Let $M$ be a (connected) complex manifold, $L$ be a local system on $M$ and $\mathcal{L}$ the vector bundle associated to $L$. If $L$ is indecomposable, does it imply that $\mathcal{L}$ is also indecomposable?



Indecomposability of local systems and vector bundles
No, this does not hold in general. For example, there do exist irreducible flat connections on the trivial holomorphic bundle of rank two over a compact Riemann surface of genus $g\geq2.$
On the other hand, if you consider irreducible local systems with unitary monodromy over compact Riemann surfaces then the associated vector bundle will be stable, hence indecomposable.


