2. Chern Connections and Chern Curvatures1

2. Chern connections and Chern curvatures1

Let V be a complex vector space with dimC V = n. A hermitian metric h on V is

h : V × V −−→ C such that h(av, bu) = abh(v, u)

h(a1v1 + a2v2, u) = a1h(v1, u) + a2h(v2, u) h(v, u) = h(u, v) h(u, u) > 0, u 6= 0 where v, v1, v2, u ∈ V and a, b, a1, a2 ∈ C. If we ﬁx a basis {ei} of V , and set

hij = h(ei, ej) then ∗ ∗ ∗ ∗ h = hijei ⊗ ej ∈ V ⊗ V ∗ ∗ ∗ ∗ where ei ∈ V is the dual of ei and ei ∈ V is the conjugate dual of ei, i.e. X ∗ ei ( ajej) = ai

It is obvious that (hij) is a hermitian positive matrix. Deﬁnition 0.1. A complex vector bundle E is said to be hermitian if there is a positive deﬁnite hermitian tensor h on E.

r Let ϕ : E|U −−→ U × C be a trivilization and e = (e1, ··· , er) be the corresponding frame. The r hermitian metric h is represented by a positive hermitian matrix (hij) ∈ Γ(Ω, EndC ) such that

hei(x), ej(x)i = hij(x), x ∈ U Then hermitian metric on the chart (U, ϕ) could be written as X ∗ ∗ h = hijei ⊗ ej

For example, there are two charts (U, ϕ) and (V, ψ). We set

g = ψ ◦ ϕ−1 :(U ∩ V ) × Cr −−→ (U ∩ V ) × Cr and g is represented by matrix (gij). On U ∩ V , we have X X X −1 −1 −1 −1 −1 ei(x) = ϕ (x, εi) = ψ ◦ ψ ◦ ϕ (x, εi) = ψ (x, gijεj) = gijψ (x, εj) = gije˜j(x) j j

For the metric X ˜ hij = hei(x), ej(x)i = hgike˜k(x), gjle˜l(x)i = gikhklgjl k,l that is h = g · h˜ · g∗

12008.04.30 If there are some errors, please contact to: [email protected] 2

Example 0.2 (Fubini-Study metric on holomorphic tangent bundle T 1,0Pn). On the n ∂ trivialization of T P , for example U0 = {Z0 6= 0}, and basis of the ﬁber ei = , we could set ∂zi

2 Pn 2 ∂ log(1 + i=1 |zi |) hij(z) = hei(z), ej(z)i = ∂zi∂zj

Or X h(z) = hijdzi ⊗ dzj When E is a hermitian vector bundle, there is a natural sequilinear map

Ap(M,E) × Aq(M,E) −−→ Ap+q(M, C), (s, t) −−→{s, t}

P i P j In local coordinate, if s = σ ⊗ ei and t = τ ⊗ ej, X X i j i j {s, t} = σ ∧ τ hei, eji = hijσ ∧ τ i,j i,j

Deﬁnition 0.3. A connection ∇ on E is compatible with the hermitian structure of E, or a metric connection if for any s ∈ Ap(M,E) and t ∈ Aq(M,E) we have

d{s, t} = {∇s, t} + (−1)p{s, ∇t}

Proposition 0.4. If ∇ is a metric connection, then

Θ∗ = −Θ with respect to any trivialization. Proof. It is obvious that

0 = d2{s, t} = {∇2s, t} + {s, ∇2t} =⇒ Θ∗ = −Θ

In the local holomorphic coordinates {zi} of M, we have

∂ p ∂ ∂ i {s, t} = {∇ s, t} + (−1) {s, ∇ t} ∂z ∂zi ∂zi

If (E, ∇, h) is a Hermitian complex vector bundle with a metric connection, there is a decomposition ∇ = ∇0 + ∇00, where

