We give a general
definition of the exponents of a meromorphic connection ∇ on a holomorphic
vector bundle ℰ of rank n over a compact Riemann surface X. We prove
that they can be computed as invariants of a vector bundle ℰL canonically
attached to ℰ, which we construct and call the Levelt bundle of ℰ, and whose
degree (equal to the sum of the exponents) we estimate by upper and lower
bounds (Fuchs’ relations). We use this definition to construct, for every
linear differential equation on a compact Riemann surface (with regular
or irregular singularities), the companion bundle of the equation, a vector
bundle endowed with a meromorphic connection that is equivalent to the
given equation and has precisely the same singularities and the same set of
exponents.
