COMPLEX GEOMETRY Course notes

2.13 Analysis on the Hilberts space of differentials
There exists a Hermitian inner product for 1-forms ω 1 and ω 2 (at least one which is compactly supported)
(ω 1 , ω 2 ) = ∫ Z ω 1 ∧ ∗ω 2 < ∞ .
Locally, ω i = p i dx + g i dy, with i = 1, 2. Then
ω 1 ∧ ∗ω 2 = (p 1 p 2 + g 1 g 2 )dx ∧ dy.
Hence we can define
∫
||ω|| 2 L = ω ∧ ∗ω < ∞.
2
Definition 2.13.1. Let B ′ be the space of bounded 1-forms ω such that ||ω|| 2 < ∞.
Z
With respect to the L 2 -norm, B 1 is a Hilbert space.
Let E be the closure in B 1 of dA 0 C , where A0 C is the space of C∞ -functions with compact support.
Theorem 2.13.1 (Orthogonal decomposition). Let ω ∈ B 1 . Then there exists a unique orthogonal decomposition
ω = ω h + df + ∗dg
where ω h is bounded harmonic and f, g ∈ A 0 , and df, dg ∈ E.
Proof: The essential point is that the space H is orthogonal to both E and ∗E, and E ⊥ ∗E.
ψ, ϕ ∈ A 0 then
∫
∫ ∫
〈 〉
dϕ, ∗dψ = − dϕ ∧ dψ = ψddϕ + d(ψdϕ)
Z
Z
Z
= 0 + 0.
If
Similarly, saying that ω is closed means that it is orthogonal to ∗E, and coclosed means that it is
orthogonal to E. For example,
∫
∫
0 = 〈d ∗ ω, ϕ〉 = d ∗ ω ∧ ϕ = dϕ ∧ ∗ω = 〈dϕ, ω〉
= 〈ω, dϕ〉 .
Hence
H ⊕ ⊥ E ⊕ ⊥ ∗E ↩→ B 1 .
To show the equality, we go to the L 2 -completion of B ′ first.
Theorem 2.13.2 (Regularity). H = (Ě ⊕ ∗Ě)⊥ in ˇB 1 where ( ∨ ) means L 2 -completion.
34