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Given a holomorphic line bundle $L$ on a Riemann surface $M\,,$ with $L^2$ nontrivial, when is it true that the only holomorphic subbundles of $$L\oplus L^{-1}$$ are $L\oplus\{0\}$ and $\{0\}\oplus L^{-1}\,?$. I have seen things like this stated for genus $\ge 2$ but I'm not sure how to see that it is true.

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  • $\begingroup$ In general, this is false. Given any rank two vector bundle, you can find line sub bundles of arbitrarily large (or should I say small?) negative degrees. $\endgroup$ – Mohan Jun 17 '17 at 18:45

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