17 Chern connection on Hermitian vector bundles

Hermitian connection A Hermitian structure hE in a smooth complex vector bundle E is a smooth field of Hermitian inner products , in the fibres of E. With respect hE to a local frame, a Hermitian structure is given by a Hermitian matrix-valued function t 39 H =(H ), with H = s ,s E which transforms according to H = A H′A . h is also ij ij i jh E called a hermitian metric on E. We call (E, hE) a Hermitian vector bundle.

If hE is a Hemitian metric of a complex vector bundle E on a smooth manifold X. A connection is called a Hermitian connection if for any s1,s2 ∞(X, E), ▽E ∈ C

d s1,s2 = s1,s2 + s1, s2 . hE ▽E hE ▽E hE There always exist Hermitian connections.

Chern connection on E Let E be a holomorphic vector bundle over a complex manifold (X,J) where J is the complex structure. Let (E, hE)bea Hermitian vector bundle over X.

2 There exists a natural C-linear operator ∂ : Ap,q(E) Ap,q+1(E) with ∂ = 0 and which is defined locally by ∂(f α)= ∂f α. → ⊗ Using the decomposition 1(E)= 1,0(E) 0,1(E) we can decompose any connection D on E in its two componentsA D1,00 andA D0,1,⊕A i.e. D = D1,0 D0,1 with ⊕ D1,0 : 0(E) 1,0(E) A →A and D0,1 : 0(E) 0,1(E). A →A Note that D0,1 satisfies D0,1(fs) = ∂f s + f D0,1(s), i.e. it behaves similarly to the ⊗ 1 0 0 1 operator ∂. Indeed, the decomposition D = D , D , makes sense even when E is not holomorphic. A connection D on a holomorphic⊕ vector bundle E is compatible with the holomorphic structure if D0,1 = ∂.

If a connection hE on a holomorphic vector bundle E is both Hermitian and compatible with the holomorphic structure is called Chern connection.

Proposition 17.1 Chern connection exists uniquely.

39 A Hermitian structure in TX is exactly a Hermitian metric on X.

104 Proof: With respect to a local holomorphic frame S = (sα)=(s1, ..., sq) on U, We have γ 1,0 0,1 DS = ωS, i.e., Dsα = ωαsγ on U. By the facts that sα are holomorphic, D = D + D = 1,0 1,0 γ γ D + ∂ and ∂sα = 0, we have Dsα = D sα = ωαsγ so that ωα are all (1, 0)-forms.

By the condition, we have

d s ,s = Ds ,s + s ,Ds = ωγ s ,s + s ,ωγs = h ωγ + h ωγ α β α β α β α γ β α β γ βγ α αγ β where we denote the smooth function h := s ,s . αβ α β On the other hand, d s ,s = dh = (∂ + ∂)h = ∂h + ∂h . Since all ωα are α β αβ αβ αβ αβ β (1, 0)-form, it implies γ ∂hαβ = ωαhγβ, i.e., γ βγ ωα = ∂hαβ h

1 βγ 1 where H− =(h )=(hαβ)− is the inverse matrix of H =(hαβ), i.e.,

1 ω = ∂H H− is uniquely determined by H, i.e., the connection matrix is completely determined by the metric. 

[Example] Let (L, h) be a Hermitian holomorphic line bundle over a complex manifold X. For any point a X, let s be a local frame on a neighborhood U of a in X, then the corresponding curvature∈ form is Ω= ∂∂log H. − i where H = s,s . We call c1(L, h)= ∂∂log H the Chern curvature form of (L, h). − 2π ∂H ∂∂H H ∂H In fact, ω = H = −H2 ∧ ,

∂H ∂∂H ∂H ∂H ∂H ∂∂log H = ∂ = + ∧ = d = dω = dω ω ω = Ω. − − H − H H2 H − ∧ If we denote s = √h s , we have H = h s 2 so that h | | | |

i i 2 i Ω= ∂∂log h s = ∂∂log h = c1(L, h). 2π −2π | | −2π

105 Chern connection and Levi-Civita connection Recall that a Hermitian structure on a complex manifold X is just a Riemannian metric g on the underlying real manifold compatible with the complex structure J defining X. Recall that the complexified tangent 1,0 0,1 C bundle TCX decomposes as TCX = TX TX . We still denote by g the -bilinear form on ⊕ 1,0 1,0 0,1 0,1 TCX induced by g. Notice that g( , ) vanishes on T X T X and on T X T X. Recall 1,0 × × that the bundle TX is the complex bundle underlying the holomorphic tangent bundle and hence is the same as the holomorphic tangent bundle. Moreover, the Hermitian extension l,0 gC of g to TCX restricted to TX is 1 (g iω), 2 − 1,0 where ω is the fundamental form g(J( ), ( )). The complex vector bundles TX and (TX ,J) are identified via the isomorphism

1,0 1 ξ : TX T , u u iJ(u) . → X → 2 −

n 1,0 1 √ 1 Let w be a local orthnomal frame of T . Then (w + w ), − (w w ) form { j}j=1 X { √2 j j √2 j − j } an orthonormal frame of TX .

1,0 Under the natural isomorphism ξ, any Hermitian connection V on TX induces a metric connection V on the Riemannian manifold (X,g). In general, a Hermitian connection V on 1,0 (TX ,g) will not necessarily induce the Levi-Civita connection on the Riemannian manifold (X,g). In fact, this could hardly be true, as the Levi-Civita connection is unique, but there are many hermitian connections (T 1,0,g). But even for the Chern connection on the holomorphic tangent bundle (TX ,g), which is unique, the induced connection may not be the Levi-Civita connection in general.

40 Theorem 17.2 Let (X,g) be a K¨ahler manifold. Then under the isomorphism ξ : TX l,0 l,0 → TX the Chern connection D on the holomorphic tangent bundle TX corresponds to the Levi-Civita connection . ▽

40Daniel Huybrechts, Complex Geometry - an introduction, Springer, 2005. , p. 219.

106 18 Chern Classes

Bianchi Identity Let M be an m-dimensional smooth manifold and (E,M,π) a q- dimensional complex vector bundle on M. We’ll define the important invariants: Chern classes.

Suppose sα, 1 α q is a local frame field of the complex vector bundle E in a neighborhood{ U M≤ . Then≤ } the action of a connection D on E can be expressed in U by ⊂ Ds = ωβs , α α β β

β where the connection matrix ωα is a complex-valued differential 1-form. If we use the matrix notation, then the above can be written as

DS = ω S, where t S = (s1, ..., sq), where 1 q ω1 ... ω1  . . .  ω = . .. .  1 q ωq ... ωq  The curvature matrix for the connection is

Ω=(Ωβ )= dω ω ω. α − ∧ Taking exteriorly differential, we then get the Bianchi identity:

dΩ= ω Ω Ω ω. (55) ∧ − ∧

If we choose another local frame field S′ and assume that

S′ = A S, where detA = 0, then we have the transformation formula for curvature matrices 1 Ω′ = A Ω A− . (56)

107