COMPLEX GEOMETRY Course notes

3.4 Review
A complex structure on a real manifold M of dimension 2n is an endomorphism J of T M such that J 2 = −1.
If M is complex, normally we take J = √ −1. A real 2-form h is Hermitian if h(u, v) = h(v, u). It is know
that a form h is Hermitian if and only if h is a positive (1, 1)-form.
Theorem 3.4.1. There exists a bijective correspondence between real alternating forms of type (1, 1) and
Hermitian metrics. Such a correspondence is given by
where Im(h) is a symplectic 2-form.
W 1,1
H
∼
−→ W 1,1
R
h ↦→ Im(h)
Theorem 3.4.2. The following conditions are equivalent for a complex Hermitian manifold (X, h):
(i) h is a Kähler metric, i.e., dw h = 0.
(ii) J is flat for the Levi-Civita connection of h.
(iii) The Chern connection of h on T 1,0
M equals the Levi-Civita connection on T R M .
Proof:
• (iii) =⇒ (ii): It is clear because the Chern connection is C-linear by definition.
• (ii) =⇒ (i): Condition (ii) means that the Levi-Civita connection commutes with J. Then
dω(ϕ 1 , ϕ 2 ) = ω(∇ϕ 1 , ϕ 2 ) + ω(ϕ 1 , ∇ϕ 2 ).
Let C ∞ (M) ∋ ϕ[ω(ϕ 1 , ϕ 2 )] = ω(∇ ϕ ϕ 1 , ϕ 2 ) + ω(ϕ 1 , ∇ ϕ ϕ 2 ). Since
dω(ϕ, ϕ 1 , ϕ 2 ) = ϕω(ϕ 1 , ϕ 2 ) − ϕ 1 ω(ϕ, ϕ 2 ) + ϕ 2 ω(ϕ 1 , ϕ) − ω([ϕ, ϕ 1 ], ϕ 2 )
the result follows from [ϕ i , ϕ j ] = ∇ ϕ1 ϕ j − ∇ ϕ2 ϕ 1 .
• (i) =⇒ (iii): The Chern connction equals the Levi-Civita connection for the flat metric ∑ i dz i ∧dz i .
The result follows from the following proposition.
Proposition 3.4.1. If (X, h) is a Kähler manifold and if x ∈ X, then there exists a holomorphic coordinate
(z 1 , . . . , z n ) centred at x such that
( ) ∂ ∂
h ij = h , = Im + O( ∑ |z i | 2 ).
∂z i ∂z j
The converse is also true.
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