COMPLEX GEOMETRY Course notes

3.2 Kähler manifolds
Let V be a complex vector space with J = √ −1 and W = Hom R (V, R). Recall V C := V ⊗ C = V 1,0 ⊕ V 0,1 .
Hence also W C = W 1,0 ⊕ W 0,1 ⊇ W .
Definition 3.2.1. Let W 1,1 = W 1,0 ⊗ W 0,1 ⊆ ∆ 2 W C ⊇ ∆ 2 W ,
W 1,1 = {(1, 1)-forms} = {sesqui-linear forms on V }.
Let W 1,1
R
= W 1,1 ∩ ∆ 2 W = {real (1, 1)-forms} = {real 2-forms of type (1, 1)} = {alternating forms}. A
(1, 1)-form h ∈ W 1,1 is called Hermitian if h(u, v) = h(v, u) for every u, v ∈ V . Let W 1,1
H
be the space of
such forms.
Fact 3.2.1. There exists a bijective correspondence between Hermitian forms and real alternating forms of
type (1, 1) via
W 1,1
1,1
H
∋ h ←→ Im(h) ∈ WR
Proof: Since h(u, v) = h(v, u), we have that Im(h) is alternating on V , i.e., Im(h) ∈ ∆ 2 W . Conversely,
let ω ∈ W 1,1
R
and set
g(u, v) = ω(u, Jv) = −ω(Ju, v) and
h(u, v) = g(u, v) − iω(u, v).
Then g(u, v) = g(v, u) and thus h(u, v) = h(v, u), i.e., h is Hermitian.
Locally, ω = ∑ i
2 a ijdz i ∧ dz j = −Im(h) ∈ Ω 1,1
X
∩ Ω2 X,R , where (a ij) is hermitian.
Definition 3.2.2. ω ∈ W 1,1
R
is positive if the correspondence h is positive definite.
Definition 3.2.3. A positive real (1, 1)-form on an almost complex manifold (X, J) is a C ∞ associated of
a positive real (1, 1)-form on each tangent space T X,x , x ∈ X.
Definition 3.2.4. A Hermitian metric on a complex vector bundle E over a smooth manifold M is an
element h ∈ Γ(E ⊗ E) ∗ . A Hermitian manifold is a complex manifold with a Hermitian metric on its
holomorphic tangent space. Likewise, an almost Hermitian manifold is an almost complex manifold with
a Hermitian metric on its holomorphic tangent space.
Corollary 3.2.1. There exists a bijective correspondence between real (1, 1)-forms ω on a complex manifold
M and Hermitian metrics on M.
Definition 3.2.5. Let h be a Hermitian metric. We shall say that h is Kähler if ω = Im(h) is closed.
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