\displaystyle {\overline {f}}(\Omega )}
 
 on M is independent of a choice of connection.<a class="footnote-ref" id="fnref:2" href="#fn:2">2</a> Let 
 
 
 
 
 ω
 
 0
 
 
 ,
 
 ω
 
 1
 
 
 
 
 {\displaystyle \omega _{0},\omega _{1}}
 
 be arbitrary connection forms on P and let 
 
 
 
 p
 :
 P
 ×
 
 R
 
 →
 P
 
 
 {\displaystyle p\colon P\times \mathbb {R} \to P}
 
 be the projection. Put

ω
          ′
        
        =
        t
        
        
          p
          
            ∗
          
        
        
          ω
          
            1
          
        
        +
        (
        1
        −
        t
        )
        
        
          p
          
            ∗
          
        
        
          ω
          
            0
          
        
      
    
    {\displaystyle \omega '=t\,p^{*}\omega _{1}+(1-t)\,p^{*}\omega _{0}}

where t is a smooth function on 
 
 
 
 P
 ×
 
 R
 
 
 
 {\displaystyle P\times \mathbb {R} }
 
 given by 
 
 
 
 (
 x
 ,
 s
 )
 ↦
 s
 
 
 {\displaystyle (x,s)\mapsto s}
 
. Let 
 
 
 
 
 Ω
 ′
 
 ,
 
 Ω
 
 0
 
 
 ,
 
 Ω
 
 1
 
 
 
 
 {\displaystyle \Omega ',\Omega _{0},\Omega _{1}}
 
 be the curvature forms of 
 
 
 
 
 ω
 ′
 
 ,
 
 ω
 
 0
 
 
 ,
 
 ω
 
 1
 
 
 
 
 {\displaystyle \omega ',\omega _{0},\omega _{1}}
 
. Let 
 
 
 
 
 i
 
 s
 
 
 :
 M
 →
 M
 ×
 
 R
 
 ,
 
 x
 ↦
 (
 x
 ,
 s
 )
 
 
 {\displaystyle i_{s}:M\to M\times \mathbb {R} ,\,x\mapsto (x,s)}
 
 be the inclusions. Then 
 
 
 
 
 i
 
 0
 
 
 
 
 {\displaystyle i_{0}}
 
 is homotopic to 
 
 
 
 
 i
 
 1
 
 
 
 
 {\displaystyle i_{1}}
 
. Thus, 
 
 
 
 
 i
 
 0
 
 
 ∗
 
 
 
 
 f
 ¯
 
 
 (
 
 Ω
 ′
 
 )
 
 
 {\displaystyle i_{0}^{*}{\overline {f}}(\Omega ')}
 
 and 
 
 
 
 
 i
 
 1
 
 
 ∗
 
 
 
 
 f
 ¯
 
 
 (
 
 Ω
 ′
 
 )
 
 
 {\displaystyle i_{1}^{*}{\overline {f}}(\Omega ')}
 
 belong to the same de Rham cohomology class by the <a href="/facts/Homotopy_invariance_of_de_Rham_cohomology/4qGKeJq0">homotopy invariance of de Rham cohomology</a>. Finally, by naturality and by uniqueness of descending,

i
          
            0
          
          
            ∗
          
        
        
          
            f
            ¯
          
        
        (
        
          Ω
          ′
        
        )
        =
        
          
            f
            ¯
          
        
        (
        
          Ω
          
            0
          
        
        )
      
    
    {\displaystyle i_{0}^{*}{\overline {f}}(\Omega ')={\overline {f}}(\Omega _{0})}

and the same for 
 
 
 
 
 Ω
 
 1
 
 
 
 
 {\displaystyle \Omega _{1}}
 
. Hence, 
 
 
 
 
 
 f
 ¯
 
 
 (
 
 Ω
 
 0
 
 
 )
 ,
 
 
 f
 ¯
 
 
 (
 
 Ω
 
 1
 
 
 )
 
 
 {\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)

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

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

is an <a href="/facts/Algebra_homomorphism/b7YL8aPR">algebra homomorphism</a>.

