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?



Short exact sequence of trivial holomorphic line bundles not splitting
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 $p$.


