ζn) be the homoge- neous coordinates on Pn

1. Fubini-Study Metric

Let Pⁿ be the n-dimensional complex projective space and (ζ₀ : · · · : ζ_n) be the homoge- neous coordinates on Pⁿ. For 0 ≤ α ≤ n, we denote the open set

U_α = {(ζ₀ : · · · : ζ_n) : ζ_α 6= 0}.

Let ϕ_α: U_α→ Cⁿ be the coordinate chart defined by ϕα(ζ0 : · · · : ζn) = ζ₀

ζ_α, · · · ,ζα−1

ζ_α ,ζα+1

ζ_α , · · · ,ζn

ζ_α

 .

In other words, we define a local coordinate system zα= (z_α¹, · · · , z_αⁿ) on Uα by zⁱ_α(ζ₀ : · · · : ζ_n) =

(_ζ

i−1

ζα if i < α

ζi

ζα if i > α.

For 1 ≤ i, j ≤ n, we define a function on Cⁿ by hij(z) = (1 +Pn

k=1|z^k|²)δ_ij− zⁱz^j (1 +P_n

k=1|z^k|²)² .

Proposition 1.1. The matrix valued function H(z) = (h_ij(z))ⁿ_i,j=1 is smooth and H(z) is a positive definite Hermitian matrix for all z ∈ Cⁿ.

Proof. Since each complex value function h_ij(z) is smooth on Cⁿ, z 7→ H(z) is smooth.

Moreover,

h_ij(z) = h_ji, 1 ≤ i, j ≤ n.

Hence H(z) is Hermitian for all z ∈ Cⁿ.

Let h·, ·i denote the standard inner product on Cⁿ and k · k the norm induced from it.

For each z, w ∈ Cⁿ, we have

hH(z)w, wi = (1 + kzk²)kwk²− |hz, wi|² (1 + kzk²)² . By the Cauchy-Schwarz inequality,

|hz, wi|² ≤ kzk²kwk². Hence we find

hH(z)w, wi ≥ kwk²

(1 + kzk²)² ≥ 0.

We find that hH(z)w, wi = 0 implies that kwk² = 0. Therefore w = 0. This shows that

H(z) is positive definite for each z ∈ Cⁿ. 

Let us define a function K on Cⁿ by

K(z) = log 1 +

n

X

k=1

|z^k|²

! . Let us compute ∂K:

∂K = Pn

j=1z^jdz^j 1 +Pn

k=1|z^k|².

1

2

Hence ∂∂K is given by

∂∂K =

n

X

i,j=1

δ_ij(1 +Pn

k=1|z^k|²) − zⁱz^j (1 +Pn

k=1|z^k|²)² dzⁱ∧ dz^j

=

n

X

i,j=1

hij(z)dzⁱ∧ dz^j.

For each 0 ≤ α ≤ n, we define a (1, 1)-form on U_α by ω_α =√

−1

n

X

i,j=1

(h_ij ◦ ϕ_α)dz_αⁱ ∧ dz^j_α. This implies that if we denote K_α = K ◦ ϕ_α on U_α, then

ω_α =√

−1∂∂K_α. Lemma 1.1. ∂² = 0 and ∂² = 0. We also have ∂∂ = −∂∂.

Proof. This can be proved by direct computation. 

Using the decomposition d = ∂ + ∂, and Lemma 1.1, we find dωα =√

−1∂²∂Kα+√

−1∂∂∂K_α

= −√

−1∂∂²K_α

= 0.

This shows that ω_α is a closed (1, 1) one-form on U_α for each α. In fact, we can show that the (1, 1)-form on each U_α can be glued together to obtain a closed global (1, 1)-form ω on Pⁿ.

On U_α, let us rewrite K_α as follows:

Kα(ζ0: · · · : ζn) = log 1 +

n

X

k=1

|z_α^k(ζ0: · · · : ζn)|²

!

= log

n

X

i=0

|ζ_i|²

!

− log |ζ_α|². Assume that α < β. Then on Uα∩ U_β,

Kα(ζ0 : · · · : ζn) − Kβ(ζ0: · · · : ζn) = log |ζβ|²− log |ζ_α|²= log    

ζ_β ζα

2

= log  

z_α^β(ζ₀: · · · : ζ_n)   

2

. In other words, on U_α∩ U_β,

Kα− K_β = log   z_α^β

2

. On Uα∩ U_β,

∂(Kα− K_β) = 1 z_αdz^β_α

and hence ∂∂(K_α− K_β) = 0 on U_α∩ U_β. In other words, on U_α∩ U_β,

∂∂Kα = ∂∂Kβ.

