  Let G be a complex reductive group and let C be a smooth curve of genus at
least one. We prove a converse to a theorem of Atiyah-Bott concerning the
stratification of the space of holomorphic G-bundles on C. In case the genus of
C is one, we establish that one has a stratification in the strong sense. The
paper concludes with a characterization of the minimally unstable strata in
case G is simple.
