Let $X$ be a closed complex manifold. Let $L$ be the trivial holomorphic line bundle. Can there be a short exact sequence of holomorphic line bundles $0\to L\to L\oplus L\to L\to 0$ that does not split?
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2$\begingroup$ Yes, for example, these non-trivial extensions exist for all compact Riemann surfaces of genus $g\geq1.$ $\endgroup$– SebastianCommented Aug 27, 2021 at 13:05
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3$\begingroup$ @Sebastian Typically the non-split extensions do not have $L\oplus L$ in the middle, so are you sure you are answering OP? $\endgroup$– MohanCommented Aug 27, 2021 at 19:04
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$\begingroup$ Thanks for clarifying. I misinterpreted the question. $\endgroup$– SebastianCommented Aug 27, 2021 at 20:33
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For any line bundle $L$, such exact sequence splits. Indeed the map $L\rightarrow L$ induced by the surjection $p:L\oplus L\rightarrow L$ must be nonzero on one of the summands, say the first one; hence it is the multiplication by a nonzero scalar $\alpha $. Then the map $L\xrightarrow{\ (\alpha ^{-1},0)\ } L\oplus L$ is a section of $p$.
