Exponents of a meromorphic connection on a compact Riemann surface

Abstract

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 E L 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.