∇0 : Ap,q(M,E) −→ Ap+1,q(M,E), ∇00 : Ap,q(M,E) −→ Ap,q+1(M,E) under the decomposition M Al(M,E) = Ap,q(M,E), p+q=l and ∇0(f ∧ s) = ∂f ∧ s + (−1)deg f f ∧ ∇0s, ∇00(f ∧ s) = ∂f ∧ s + (−1)deg f f ∧ ∇00s. 3

i In the local holomorphic coordinates {z }i=1,··· ,n on M, and local holomorphic frame {eα}α=1,··· ,r, we set ∇e = (Γβ dzi + Γβ dzl) ⊗ e α αi αl β then ∇0e = Γβ dzi ⊗ e , ∇00e = Γβ dzl ⊗ e α αi β α αl β The formal adjoint operators are denoted by

δ0 : Ap+1,q(M,E) −→ Ap,q(M,E), δ00 : Ap,q+1(M,E) −→ Ap,q(M,E) with respect to the inner product induced by h. The Laplacian operators are deﬁned by

∆0 = ∇0δ0 + δ0∇0; ∆00 = ∇00δ00 + δ00∇00; ∆ = ∇δ + δ∇

For the metric connection ∇ on the Hermitian complex vector bundle (E, h), we have the decomposition of the curvature Θ = Θ2,0 + Θ1,1 + Θ0,2 where Θ2,0 = ∇02 ∈ Γ(M, Λ2,0T ∗M ⊗ E∗ ⊗ E) Θ1,1 = ∇0∇00 + ∇00∇0 ∈ Γ(M, Λ1,1T ∗M ⊗ E∗ ⊗ E) Θ2,0 = ∇002 ∈ Γ(M, Λ0,2T ∗M ⊗ E∗ ⊗ E) i In the local holomorphic coordinates (z ) of M and coordinates (eα) of E, we can write

Θ2,0 = Rβ dzi ∧ dzj ⊗ eα∗ ⊗ e ;Θ0,2 = Rβ dzi ∧ dzj ⊗ eα∗ ⊗ e ijα β ijα β

Θ1,1 = Rβ dzi ∧ dzj ⊗ eα∗ ⊗ e ijα β Here and henceforth we sometimes adopt the Einstein convention for summation. On the holomorphic vector bundle E −→π M, we can deﬁne the ∂-operator:

∂ : Ap(M,E) −→ Ap+1(M,E)

p If we take the holomorphic frame e = (e1, ··· , en) for E over U, then write s ∈ A (M,E) as X i s = σ ⊗ ei and set i ∂s = ∂σ ⊗ ei Proposition 0.5. The ∂-operator is well deﬁned.

0 0 0 Proof. If e = (e1, ··· , en) is any other holomorphic frame of E over U, with

0 ei = gijej

i i0 0 i0 P j If s = σ ⊗ ei = σ ⊗ ei, then σ = j gjiσ then we have i0 0 j 0 j 0 j ∂s = ∂σ ⊗ ei = ∂(gjiσ ) ⊗ ei = gij∂σ ⊗ ei = ∂σ ⊗ ej since gij is holomorphic. 4

Lemma 0.6. If E is a hermitian holomorphic vector bundle, then there exists a unique connection such that: i. ∇ is compatible with the complex structure, i.e. ∇00 = ∂ ii. ∇ is compatible with the hermitian structure, i.e.

d{s, t} = {∇s, t} + (−1)deg s{s, ∇t}

for any s, t ∈ A•(M,E).

Proof. Let e = (e1, ··· , er) be a holomorphic frame for E under some trivialization, and let hαβ = heα, eβi. If the metric connection ∇ exists, then its connection matrix ω must be type (1, 0), for the (0, 1) part of ω for the ∂ is zero, i.e. Γβ = 0. Then we have: αl γ γ dhαβ = dheα, eβi = ωαhγβ + hαγωβ so we have γ γ γβ ∂hαβ = ωαhγβ, i.e. ωα = ∂hαβh γ γ γβ ∂hαβ = hαγωβ, i.e. ωα = ∂hαβh If we set ω = h−1 ·∂h, we know ω satisﬁes the condition above. Then the connection is determined by ω. Remark 0.7. Here h−1 = (hαβ) which is the inverse of ht, that is X αγ γ hαβh = δβ

