COMPLEX GEOMETRY Course notes

3.5 The Fubini Study metric
Let L = {(l, v) ∈ CP n × C n / v ∈ l}. Consider the diagram
i π
L CP n 2
× C n+1 C n+1
π 1 ◦ i
π 1
CP n
The composite map π 2 ◦ i is called the blow up at 0. We have that L is a holomorphic line bundle over CP n
and is denoted by O(−1).
Definition 3.5.1. O CP n(h) := L −k where L −k := (L ∨ ) ⊗k for k > 0, is a holomorphic line bundle over CP 1 .
The standard metric ∑ |z i | 2 on CP n+1 restricts to a Hermitian metric on L. Its curvature (Ricci or Chern
form) is given by
ω = σ ∗ 2
2π ∂∂log|z i| i =
i
2π ∂∂log|σ|2
for any choice of a holomorphic section σ of L over CP n . Therefore, σ ′ = σf, for f ∈ O and so
where log|f| 2 = logf + logf and it is ∂∂-closed.
log(σ ′ ) 2 = log|σ| 2 + log|f| 2 ,
Lemma 3.5.1. ω is a positive (1, 1)-form.
Proof: We prove only the case n = 1. We have
So
ω =
∂log(1 + |z| 2 ) = ∂(1 + |z|2 )
1 + |z| 2 = zdz
1 + |z| 2 .
i [(1 + |z| 2 )dz ∧ dz − zdz ∧ zdz]
2π (1 + |z| 2 ) 2 = i dz ∧ dz
2π (1 + |z| 2 ) 2
and the conclusion follows from the transitivity of SU(n + 1) on T CP n .
Definition 3.5.2. ω is called the Fubini study metric in CP n and is denoted ω F S . It depends on the
choice of coordinates on C n+1 .
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