COMPLEX GEOMETRY Course notes

Chapter 5
HARMONIC FORMS
5.1 Harmonic forms on compact manifolds
Let (X, g) be a Riemannian manifold, where X compact is a blanket assumption. Then we have a metric
( , ) on ∆ ∗ TX,x ∨ . We assume X is oriented. Let α, β ∈ Ak : C ∞ (A k TX ∨ ). Then
∫
〈α, β〉 =
X
〈α, β〉 x
dVol(x)
gives an L 2 -metric on A k . We also have a pointwise isomorphism p : ∆ n−k Tx
∨ −→ Hom(∆ k Tx ∨ , ∆ n Tx ∨ ) given
by v ↦→ v ∧ −, where ∆ n Tx
∨ = RdVol(x), and an isomorphism m : ∆ k Tx
∨ ∼
−→ Hom(∆ k T ∨ , R) given by
e ↦→ 〈e, 〉 ∆ k T ∨.
∼
Definition 5.1.1. The Hodge Star Operator is given by
and the associated global isomorphism by
∗ = p −1 ◦ m : ∆ k T ∨ x
∗ : ∆ k T ∨
∼
−→ ∆ n−k T ∨ x
∼
−→ ∆ n−k T ∨
Ω k (X) −→ Ω n−k (X)
We extend ∗ to complex-valued forms by extending 〈 , 〉 to Hermit metrics on ∆ k C (T ∨ ⊗ C) = (∆ k R T ∨ ) ⊗ C.
We get (α, β) x dVol(x) = α x ∧ β x and so
∫
〈α, β〉 = α ∧ ∗β
is the L 2 -metric on A k C = Ak ⊗ C. In the case X is complex, A k C = ⊕ p+q=k ⊕ A p,q , A p,q = C ∞ (Ω p,q
X ).
X
Fact 5.1.1. The Stokes Theorem implies that (α, d ∗ β) = (dα, β) where d ∗ := (−1) k ∗ −1 d∗.
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