Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 115 / 4 / d

Table of contents
- Solution d

Solution

A form is harmonic when

Δ α = (d δ + δ d) α = 0.

Adjointness gives

⟨ Δ α, α ⟩_{X} = ∥ d α ∥^{2} + ∥ δ α ∥^{2} .

Therefore

Δ α = 0

implies

d α = 0

and

δ α = 0

; the converse follows immediately from the definition of

Δ

.

The Hodge decomposition theorem states in particular that every de Rham cohomology class has a unique harmonic representative. Hence

H^{p} (X) \sim H_{dR}^{p} (X), α ⟼ [α] .

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