3

Hence ωα = ω_β on Uα∩ U_β. We obtain a global (1, 1)-form ω. Since dωα = 0 on each Uα, the (1, 1)-form ω is a closed form on Pⁿ. Since ω is globally defined, the 2-tensor

(1.1)

n

X

i,j=1

(h_ij ◦ ϕ_α)dzⁱ_α⊗ dz^j_α

is also globally defined. Since we have already verify that H(z) is a positive definite matrix on Cⁿ, (1.1) defines a Hermitian metric on Pⁿ whose associated (1, 1)-form is ω. Thus we conclude that

Theorem 1.1. Pⁿ is a Kahler manifold.

The Kahler metric on Pⁿ defined in (1.1) is called the Fubini-Study metric.

2. Case for P¹.

Let (ζ₀, ζ₁) be the standard coordinate on C²and (ζ₀ : ζ₁) be the corresponding coordinate on P¹. Denote U = {(ζ0 : ζ1) : ζ0 6= 0} and V = {(ζ₀ : ζ1) : ζ1 6= 0}. {U, V } forms an open covering of P¹. Let z be the coordinate function on U and w be the coordinate function on V. In other words,

z(ζ0 : ζ1) = ζ1

ζ0

, w(ζ0 : ζ1) = ζ0

ζ1

. Hence on U ∩ V, we have the coordinate transformation

w = 1 z.

Let K_U = log(1 + |z|²) on U and K_V = log(1 + |w|²) on V. Observe that

KU(ζ0 : ζ1) = log(|ζ0|²+ |ζ1|²) − log |ζ0|², KV(ζ0: ζ1) = log(|ζ0|²+ |ζ1|²) − log |ζ1|². We see that on U ∩ V,

K_U(ζ₀ : ζ₁) − K_V(ζ₀: ζ₁) = log    

ζ₁ ζ0

2

= log |z(ζ₀: ζ₁)|². Hence we obtain that on U ∩ V

K_U− K_V = log |z|². By |z|² = zz, we have

∂(KU − K_V) = 1 zdz.

Hence we find

∂∂(KU− K_V) = ∂ 1 z



∧ dz = 0.

This shows that on U ∩ V,

∂∂K_U = ∂∂K_V.

This {(∂∂K_U, U ), (∂∂K_V, V )} gives us a globally defined (1, 1)-form ω defined by ω =

(√−1∂∂K_U on U ,

√−1∂∂K_V on V .

4

Let us compute ω. We know ∂K_U = z

1 + |z|²dz, and hence

∂∂K_U = ∂

 z

1 + |z|²



∧ dz

= (1 + |z|²)∂z − z∂(1 + |z|²) (1 + |z|²)² ∧ dz

= (1 + |z|²)dz − |z|²dz (1 + |z|²)² ∧ dz

= 1

(1 + |z|²)²dz ∧ dz.

Similarly,

∂∂K_V = 1

(1 + |w|²)²dw ∧ dw.

Hence the (1, 1)-form ω is given by ω =

( ^√₋₁

(1+|z|²)²dz ∧ dz on U ,

√−1

(1+|w|²)²dw ∧ dw on V . The Kahler metric ds² on Pⁿ is given by

ds² =

( ₁

(1+|z|²)²dz ⊗ dz on U ,

1

(1+|w|²)²dw ⊗ dw on V .

Let us verify this metric is globally defined. On U ∩ V, we have w = 1/z and w = 1/z.

Hence on U ∩ V,

dw ⊗ dw = dz z² ⊗dz

z² = 1

|z|⁴dz ⊗ dz.

Moreover, on U ∩ V,

1

(1 + |w|²)² = 1 (1 +

¹_z  

2)² = |z|⁴ (|z|²+ 1)². On U ∩ V,

1

(1 + |z|²)²dz ⊗ dz = |z|⁴ (1 + |z|²)²

dz ⊗ dz

|z|⁴ = 1

(|w|²+ 1)²dw ⊗ dw.

We prove that ds² is a globally defined Hermitian metric.