Now we compute the curvature in the local coordinates. In the local holomorphic coordinates i {z }i=1,··· ,n on M, and local holomorphic frame {eα}α=1,··· ,r, we have X X Θ = Θβ eα∗ ⊗ e = Rβ dzi ∧ dzj ⊗ eα∗ ⊗ e ∈ Γ(M, Λ1,1T ∗M ⊗ E∗ ⊗ E) α β ijα β α,β By the relation Θ = dω − ω ∧ ω, we get

β β γ β Θα = dωα − ωα ∧ ωγ On the other hand ∂h ωβ = hβγ αγ dzi α ∂zi we get ∂2h ∂h ∂h ∂h ∂h dωβ = −hβγ( αγ − hδµ αµ δγ )dzi ∧ dzj + hγδ αδ hβλ γλ dzi ∧ dzj α ∂zi∂zj ∂zi ∂zj ∂zi ∂zj and ∂h ∂h ωγ ∧ ωβ = hγδ αδ hβλ γλ dzi ∧ dzj α γ ∂zi ∂zj then ∂2h ∂h ∂h Rβ = −hβγ( αγ − hδµ αµ δγ ). ijα ∂zi∂zj ∂zi ∂zj where we use the formula αβ αγ δγ ∂h = −h ∂hδγh 5

Here use the formal expression, we have

ω = h−1 · ∂h then dω = ∂(h−1∂h) + ∂h−1 ∧ ∂h For ω ∧ ω = ∂h−1 ∧ ∂h Then Θ = ∂(h−1 · ∂h) If r = 1, that is E is a line bundle, then

Θ = ∂∂ log h

Now we set R = Rγ h ijαβ ijα γβ then ∂h ∂h h R = − αβ + hδγ αγ δβ ijαβ ∂zi∂zj ∂zi ∂zj

Theorem 0.8 (Normal frames on holomorphic vector bundles). For every point x0 of X and every local holomorphic coordinates (z1, ··· , zn) at x0, there exists a holomorphic frame (e1, ··· , er) in a neighborhood Ω of x0 such that Xn 3 heλ(z), eµ(z)i = δλµ − Rjkλµzjzk + O(|z| ) j,k=1

E where the (Rjkλµ) are the coeﬃcients of the Chern curvature tensor Θx0 under the above local frame. Proof. By linear transformations.

Lemma 0.9. If L is a holomorphic line bundle over a complex manifold M with trivialization ∗ functions ϕi : L|Ui −−→ Ui × C and transition functions gij : Ui ∩ Uj −−→ C .

+ −1 2 (1) If ( , )L is a hermitian metric on L and µj : Uj −−→ R is given by µj(p) = kϕj (p, 1)k . Then 2 µj = µi|gij| , on Ui ∩ Uj

+ (2) Any collections µi : Ui −−→ R satisﬁes

2 µj = µi|gij| , on Ui ∩ Uj

deﬁnes a hermitian metric on L.

(3) The unique Chern connection on L has the curvature √ Θ = − −1∂∂ log µi on Ui 6

Proof. (1) If p ∈ Ui ∩ Uj,

−1 2 −1 −1 2 µj(p) = khj (p, 1)k = khi ◦ hi ◦ hj (p, 1)k −1 2 2 −1 2 = khi (p, gij(p))k = |gij(p)| khi (p, 1)k 2 = µi(p)|gij(p)|

−1 (2) If p ∈ Ui and v1, v2 ∈ Lp, then vi = ϕj (p, ti) for i = 1, 2. We can deﬁne a hermitian metric ( , )L on L by (v1, v2)L = µi(p)t1t2

It is a well-defined hermitian metric. In fact, if p ∈ Ui ∩ Uj, by the definition of line bundles, 0 0 there exists t1, t2 such that −1 −1 0 vi = ϕi (p, ti) = ϕj (p, ti) for i = 1, 2 where 0 ti = gij(p)ti By definition 0 0 −2 (v1, v2)L = µj(p)t1t2 = µj(p)|gij(p)| t1t2 = µi(p)t1t2 So the hermitian metric is well-defined on L.

