Let $\pi:E\to X$ be a holomorphic vector bundle of degree 0 over a compact Riemann surface $X$. Why does $E$ admit a flat connection. I could work this out in the case of line bundles, where one starts with the natural logarithmic connection on $\mathscr{O}(\sum\_{i=0}^{k}n\_iP\_i)$ (here $\sum\_{i=0}^{n}n\_i=0$) and modify this by an element of $H^0(X,\Omega(\sum\_{i=0}^{k}P\_i))$ to get a holomorphic connection, which gives a flat connection.


How to prove this for a vector bundle of rank > 1?


EDIT: Thanks Richard. In the above let $E$ be an indecomposable vector bundle. Could you give a good reference for this result?



Flat connections on Bundles of degree 0 on a compact Riemann surface
Rex: A holomorphic bundle doesn't always admit a flat connection. You need to assume further that each of its indecomposable pieces has degree 0. This is the result of Weil [J. Math. Pures Appl. (9) 17 (1938), 47--87] and Atiyah [Trans. Amer. Math. Soc. 85 (1957), 181–207].


