 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2018/iii/paper-118/5/b/solution

Codex  0 Created 2026-10-03 Updated 2026-10-05

C^{2}

, centred at zero. A holomorphic function has

\partial f = 0

exactly when it is constant on this neighbourhood, giving exactness at

O_{M}

and the injection of the constant sheaf.

For exactness at

Ω_{M}^{1}

, let

α = a_{1} (z) d z_{1} + a_{2} (z) d z_{2}

be holomorphic and

\partial

-closed. Then

\partial a_{1} / \partial z_{2} = \partial a_{2} / \partial z_{1}

F (z) = \int_{0}^{1} (z_{1} a_{1} (t z) + z_{2} a_{2} (t z)) d t .

Differentiation under the integral and the closedness identity give

\frac{\partial F}{\partial z _{j}} = \int_{0}^{1} (a_{j} (t z) + t \sum_{i} z_{i} \frac{\partial a _{j}}{\partial z _{i}} (t z)) d t = [t a_{j} (t z)]_{t = 0}^{t = 1} = a_{j} (z) .

Thus

\partial F = α

For exactness at

Ω_{M}^{2}

β = b (z) d z_{1} \land d z_{2}

and is automatically

\partial

γ (z) = \int_{0}^{1} t b (t z) (z_{1} d z_{2} - z_{2} d z_{1}) d t .

\partial γ = \int_{0}^{1} (2 t b (t z) + t^{2} \sum_{i} z_{i} \frac{\partial b}{\partial z _{i}} (t z)) d t d z_{1} \land d z_{2} = [t^{2} b (t z)]_{0}^{1} d z_{1} \land d z_{2} = β .

These local primitives prove the holomorphic Poincaré lemma in the degrees needed here. Exactness on every stalk gives

0 ⟶ \underline{C} ⟶ O_{M} \partial Ω_{M}^{1} \partial Ω_{M}^{2} ⟶ 0.

This is sheaf exactness; the local primitives need not glue to global ones.

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