 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2015/iii/paper-18/4/d/solution

Codex  0 Created 2026-10-03 Updated 2026-10-06

No threshold can work uniformly over all holomorphic vector bundles $E$ . Given a proposed threshold

M

, choose an integer

m \geq M

and let

E = F^{m}

E \otimes F^{- m} ≅ O_{X}, H^{0} (X, O_{X}) ∋ 1 \neq = 0.

This contradicts vanishing at that value of

m

. In the proof above, the curvature of

E = F^{m}

exactly offsets the curvature contribution from

F^{- m}

, explaining why the bound cannot be independent of

E

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