<h2 id="example-chern-classes-and-chern-character">Example: Chern classes and Chern character</h2>
Let 
 
 
 
 G
 =
 
 GL
 
 n
 
 
 ⁡
 (
 
 C
 
 )
 
 
 {\displaystyle G=\operatorname {GL} _{n}(\mathbb {C} )}
 
 and 
 
 
 
 
 
 g
 
 
 =
 
 
 
 g
 l
 
 
 
 n
 
 
 (
 
 C
 
 )
 
 
 {\displaystyle {\mathfrak {g}}={\mathfrak {gl}}_{n}(\mathbb {C} )}
 
 its Lie algebra. For each x in 
 
 
 
 
 
 g
 
 
 
 
 {\displaystyle {\mathfrak {g}}}
 
, we can consider its <a href="/facts/Characteristic_polynomial/7m8roDqo">characteristic polynomial</a> in t:<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a>

det
        
          (
          
            I
            −
            t
            
              
                x
                
                  2
                  π
                  i
                
              
            
          
          )
        
        =
        
          ∑
          
            k
            =
            0
          
          
            n
          
        
        
          f
          
            k
          
        
        (
        x
        )
        
          t
          
            k
          
        
        ,
      
    
    {\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 
 
 
 
 
 f
 
 k
 
 
 
 
 {\displaystyle f_{k}}
 
 are invariant polynomials on 
 
 
 
 
 
 g
 
 
 
 
 {\displaystyle {\mathfrak {g}}}
 
, since the left-hand side of the equation is. The k-th <a href="/facts/Chern_class/Tw2WD5hM">Chern class</a> of a smooth complex-vector bundle E of rank n on a manifold M:

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

is given as the image of 
 
 
 
 
 f
 
 k
 
 
 
 
 {\displaystyle 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
 )
 
 
 {\displaystyle \det \left(I-{x \over 2\pi i}\right)=1+f_{1}(x)+\cdots +f_{n}(x)}
 
 is an invariant polynomial. The <a href="/facts/Total_Chern_class/Tw2WD5hM">total Chern class</a> of E is the image of this polynomial; that is,

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

Directly from the definition, one can show that 
 
 
 
 
 c
 
 j
 
 
 
 
 {\displaystyle c_{j}}
 
 and 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
            
          
          )
        
        ]
        ,
      
    
    {\displaystyle c_{t}(E)=[\det \left(I-t{\Omega /2\pi i}\right)],}

where we wrote 
 
 
 
 Ω
 
 
 {\displaystyle \Omega }
 
 for the <a href="/facts/Curvature_form/LWp7DMDM">curvature 2-form</a> 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
 
 
 {\displaystyle \Omega }
 
. Now, suppose E is a direct sum of vector bundles 
 
 
 
 
 E
 
 i
 
 
 
 
 {\displaystyle E_{i}}
 
's and 
 
 
 
 
 Ω
 
 i
 
 
 
 
 {\displaystyle \Omega _{i}}
 
 the curvature form of 
 
 
 
 
 E
 
 i
 
 
 
 
 {\displaystyle E_{i}}
 
 so that, in the matrix term, 
 
 
 
 Ω
 
 
 {\displaystyle \Omega }
 
 is the block diagonal matrix with ΩI'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
 
 
 
 )
 
 
 {\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:

c
          
            t
          
        
        (
        E
        )
        =
        
          c
          
            t
          
        
        (
        
          E
          
            1
          
        
        )
        ⋯
        
          c
          
            t
          
        
        (
        
          E
          
            m
          
        
        )
      
    
    {\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: <a href="/facts/Cup_product/SGWYp5cF">cup product</a>. For the normalization property, one computes the first Chern class of the <a href="/facts/Complex_projective_line/4XgBzplC">complex projective line</a>; see <a href="/facts/Chern_class/Tw2WD5hM">Chern class#Example: the complex tangent bundle of the Riemann sphere</a>.
Since 
 
 
 
 
 Ω
 
 E
 ⊗
 
 E
 ′
 
 
 
 =
 
 Ω
 
 E
 
 
 ⊗
 
 I
 
 
 E
 ′
 
 
 
 +
 
 I
 
 E
 
 
 ⊗
 
 Ω
 
 
 E
 ′
 
 
 
 
 
 {\displaystyle \Omega _{E\otimes E'}=\Omega _{E}\otimes I_{E'}+I_{E}\otimes \Omega _{E'}}
 
,<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a> we also have:

c
          
            1
          
        
        (
        E
        ⊗
        
          E
          ′
        
        )
        =
        
          c
          
            1
          
        
        (
        E
        )
        rank
        ⁡
        (
        
          E
          ′
        
        )
        +
        rank
        ⁡
        (
        E
        )
        
          c
          
            1
          
        
        (
        
          E
          ′
        
        )
        .
      
