We consider filtered holomorphic vector bundles on a compact
Riemann surface X equipped with a holomorphic connection
satisfying a certain transversality condition with respect to the
filtration. If Q is a stable vector bundle of rank r and
degree (1−genus(X))nr, then any holomorphic connection on the
jet bundle Jn(Q) satisfies this transversality condition for
the natural filtration of Jn(Q) defined by projections to
lower-order jets. The vector bundle Jn(Q) admits holomorphic
connection. The main result is the construction of a bijective
correspondence between the space of all equivalence classes of
holomorphic vector bundles on X with a filtration of length n
together with a holomorphic connection satisfying the
transversality condition and the space of all isomorphism classes
of holomorphic differential operators of order n whose symbol
is the identity map.