Past exam of the mathematics course of the University of Cambridge / 2015 / iii / Paper 18 / 4 / d

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- Solution d

Solution

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}

. The canonical trivialization of a line bundle tensored with its dual gives

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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