COMPLEX GEOMETRY Course notes

Corollary 3.2.2. If a symplectic structure ω on a complex manifold is positive of type (1, 1) (i.e., it vanishes
on Ω 2,0 and hence also on Ω 0,2 and its associated h is positive definite), then it is −Im(h) for a Kähler metric.
Corollary 3.2.3. A Hermitian metric can always be written as h = g + iω where g is a Riemannian metric
invariant under J (g(u, v) = g(Ju, Ju)) and ω is a positive (1, 1)-form, g(u, v) = ω(u, Jv).
Definition 3.2.6. A pair (X, ω) formed by a complex manifold X and a positive (1, 1)-form ω is called a
Kähler manifold.
Lemma 3.2.1. dVol h = ωn
n!
for (X, h), h = h ω = g(u, v), where ω n = ω ∧ · · · ∧ ω of type (n, n).
Proof: Let {e i } be an orthonormal basis of T X,x with respect to h. Then {e i , Je i } is a real orthonormal
basis for TX,x R ωn
with respect to g with positive orientation. It suffices to check
n!
= dx 1 ∧dy 1 ∧· · ·∧dx n ∧dy n
where
∑
{dx 1 , dy 1 , . . . , dx n , dy n } is the dual basis to (e i , Je i ). Let dz j = dx j + dy j . Then we have ω x =
j dz j ∧ dz j and
i
2
ω n x
n!
( ) n i ∏
= dz j ∧ dz j at x,
2
i
2 dz j ∧ dz j = dx i ∧ dy j .
j
Corollary 3.2.4. If X (n) is a compact Kähler manifold then [ω k ] ∈ HdR 2k (X) is nonzero for every k < n.
Proof: ω k = dγ implies ω n = d(ω n−k ∧ γ). The last implies 0 ≠ ∫ X ωn = 0, getting a contradiction.
Corollary 3.2.5. Let X (k) be a compact Kähler submanifold M. Then [x] ∈ H 2k (X) is nonzero.
Proof: Clearly h M | T X = h X and i ∗ ω (M,h) = ω (X,hX ), where i : X ↩→ M is the inclusion. If i(X) = ∂Γ,
then by the Stokes Theorem
∫ ∫
Volume X = i ∗ ωM h = dωM h = 0 since dωM h ≡ 0.
X
Γ
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