(Ir)reducibility

A pair (H, ∇), where H → M is a holomorphic vector bundle on a complex manifold M, and ∇ is a (flat) meromorphic connection, is said to be reducible if there exists a subbundle G ⊂ H with 0 < rank G < rank H which is (at all nonsingular points of the connection) a flat subbundle. Such a G will simply be called a flat subbundle. A pair (H, ∇) is completely reducible if it decomposes into a sum of flat rank 1 subbundles.