    
    {\displaystyle c_{1}(E\otimes E')=c_{1}(E)\operatorname {rank} (E')+\operatorname {rank} (E)c_{1}(E').}

Finally, the <a href="/facts/Chern_character/Tw2WD5hM">Chern character</a> of E is given by

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

where 
 
 
 
 Ω
 
 
 {\displaystyle \Omega }
 
 is the curvature form of some connection on E (since 
 
 
 
 Ω
 
 
 {\displaystyle \Omega }
 
 is nilpotent, it is a polynomial in 
 
 
 
 Ω
 
 
 {\displaystyle \Omega }
 
.) Then ch is a <a href="/facts/Ring_homomorphism/UptcMpFk">ring homomorphism</a>:

ch
        ⁡
        (
        E
        ⊕
        F
        )
        =
        ch
        ⁡
        (
        E
        )
        +
        ch
        ⁡
        (
        F
        )
        ,
        
        ch
        ⁡
        (
        E
        ⊗
        F
        )
        =
        ch
        ⁡
        (
        E
        )
        ch
        ⁡
        (
        F
        )
        .
      
    
    {\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 
 
 
 
 
 H
 
 ∗
 
 
 (
 M
 ,
 
 C
 
 )
 
 
 {\displaystyle H^{*}(M,\mathbb {C} )}
 
, there is the factorization of the polynomial in t:

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

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

<h2 id="example-pontrjagin-classes">Example: Pontrjagin classes</h2>
If E is a smooth real vector bundle on a manifold M, then the k-th <a href="/facts/Pontrjagin_class/TIFceC5j">Pontrjagin class</a> of E is given as:

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

where we wrote 
 
 
 
 E
 ⊗
 
 C
 
 
 
 {\displaystyle E\otimes \mathbb {C} }
 
 for the <a href="/facts/Complex_vector_bundle/0nE8TEIn">complexification</a> of E. Equivalently, it is the image under the Chern–Weil homomorphism of the invariant polynomial 
 
 
 
 
 g
 
 2
 k
 
 
 
 
 {\displaystyle g_{2k}}
 
 on 
 
 
 
 
 
 
 g
 l
 
 
 
 n
 
 
 (
 
 R
 
 )
 
 
 {\displaystyle {\mathfrak {gl}}_{n}(\mathbb {R} )}
 
 given by:

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

<h2 id="the-homomorphism-for-holomorphic-vector-bundles">The homomorphism for holomorphic vector bundles</h2>
Let E be a <a href="/facts/Holomorphic_vector_bundle/FeRrK9Bg">holomorphic (complex-)vector bundle</a> on a complex manifold M. The curvature form 
 
 
 
 Ω
 
 
 {\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 <a href="/facts/Holomorphic_vector_bundle/FeRrK9Bg">holomorphic vector bundle#Hermitian metrics on a holomorphic vector bundle</a>). Hence, the Chern–Weil homomorphism assumes the form: with 
 
 
 
 G
 =
 
 GL
 
 n
 
 
 ⁡
 (
 
 C
 
 )
 
 
 {\displaystyle G=\operatorname {GL} _{n}(\mathbb {C} )}
 
,

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

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/%E2%88%9E-Chern%E2%80%93Weil_theory/F5TJbGNc">∞-Chern–Weil theory</a></li></ul>
<h2 id="notes">Notes</h2>