(3) We just need to check that the curvature matrix is globally deﬁned. We have

∂∂ log µj = ∂∂ log µi

for gij is holomorphic.

n Example 0.10 (Universal line bundle OPn (−1) of P ). We have a construction for the universal bundle J as follows:

n n+1 J = {[Z0,Z1, ··· ,Zn], λ(Z0,Z1, ··· ,Zn)) ∈ P × C | λ ∈ C}

n n+1 Proof. Set L = {([Z0,Z1, ··· ,Zn], λ(Z0,Z1, ··· ,Zn)) ∈ P × C | λ ∈ C}. We have [n n P = Ui i=0

n where Ui = {Zi 6= 0} ⊂ P . Then the trivialiZation functions

∼ n+1 ϕi : L|Ui −−→ Ui × C = C where ϕi([Z0,Z1, ··· ,Zn], λ(Z0,Z1, ··· ,Zn))) = ([Z0,Z1, ··· ,Zn], λZi) or equivalently

\ ϕi([Z0,Z1, ··· ,Zn], λ(Z0,Z1, ··· ,Zn))) = (Z0/Zi,Z1/Zi, ··· , Zi/Zi, ··· Zn/Zi, λZi)

It is a biholomorphic map and the inverse

−1 ϕi : Ui × C −−→ L|Ui is given by −1 ϕi ([Z0,Z1, ··· ,Zn], λZi) = ([Z0,Z1, ··· ,Zn], λ(Z0,Z1, ··· ,Zn))) 7 or equivalently −1 ϕi (Z1,Z2, ··· ,Zn, λ) = ([Z1,Z2, ··· , 1, ··· Zn], λ(Z1,Z2, ··· , 1, ··· Zn)) So we get −1 ϕi ◦ ϕj ([Z0,Z1, ··· ,Zn], λZj) = ([Z0,Z1, ··· ,Zn], λZi) The transition functions are given by

gij([Z0,Z1, ··· ,Zn]) = Zi/Zj on Ui ∩ Uj. n Example 0.11 (Metric on the tautological line bundle OPn (1) of P ). The tautological line bundle L is the dual line bundle of the universal line bundle J = OPn (−1), and we denote it by L = OPn (1). The line bundle L, dual bundle of J, by deﬁnition, it represented the transition function Zj g˜ij = Zi Now we construct a hermitian metric on L. On the chart Ui, we set 1 hi = ¯ ¯2 Pn ¯ ¯ ¯ Zk ¯ k=0 Zi Then ¯ ¯ ¯ ¯2 ¯Zj ¯ 2 hj = hi ¯ ¯ = hi|gij| Zi So we deﬁne a metric on the line bundle L. The curvature of this metric is given by ¯ ¯ Xn ¯ ¯2 ¯Zk ¯ ω = ∂∂ log( ¯ ¯ ) Zi k=0

For example, we choose i = 0 and set zi = Zi/Z0, i = 1, ··· , n. Then Xn 2 ω = ∂∂ log(1 + |zi| ) i=1 which is the Fubini-Study metric on T 1,0Pn. 1,0 n Example 0.12 (Curvature of the Fubini-Study metric on T P ). On the chart U0 2 Pn 2 ∂ log(1 + i=1 |zi| ) hij = ∂zi∂zj

Now we choose the point P = [1, 0, ··· , 0] = (0, ··· , 0) ∈ U0. By Taylor’s expansion, we have X X X 2 2 2 2 6 log(1 + |zi| ) = |zi| − ( |zi| ) + O(|z| ) Then, around the point P , 4 hij = δij − (δijδkl + δilδjk)zkzl + O(|z| ) that is

Rijkl(P ) = δijδkl + δilδjk = hij(P )hkl(P ) + hil(P )hkj(P ) In fact, by a linear transformation,

Rijkl = hijhkl + hilhkj for any P ∈ U0, and we will prove it later.