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

No threshold can work uniformly over all holomorphic vector bundles $E$. Given a proposed threshold $M$, choose an integer $m\ge M$ and let $E=F^m$. The canonical trivialization of a line bundle tensored with its dual givesThis 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 $E$.