<ul><li><a href="/facts/Raoul_Bott/5JHkhPx8">Bott, Raoul</a> (1973), "On the Chern–Weil homomorphism and the continuous cohomology of Lie groups", <a href="/facts/Advances_in_Mathematics/kXtcl4pM">Advances in Mathematics</a>, 11 (3): 289–303, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1016%2F0001-8708%2873%2990012-1">10.1016/0001-8708(73)90012-1</a>.</li>
<li><a href="/facts/S.-S._Chern/FpfIQsEH">Chern, Shiing-Shen</a> (1951), Topics in Differential Geometry, Institute for Advanced Study, mimeographed lecture notes.</li>
<li><a href="/facts/S.-S._Chern/FpfIQsEH">Chern, Shiing-Shen</a> (1995), Complex Manifolds Without Potential Theory, <a href="/facts/Springer_Science%2BBusiness_Media/nAesf6nT">Springer-Verlag</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-387-90422-0, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 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.)</li>
<li><a href="/facts/S.-S._Chern/FpfIQsEH">Chern, Shiing-Shen</a>; <a href="/facts/James_Harris_Simons/5bsjBRyz">Simons, James</a> (1974), "Characteristic forms and geometric invariants", <a href="/facts/Annals_of_Mathematics/CUYktUqD">Annals of Mathematics</a>, Second Series, 99 (1): 48–69, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.2307%2F1971013">10.2307/1971013</a>, <a href="/facts/JSTOR_(identifier)/YTeVmaJ7">JSTOR</a> <a href="https://www.jstor.org/stable/1971013">1971013</a>.</li>
<li><a href="/facts/Shoshichi_Kobayashi/P3rHgfGs">Kobayashi, Shoshichi</a>; <a href="/facts/Katsumi_Nomizu/Xhs8rgXm">Nomizu, Katsumi</a> (1969), <a href="/facts/Foundations_of_Differential_Geometry/lgGBjhhd">Foundations of Differential Geometry</a>, vol. 2 (new ed.), Wiley-Interscience (published 2004), <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0152974">0152974</a>.</li>
<li><a href="/facts/M._S._Narasimhan/QryQRuVu">Narasimhan, M. S.</a>; <a href="/facts/S._Ramanan/VVLKyYUY">Ramanan, S.</a> (1961), <a href="http://dml.cz/bitstream/handle/10338.dmlcz/700905/Toposym_01-1961-1_68.pdf">"Existence of universal connections"</a> (PDF), <a href="/facts/American_Journal_of_Mathematics/lkgftcD7">American Journal of Mathematics</a>, 83 (3): 563–572, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.2307%2F2372896">10.2307/2372896</a>, <a href="/facts/Hdl_(identifier)/rdebSxmC">hdl</a>:<a href="https://hdl.handle.net/10338.dmlcz%2F700905">10338.dmlcz/700905</a>, <a href="/facts/JSTOR_(identifier)/YTeVmaJ7">JSTOR</a> <a href="https://www.jstor.org/stable/2372896">2372896</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0133772">0133772</a>.</li>
<li>Morita, Shigeyuki (2000), "Geometry of Differential Forms", Translations of Mathematical Monographs, 201, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1851352">1851352</a>.</li></ul>
<h2 id="further-reading">Further reading</h2>
<ul><li><a href="/facts/Dan_Freed/5EF4xX8G">Freed, Daniel S.</a>; <a href="/facts/Michael_J._Hopkins/nM5Jqqnx">Hopkins, Michael J.</a> (2013). "Chern-Weil forms and abstract homotopy theory". <a href="/facts/Bulletin_of_the_American_Mathematical_Society/8ER5KY5z">Bulletin of the American Mathematical Society</a>. (N.S.). 50 (3): 431–468. <a href="/facts/ArXiv_(identifier)/H6EtgnBe">arXiv</a>:<a href="https://arxiv.org/abs/1301.5959">1301.5959</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1090%2FS0273-0979-2013-01415-0">10.1090/S0273-0979-2013-01415-0</a>. <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=3049871">3049871</a>. <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:51755613">51755613</a>.</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Kobayashi & Nomizu 1969, Ch. XII. - Kobayashi, Shoshichi; Nomizu, Katsumi (1969), Foundations of Differential Geometry, vol. 2 (new ed.), Wiley-Interscience (published 2004), MR 0152974 <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0152974" target="_blank">https://mathscinet.ams.org/mathscinet-getitem?mr=0152974</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">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.{{cite web}}: CS1 maint: archived copy as title (link). Kobayashi-Nomizu, the main reference, gives a more concrete argument. <a href="https://web.archive.org/web/20141217025038/https://math.berkeley.edu/~amathew/kodaira.pdf" target="_blank">https://web.archive.org/web/20141217025038/https://math.berkeley.edu/~amathew/kodaira.pdf</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn: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. <a href="/wiki/Chern_class" target="_blank">/wiki/Chern_class</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Proof: By definition, 
 
 
 
 
 ∇
 
 E
 ⊗
 
 E
 ′
 
 
 
 (
 s
 ⊗
 
 s
 ′
 
 )
 =
 
 ∇
 
 E
 
 
 s
 ⊗
 
 s
 ′
 
 +
 s
 ⊗
 
 ∇
 
 
 E
 ′
 
 
 
 
 s
 ′
 
 
 
 {\displaystyle \nabla ^{E\otimes E'}(s\otimes s')=\nabla ^{E}s\otimes s'+s\otimes \nabla ^{E'}s'}
 
. Now compute the square of 
 
 
 
 
 ∇
 
 E
 ⊗
 
 E
 ′
 
 
 
 
 
 {\displaystyle \nabla ^{E\otimes E'}}
 
 using Leibniz's rule. <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
</ol>

Chern–Weil homomorphism open-in-new

Chern–Weil homomorphism